70- 13,980

BENTZ, William Frederick, 1940- THE MEASUREMENT OF COMMON STOCK PRICE RELATIVE f UNCERTAINTY: AN EMPIRICAL STUDY. f I I The Ohio State University, Ph.D., 1969 | Accounting £

University Microfilms, Inc., Ann Arbor, Michigan j t ...... ____ ...... _...... -i

<© Copyright by

William Frederick Bentz

I1970i I i

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED THE MEASUREMENT OF COMMON STOCK PRICE RELATIVE UNCERTAINTY:

AN EMPIRICAL STUDY

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

William Frederick Bentz, B.A., M.Acc.

******

The Ohio State University 1969

Approved by

Adviser Department of Accounting PLEASE NOTE:

Not original copy. Some -pages have very light type. Filmed as received.

University Microfilms ACKNOWLEDGMENTS

I gratefully acknowledge the constructive criticisms and suggestions provided by my dissertation reading committee, Professors Thomas J. Burns

(chairman), Diran Bodenhorn and Melvin Greenball. They have never failed to give prompt attention to this work in spite of the many demands on their time. Professor Burns deserves special thanks for creating a very favor­ able environment in which I could improve my research skills and pursue my research interests. As a dissertation advisor, he has been instrumental

in obtaining data, computer time, financial assistance, and all the other resources necessary to undertake an extensive empirical study. As an aca­ demician, Professor Burns has constantly demanded that I improve the rea­ soning behind each step in the dissertation, as well as suggesting ways in which the exposition of theory and results could be improved. Professor

Bodenhorn is responsible for much of the improvement in logic and internal

consistency which distinguish the dissertation from earlier drafts. Pro­

fessor Greenball has made many substantive and methodological suggestions which have improved the dissertation, even though it was well in progress when he joined the committee.

Helpful suggestions have been received from Professors Cunnyngham, Cole

Nestel, and Lyle during various stages of the development of the disserta­

tion.

Preliminary analyses of the data were run on the Ohio State University

7094 system, with the assistance of the College of Administrative Science

ii Data Center staff. Jim Boltz, Anita Gehr, and Marjorie Brundage provided

programming assistance as well as technical information.

Final computations and statistical analyses were run at the Kansas

University Computation Center. Jeff Bangert has provided assistance in the use of the program library, and in the operation of the GE 635 system.

Financial assistance has been provided by the Department of Accounting

at The Ohio State University in the form of teaching opportunities and re­

search appointments. Support during much of my dissertation work was pro­ vided by The Haskins & Sells Foundation in the form of two Faculty Assis­

tance Grants (1965-66 and 1966-67). Some financial support was also pro­

vided by the Ohio Society of Certified Public Accountants in an award

called the Herman Miller Price (1966). The financial aid provided by these

sources is gratefully acknowledged since a doctoral degree would not have

been feasible without this support.

The secretarial assistance provided by the School of Business of The

University of Kansas is gratefully acknowledged. Mrs. Betty Bovee has

typed the dissertation with great care and cheerfulness. Mrs. Marcia Brown

typed many of the tables, and was responsible for duplication of the dis­

sertation. Without the cooperation of Mrs. Bovee and Mrs. Brown, the dis­

sertation would have been a much more difficult undertaking.

I am most thankful to my wife, Janet, and our children, Michael and

Jennifer, who provided encouragement and understanding throughout my grad­

uate studies. In terms of unfilled needs, a family's investment in a doc­

toral degree is never quite repaid.

iii VITA

June 14, 194C . . . Born - Dayton, Ohio

1962 ...... B.A., Economics, The University of Cincinnati, Cincinnati, Ohio

1964 ...... Public Accounting Internship, Ernst and Ernst, Columbus, Ohio

1964 ...... Internship, General Motors Corporation, Detroit, Michigan

1965 ...... M.Acc., The Ohio State University, Columbus, Ohio

1965-1967 ..... Teaching Associate, Department of Accounting, The Ohio State University, Columbus, Ohio

1967-1968 ...... Research Associate, Department of Accounting, The Ohio State University, Columbus, Ohio

1968-1969 ...... Assistant Professor of Business, School of Business, The University of Kansas, Lawrence, Kansas

PUBLICATIONS

Magazine Review: "The Mathematical Content of the Business School Curriculum" by David Novick, California Manage­ ment Review, Spring, 1966, which was reviewed in Management Services, July-August, 1966

Magazine Review: "Capital Budgeting of Interrelated Projects: Survey and Synthesis" by H. Martin Weingartner, Management Science, March, 1966, which was re­ viewed in Management- Services, September-October, 1966. TABLE OF CONTENTS

Page ACKNOWLEDGMENTS ...... ii

VITA ...... iv

LIST OF TABLES ...... vii

Chapter

I. INTRODUCTION AND SUMMARY ...... 1

A Statement of Objectives Outline of the Study Summary of the Results

II. THE INVESTMENT PROCESS ...... 28

Individual Investment: A Postulated Process

III. UNCERTAINTY AND PREDICTION MODEL ERROR ...... 45

Uncertainty: Some General Notions Uncertainty and Prediction Uncertainty Defined A Measure of Uncertainty Price Relative Variance as a Measure of Uncertainty Analytical Arguments for Attempting to Predict Price Relatives Summary

IV. THE PREDICTIVENESS OF COMMON STOCK PRICES AND PRICE RELATIVES 75

Introduction The Random Walk Hypothesis: A Review The Potentiality of Stock Price Prediction Naive Prediction Models An Economic Model of Stock Value

V. ANALYSIS AND INTERPRETATION OF THE PREDICTION MODEL’S PERFORMANCE ...... 122

Performance of the Exponential Smoothing Models Performance of the Growth Model

v Page

Implications for Measuring Uncertainty Potential Improvements in the Growth Model Predictions

APPENDIX

A ...... 156

B ...... 159

BIBLIOGRAPHY ...... 230

vi LIST OF TABLES

Table Page 1. Naive Prediction Model No. 1 - Prediction. Results for the Years 1958 Through 1967 Using the Best Smoothing Constant for the Ten Years Preceding the Year for which a Prediction is Being M a d e ...... 162

2. Naive Prediction Model No. 2 - Prediction Results for the Years 1958 Through 1967 Using the Best Smoothing Con­ stant for the Ten Years Preceding the Year for which a Prediction is Being Made ...... 164

3. Naive Prediction Model No. 3 - Prediction Results for the Years 1958 Through 1967 Using the Best Smoothing Con­ stant for the Ten Years Preceding the Year for which a Prediction is Being M a d e ...... 166

4. Naive Prediction Model No. 1 - Prediction Results for the Years 1958 Through 1967 Using the Smoothing Constants I n d i c a t e d ...... 168

5. Naive Prediction Model No. 2 - Prediction Results for the Years 1958 Through 1967 Using the Smoothing Constants I n d i c a t e d ...... 170

6 . Naive Prediction Model No. 3 - Prediction Results for the Years 1958 Through 1967 Using the Smoothing Constants Indicated ...... 172

7. Price Relative Prediction Performance with Respect to Mean-Squared Error for Sixty-one Companies over the Ten Year Period 1958 Through 1967 Using Three Exponen­ tial Smoothing Models with Both Shifting Smoothing Constants and Fixed Smoothing Constants ...... 174

8 . Price Relative Prediction Performance with Respect to Constant Bias for Sixty-one Companies over the Ten Year Period 1958 Through 1967 Using Three Exponen­ tial Smoothing Models with Both Shifting Smoothing Constants and Fixed Smoothing Constants ...... 176

vii Table page 9. Price Relative Prediction Performance with Respect to Proportional Bias for Sixty-one Companies over the Ten Year Period 1958 Through 1967 Using Three Expo­ nential Smoothing Models with Both Shifting Smooth­ ing Constants and Fixed Smoothing Constants ...... 178

10. Price Relative Prediction Performance with Respect to Correlation for Sixty-one Companies over the Ten Year Period 1958 Through 1967 Using Three Exponen­ tial Smoothing Models with Both Shifting Smoothing Constants and Fixed Smoothing Constants ...... 180

11. Price Relative Prediction Performance Ranking of Three Exponential Smoothing Models, Incorporating Shifting as Well as Fixed Smoothing Constants, by Industry for the Ten Year Period 1958 Through 1967 182

12. Comparative Price Relative Prediction Results Using Fixed Smoothing Constants versus Shifting Smoothing Constants for Sixty-one Companies from 1958 Through 1967 ...... 184

13. Comparative Absolute Prediction Errors Using Fixed Smoothing Constants versus Shifting Smoothing Con­ stants for Sixty-one Companies from 1958 Through 1967 ...... 186

14. Kolmogorov-Smirnov One Sample Test for the Normality of the Constant Exponent Smoothing Model's Pre­ dictions for Sixty-one Companies from 1958 Through 1967 where Shifting Smoothing Constants are Used ...... 188

15. Kolmogorov-Smirnov One Sample Test for the Normality of the Constant Exponent Smoothing Model's Pre­ diction Errors for Sixty-one Companies from 1958 Through 1967 where Shifting Smoothing Constants are Used ...... 190

16. Kolmogorov-SmirnovOne Sample Test for the Normality of the Constant Exponent Smoothing Model's Pre­ dictions for Sixty-one Companies from 1958 Through 1967 where Fixed Smoothing Constants are Used ...... 192

17. Kolmogorov-Smirnov One Sample Test for the Normality of the Constant Exponent Smoothing Model's Pre­ diction Errors for Sixty-one Companies from 1958 Through 1967 where Fixed Smoothing Constants are U s e d ...... 194

viii Table Page 18. Constant Exponent Smoothing Model's Mean-Squared Error versus Price Relative Variances as Measures of Uncertainty ...... 196

19. Analysis of Covariance Table for the Price Relative Variance and the Constant Exponent Smoothing Model's Mean-Squared Prediction Error for Sixty- one Companies Using Shifting Smoothing Constants ...... 198

20. Analysis of Covariance Table for the Price Relative Variance and the Constant Exponent Smoothing Model's Mean-Squared Prediction Error for Sixty- one Companies Using Fixed Smoothing Constants ...... 199

21. Analysis of Covariance Table for Price Relatives and the Constant Exponent Smoothing Model's Price Relative Predictions for Sixty-one Companies from 1958 Through 1967 Using Shift­ ing Smoothing Constants ...... 200

22. Analysis of Covariance Table for Price Relatives and the Constant Exponent Smoothing Model's Price Relative Predictions for Sixty-one Companies from 1958 Through 1967 Using Fixed Smoothing Constants ...... 201

23. Kolmogorov-Smirnov One Sample Test for the Nor­ mality of Growth Model Prediction Errors for Sixty-one Companies from 1958 Through 1967 ...... 202

24. Kolmogorov-Smirnov One Sample Test for the Nor­ mality of Growth Model Price Relative Pre­ dictions for Sixty-one Companies from 1958 Through 1967 ...... 204

25. Kolmogorov-Smirnov One Sample Test for the Nor­ mality of Price Relatives for Sixty-one Companies from 1958 Through 1967 ...... 206

26. Analysis of Covariance Table for Price Relative Variances and the Growth Model's Mean-Squared Errors for Sixty-one Companies ...... 208

27. Analysis of Covariance Table for Price Relatives and the Price Relative Predictions Produced by the Growth Model for Sixty-one Companies from 1958 Through 1967 ...... 209

ix Table Page 28. Analysis of Variance Table for Price Relatives Regressed on Price Relative Predictions from the Growth Model for Sixty-one Companies from 1958 Through 1967 ...... 211

29. The Optimal Linear Correction of the Price Rela­ tive Predictions Made by the Growth Model for Sixty-one Companies from 1958 Through 1967 ...... 212

30. Price Relative Prediction Error Decomposition for the Postulated Growth Model and for an Exponen­ tial Smoothing Model Tested on Sixty-one Com­ panies over the Ten Year Period 1958 Through 1967 ...... 213

31. Price Relative Prediction Performance of the Postulated Growth Model versus an Exponential Smoothing Model for Sixty-one Companies over the Ten Year Period 1958 Through 1967 ...... 215

32. Price Relative Prediction Error Decomposition for the Postulated Growth Model and for an Exponential Smoothing Model Using Shifting Smoothing Constants for Sixty-one Companies over the Ten Year Period 1958 Through 1967 ...... 217

33. Price Relative Prediction Performance of the Postulated Growth Model versus an Exponential Smoothing Model with Shifting Smoothing Con­ stants for Sixty-one Companies over the Ten Year Period 1958 Through 1967 219

34. Prediction Performance of Prediction Model No. 1 versus Prediction Model No. 3 Using Shifting Smoothing Constants ...... 221

35. Performance of Prediction Model No. 1 versus Prediction Model No. 3 Using Fixed Smooth­ ing C o n s t a n t s ...... 222

36. Price Relative Variance versus the Mean- Squared Prediction Error of Model No. 3 Using Shifting Smoothing Constants ...... 223

37. Mean-Squared Error versus Variance for Sixty- one Companies...... 224

38. Prediction Performance of Prediction Model No. 3 Using Shifting Smoothing Constants versus the Postulated Growth Model ...... 225

x Table Page 39. Mean-Squared Prediction Errors Using the Mean of Past Price Relatives versus the Price Relative Variance as a Measure of Uncertainty...... 227

40. Prediction Performance of the Arithmetic Mean of Past Price Relatives versus Model No. 3 Using Fixed Smoothing Constants ...... 228

41. Prediction Performance of the Arithmetic Mean of Past Price Relatives versus Model No. 3 Using Shifting Smoothing Constants ...... 229

xi CHAPTER I

INTRODUCTION AND SUMMARY

A Statement of Objectives

Rate of return on investment and the uncertainty associated with that rate of return are two major elements in capitai investment theory, mathe­ matical algorithms for selecting portfolios, financial utility theory, and the evaluation of prior investment performance. This study is concerned with the prediction of common stock price relatives and with the estimation of the uncertainty associated with those predictions.'*'

According to the neo-classical economic theory of the firm, a corpora­ tion should select investments so as to maximize stockholder wealth, which 2 is a function of return on investment and the uncertainty of that return.

In portfolio selection theory, the objective is to allocate an investor's 3 resources so as to maximize utility. In the economic and the finance literature, utility is a function of rate of return on investment and the .

*A common stock price relative is equal to the annual rate of return on investment plus 1.0. Annual price relatives are calculated by dividing the sum of cash dividends received during the year plus the cash value of shares held at year end, by the cash value of shares held at the beginning of the year, assuming no new investment. In this study, rate of return is used synonymously with price relative. 2 John Lintner, "Dividends, Earnings, Leverage, Stock Prices and the Supply of Capital to Corporations," The Review of Economics and Statistics, Vol. XLIV, No. 3 (August, 1961), pp. 243-269. 3 The classic work in this area is by Harry M. Markowitz, Portfolio Selection (New York: John Wiley and Sons, Inc., 1959), Part IV.

1 4 level of uncertainty associated with that rate of return. Therefore, when evaluating the economic success of a corporation, of an investor’s port­ folio, or of an investment fund for a large number of investors, the rate of return on investment and the uncertainty of that rate of return are two very important measures of performance.^ It seems clear that the measure­ ment and prediction of rate of return on investment, and the measurement and estimation of the uncertainty of the rate of return on investment, are significant elements of financial planning and financial performance mea­ surement .

The objective of this study.— Our purpose is the development and ap­ plication of the methodology necessary to measure the uncertainty of the common stock price relatives of individual corporations. The study is of practical importance because our measures of uncertainty can be used in other research studies, as inputs for the development of subjective prob­ ability distribution estimates made by financial analysts, or in evaluating the historical performance of common stocks. The uncertainty measurement methodology developed herein can be used to measure the past uncertainty of those common stocks for which we do not provide uncertainty estimates.

Our analysis begins with the development of a conceptual model of the investment process, because we must have a set of standard conditions for 6 which the analysis makes sense. A methodology for measuring the uncertain-

^See James Tobin, "Liquidity Preference as Behavior Toward Risk," Review of Economic Studies, Vol. XXVI, No. 67 (February, 1958), pp. 65-86.

^Peter 0. Dietz, "Measurement of Performance of Security Portfolios," Journal of Finance, Vol. XXIII, No. 2 (May, 1966), pp. 267-274.

^The concept of a set of standard conditions in measurement theory is provided by C. West Churchman, Prediction and Optimal Decision (Englewood Cliffs: Prentice-Hall, Inc., 1961). 3 ty of common stock price relatives, which is consistent with the conceptual model of the investment process, is then developed and applied to create measures of uncertainty for each of sixty-one major corporations.

Motivation for the study.— The two measures of uncertainty usually discussed in connection with common stock investment are (1) subjective estimates of the probability density function of expected returns on in­ vestment, and (2) the variance of observed rates of return about the mean rate of return for some block of investment periods.^ Subjective probabil­ ity estimates should be based predominately on past experience unless the investor has some reason to believe a shift has occurred in the level of uncertainty associated with a particular common stock, or set of common stocks. In any case, historical evidence of uncertainty is the usual starting point in formulating expectations about the future.

The variance of past price relatives about the arithmetic mean of a series of past price relatives is an appropriate measure of uncertainty whenever the price relatives can.be described as observations from one popu­ lation. If the price relatives are independent observations from a single population, then it is proper to use sample statistics to estimate the mo­ ments of the population's probability density function.

However, the variance of price relatives about the mean of a series of past price relatives is not believed to be a complete measure of uncer­ tainty. The variance is an incomplete measure because (1) the moments of the population's probability density function are not known, so there always

^Charles G. Ferreira, "Quantification and Measurement of Risk: An Empirical Study of Selected Common Stocks" (unpublished Ph.D. dissertation, School of Business, University of Washington, 1966). 4 exists a sampling error in the estimation of population moments, and be­ cause (2) the price relatives may not behave as if they are represented by a stable probability density function.

At this stage we might hypothesize either that the variance of past price relatives tends to overstate uncertainty, or that the variance of past price relatives tends to understate uncertainty. In Chapter III we argue that the mean-squared prediction error is the more meaningful measure of the uncertainty associated with a set of predictions. The mean-squared prediction error computed over our ten-year test period is equal to the sum of the squared differences between the ten actual annual price relatives of a company and the respective predicted price relatives, divided by the num­ ber of predictions, ten. Accordingly, the mean-squared price relative pre­ diction error is the standard by which we evaluate the variance of past price relatives as a measure of uncertainty.

On the one hand, we can argue that, on the average, squared errors of ex-ante price relative predictions may tend to exceed the ex-post dif­ ferences squared between the price relatives and the mean of these price relatives because (1) of sampling errors in the ex-ante estimates of the price relative mean, or because (2) the price relatives are not properly described by a stable probability density function. Both situations lead to a difference between the ex-post measure of the mean of a series of an­ nual price relatives and the ex-ante prediction of that mean. Therefore, prediction errors tend to be larger than one would expect based on the variance of annual price relatives about the mean price relative (ex-post).

On the other hand, we can argue that, on the average, squared errors associated with ex-ante price relative predictions may tend to be less than 5 the ex-post squared differences between the price relatives and their mean because (1) the mean may not contain all the information relevant to pre­ diction which is contained in the price relative series, or because (2) any correlation between the price relative series and a cue (prediction) variable is ignored in the calculation of differences between the price relatives and their mean value. If there exists a trend in a common stock's price relative series, then the price relatives are not independent obser­ vation* from a single population which is described by a stable probability density function, and the trend information should be used to predict the price relatives, not just the mean. Likewise, when another variable can be used to predict price relatives, then the average prediction error can be made smaller than the average difference between past price relatives 8 and their mean value.

• Prediction models.— As defined, the study of uncertainty is in terms of prediction errors, which necessitates the formulation of predictions.

We construct five prediction models which are used to predict annual price relatives for sixty-one common stocks over the ten-year period 1958 through

1967. Four of the five prediction models are based exclusively on the in­ formation contained in the time-series of past price relatives for that company. The fifth model incorporates economic information and corporate data in the formulation of price relative predictions.

The three linear prediction models are a constant model, a linear trend model, and a constant exponent model. If price relatives vary about a normal value, which is constant with respect to time, then that normal

O See section six of Chapter III. 6 value is the best prediction of future price relatives. Similarly, the linear trend model is based on the hypothesis that price relatives are equal to a constant amount plus a constant increment each year. The con­ stant exponent model is based on the hypothesis that the natural (Naperian) logarithm of a series of price relatives is constant with respect to time.

The three models can be formulated as follows:

(1.0) Constant model: R = a. it 1

(1.1) Linear trend model: = a. + b. • t It 1 1

a. (1.2) Constant exponent model: R = e ,

th where a^ and b^ are equation constants which are unique to the i securi­ ty, and R is the price relative of security i during period t.

The equation constants a^ and b^ are estimated for each company by means of exponential smoothing. The smoothing constant is allowed to vary from 0.0 through 1.0 and a set of prediction errors is generated for each smoothing constant tested. That smoothing constant which resulted in the smallest mean-squared prediction error for the prior ten years is used to estimate the equation constants, which are used to estimate next year's price relative. This smoothing constant selection procedure is repeated ten times in order to make ten annual predictions for each security. These models are described completely in equations 4.3 through 4.18, and in the text following those equations.

These three models are designed to incorporate information which is contained in the time-series of past price relatives.

The unweighted arithmetic mean of the prior ten price relatives is 7 also used to predict future price relatives. This method resembles the

constant model, but there are two important differences. The arithmetic mean calculation is not dependent on the sequence of the price relatives

in the sample period, whereas the constant model based on exponential

smoothing is influenced by the order in which the prior price relatives were observed. Also, each past price relative is weighted equally in the

fourth method, but not in the constant model^ where exponential smoothing

is used.

In addition to the use of naive prediction models, we develop an eco­ nomic model of common stock price relatives. The model is an economic model in the sense that measures of economic performance are used to pre­

dict future price relatives, as opposed to using only, past price relatives

to predict future price relatives.

Economic growth, price stability, earnings stability, sales growth,

earnings growth, and past price relative performance are variables which

appear in many stock valuation models, and which are examined by profes­

sional security analysts. Accordingly, the postulated economic model is

of the form:

where

Rt is the expected price relative for period t,

(t = 1948, 1949, .. 1967) for any security,

is a constant (i = 0 , 1 , 2 , • • • * 6)

Gt is the expected growth rate of gross national product

from the year t-1 to the year t, 8

0 ^ is one plus the average historical geometric growth

rate in operating income per share,

'S is one plus the average historical geometric growth

rate in sales per share,

Dt_^ is the difference between the price relative for

year t-1 and the average price relative for the five

one-year periods ending with year t-1 ,

Ut_^ is the variance of the changes in operating income

over the five one-year periods #nding with the year

t-1 ,

Vt_^ is the variance of price relatives over the five one-

year periods ending with year t-1 .

The prediction model summarized in equation 1.3 is the hypothesized

form of the model. Several variations of the six independent variables

included in 1.3 were tested, so that a total of fourteen independent vari­

ables were actually subjected to empirical test. These fourteen variables

are listed on page 9.

A least-squares regression model, applied on a step-wise basis, pro­

vides preliminary information about the ultimate form to be taken by the

prediction model.^ Because it is applied on a step-wise basis, the re­

gression model tests indicate (1) how many variables are significantly cor­

related with the past common stock price relatives of each company, (2)

9 This program is a completely rewritten, double precision, FORTRAN IV version of the original UCLA BIMED 34 Stepwise Regression Program. Hodson Thornber of the Center for Mathematical Studies in Business and Economics modified the original BIMED routine, Anita Gehr adapted the program to The Ohio State University 7094 System, and the author made some minor changes to adapt the program to the GE 635 System at The University of Kansas. ______Industry______Rubber Petroleum Beverage Goods Products Cement Textiles Last year's price relative less the average for the previous five years X X X X Last year's price relative , X The variance of the company's price relatives over the prior five years The variance of changes in operating income about the average change for the prior five years X X X X The average proportional price range over the prior five years X The average proportional price range over the prior two years The geometric average growth rate in operating income (before depreciation) per share over the prior six years XX The geometric average growth rate in operating income (before depreciation) per share over the prior two years X X The geometric average growth rate in net sales per share over the prior six years X The geometric average growth rate in net sales per share over the prior two years X X The annual growth rate in gross national product over the year for which a price relative is being predicted X The annualized growth rate a fourth-quarter gross national product prior to the prediction period X X The annual growth rate in gross national personal income over the year for which price relatives are being predicted X The annualized growth rate of fourth-quarter gross national personal income X 10 which set of the fourteen variables are most highly correlated with the past price relatives of each company, and (3) the order in which the inde­ pendent variables are admitted into the prediction equation, which is a * • measure of their relative importance.

At least six variables usually enter the step-wise regressions, so we have decided to use six variables in each prediction model. On an in­ dustry by industry basis we select those six measures, out of the fourteen possible, which are the most highly correlated with common stock price relatives over the period 1948 through 1957. Five different sets of six variables are used for the five different industries (see page 9) .

Preliminary tests of the data indicate that extreme predictions are sometimes generated by the model. Accordingly, predictions were bounded by the range of past price relatives, or by 0.8 and 1.50, whichever is the more restrictive set of bounds. This bounding process significantly af­ fects the predictions of 6 out of the 61 companies tested.

Once a variable set is selected for each industry, that same set is used for the entire test period, 1958 through 1967. For each company, ten prediction equations are constructed for the ten prediction periods. As illustrated in equation 1.3, the constants Bq , B^, ..., B^ are estimated by means of a least-squares regression model applied to the variable ob­ servations over the prior ten years. For example, the equation used to predict 1958's price relative for a particular company is constructed from the data available during the prior ten years, 1948 through 1957. Like­ wise, the data from 1949 through 1958 is used to construct the equation from which the 1959 price relative is predicted. This process is repeated- ten times for each company. 11

Nimrod and Bower have used this regression model approach to predic­ tion in the futures market for hog bellies.^ Their simulated investment program was profitable, so they achieved some predictive success. Also,

Whitbeck and Kisor constructed regression equations on a cross-sectional basis for a large number of firms, then they applied these regression equation coefficients to each company's data to estimate the intrinsic value of that company's stock.^ By purchasing these stocks for which the intrinsic value estimate greatly exceeded the current price, they were able to earn a favorable return. The results reported by Shelton also suggest that price predictions are possible, and that we are not restricted to earn- 12 ing an average rate of return on common stocks, or to lucky choices.

The Value Line Investment Survey represents an objective, but prag­ matic approach to common stock valuation. Their methodology has some em­ pirical credence since it is the methodology of an investment advisory group which is selling a service. Accordingly, our prediction model in­ corporates the methodology of Nimrod and Bower and an adaptation of the variables examined by Value Line as described by the founder, Arnold

Bernhard."^

^Vance L. Nimrod and Richard S. Bower, "Commodities and Computers," Journal of Financial and Quantitative Analysis," Vol. II, No. 1 (March* 1967), pp. 61-73.

^Volkert S. Whitbeck and Manown Kisor, Jr., "A New Tool in Investment Decision-Making," Financial Analysts Journal, Vol. 19, No. 3 (May-June, 1963), pp. 55-62. 12 John P. Shelton, "The Value Line Contest: A Test of the Predict­ ability of Stock-Price Changes," The Journal of Business, Vol. L, No. 3 (July, 1967), pp. 251-269. 13 Arnold Bernhard, The Evaluation of Common Stocks (New York: Simon and Schuster, 1959). 12

Because investors have different goals and objectives, Value Line evaluates and ranks companies by three different criteria: quality grade, appreciation potentiality in the next 3 to 5 years, and probable market performance in the next 12 months. Quality grade is dependent on the sta­ bility of earnings and dividends, as well as growth. A high quality com­ pany is a financially sound, low risk investment for which there is a low probability of failure, and a relatively low probability of missing divi­ dends. High growth rate is another attribute of quality because growth of earnings is an important prerequisite to dividend growth and price appre­ ciation. To summarize, a high quality stock is one which is growing in a stable, predictable manner.

Price appreciation potential over the next 3 to 5 years is another factor by which stocks are ranked. Long-run price appreciation potential depends on the expected performance of the economy in general, the expected performance of the industry, and the sales-earnings potential of each firm.

Because individual stocks are being ranked, economic conditions and indus­ try factors are only important in so far as they influence the price per­ formance of the firms being ranked. The average level of appreciation over a three to five year period is used in order to avoid the distortions that 14 might be caused by a cyclical variation in single years.

The other criteria by which stocks are ranked is probable market per­ formance over the next twelve months.

14 Bernhard, p. 119. 13

The third method gives us an indication as to whether a stock is cheap or dear now in relation to its own Intrinsic Value this year, ^ and whether it is cheap or dear in relation to all other stocks now.

Because only the price performance in the next twelve months is being con­ sidered, the third method is based on "an estimate of earnings and dividends in the next twelve months and the likelihood that the estimated earnings and dividends will command a price capitalization similar to the past."^

The theoretical analysis and the empirical evidence which provides the logical transition from the Value Line methodology to the model postulated in 1.3 are presented in Chapter IV. The methodology used by Value Line represents but one approach to common stock selection, but the continued success of The Value Line Investment Survey is evidence of the acceptance and interest in this relatively objective approach to investment selection.

Many authors believe that one cannot earn a rate of return which is above average unless one has inside information. By definition, all inves­ tors and financial analysts have access to generally available information, so that current market prices reflect the information currently available.

Some of the relevant studies are reviewed in Chapter IV.

The analysis of the five prediction models can be described in terms of the major steps involved. First, all five prediction models are used to make predictions over the ten year test period, then the mean-squared

4 prediction error is calculated for each prediction model, for each of the

^Bernhard, p. 110 .

^Bernhard, p. 110 . 14

61 companies. Next, the mean-squared prediction errors associated with the three naive time-series models are compared in order to determine which one is the best. We then compare the average mean-squared prediction error for the best naive time-series model with the average mean-squared pre­ diction error associated with the arithmetic mean model. The results of this comparison indicate which is the best naive model as well as indicating if there is any information in the time-series of past price relatives.

Next, we compare the average mean-squared prediction error associated with the economic growth model to the average mean-squared prediction error associated with the best naive model. The research hypothesis in the latter test could easily be stated negatively, but the economic model supposedly contains more variables and more information, so the research hypothesis is stated in a positive manner.

Hy thesis 1.— The proposed economic growth model produces better price relative predictions than the best of the naive prediction models. .

The final step is a comparison of the mean-squared prediction errors associated with the best prediction model of the five tested, with the variance of the past price relatives of each company about their own mean.

The variance is measured according to the equation 10 _ a . 4) vi = ^ (Rit" V 2/9’ t=l where i denotes the firm subscript

(i = 1 , 2 ..... 61)

is the price relative variance for company i 15

is the price relative for company i (i = 1 , 2 , 61)

during period t (t = 1958, 1967)

is the arithmetic mean of the price relatives for company i

(i = 1, 2, ..., 61) over the period 1958 through 1967.

As mentioned above, we believe that the variance is an incomplete

measure of uncertainty because the mean price relative for the period

relative for the period 1958-1967 is not known in advance, and because

we do not believe the price relative series can actually be described by

a distribution which is stable over time and under all types of conditions.

Hypothesis 2 is stated accordingly.

Hypothesis 2.— The variance of a series of observed price relatives

about the mean price relative for that series tends to understate the

uncertainty of those price relatives as measured by the mean-squared

error. As we have pointed out above, it would not be too surprising to

find that the price relative variance may overstate uncertainty. However,

the time-series of price relatives is not expected to contain sufficient

information to support the hypothesis that the price relative variance

overstates uncertainty.

In summary, we argue that the proper measure of the uncertainty

associated with a price relative over some prior set of investment periods

is the mean-squared prediction error, not the variance of a set of price

relatives about the mean price relative. It is hypothesized that the mean-

squared prediction errors associated with the best of the five prediction models tested are larger than the respective price relative variances. 16

Various other hypotheses are tested in conjunction with the selection of the best prediction model, the evaluation of the resulting prediction errors, the existence of risk classes, and the form of the relationship of the mean-squared prediction error to the price relative variance of each company.

Outline of the Study

The organization of this paper parallels the thought processes and the methodology used to attack the problem of measuring the past uncer­ tainty of common stock price relatives. In Chapter II we present a con­ ceptual model of the investment process which consists of six steps:

(1) the formation of investor goals, (2) the recognition of constraints on the investment process, (3) the creation of an algorithm for select­ ing a portfolio of securities, (4) the search for and description of acceptable securities, (5) the application of a predetermined algorithm to the set of potential security investments, and (6) the selection of an efficient set of securities. The description of steps one through three represents a theory of investor behavior which is meaningful under a specified set of conditions.

The theory of investor behavior is assumed to apply to a signifi­ cant number of investors, but not to all investors. With respect to goals, we assume that investors seek to maximize utility, where utility is a function of the relative increase in wealth. The relative increase in wealth over a year is a function of the relative increase in the price of each common stock owned. 17

Next, some constraints are placed on the investment process in order to make it more amenable to formal study. Consumption, once decided upon, is treated as a constraint because increased current consumption reduces the amount of current investment in wealth creating resources. Invest­ ment periods of one calendar year, and the holding of cash dividends until year end are two other constraints placed on the investment process.

The mean, variance model is the most widely accepted and well-defined approach to the selection of efficient portfolios of common stocks. The method is operational because algorithms have been developed which per­ form the computations necessary to select the most efficient portfolio from among a set of alternative investments.

The mean, variance model is based on the assumption that investors are risk averse, and that risk can be adequately described by the variance or standard deviation of rates of return. Accordingly, expectations about common stock price relatives and the uncertainty of those expectations are the most important factors in the investment process, and thus are the most deserving of isolation for further study.

The concept of the uncertainty of common stock price relatives is discussed in Chapter III. Some general comments about traditional dis­ tinctions between risk and uncertainty are presented in section one.

Next, we define a common stock price relative and indicate how it is measured under various circumstances. The price relative is discussed in

Chapter III because it is the subject of our predictions— that variable which 18

is uncertain. In section three, uncertainty is defined as the extent to which price relatives can be predicted one year in advance. The prob­

ability density function of the prediction errors is the basis for measur­ ing uncertainty. The mean-squared prediction error is the second moment about zero of the prediction errors, which is our measure of uncertainty

(predictability) associated with past price relative predictions.

The variance of past price relatives about the mean of those price relatives is frequently used as a measure of uncertainty. We argue that

the variance is a poor measure of uncertainty because (1) the mean is not known at the time predictions are being made, (2) price relatives may not properly be represented by a stable probability density function, (3) the price relative series may contain some information, such as trend, which the variance calculation ignores, and (4) another predictor vari­ able may be highly correlated with the price relative series.

A mathematical argument for attempting to predict price relatives is presented in section six of Chapter III. The variance of price relatives about the expected price relative tends to overstate uncertainty whenever there exists a predictor variable which is correlated with the price rela­ tive series because the conditional variance, given the predictor variable, is less than the unconditional variance of the price relatives.

The improvement of price relative expectations or predictions is im­ portant from a market point of view, as well as from the point of view of individual investors. More intelligent analyses of corporate performance should produce better predictions of long-run profitability, and thus im­ prove the allocation of capital funds by the market process. Moreover, 19

individual investors can earn above average rates of return only by being

better predictors, because the probability of earning high rates of re­

turn on randomly selected portfolios is small.

A part of the literature concerning the time-series behavior of com­

mon stock price changes bears on the subject of price relative prediction.

In regard to short-term price changes in common stocks and other specula­

tive price series, the general concensus is that average price changes

tend to be zero and the variance of price changes tends to be slightly

greater than is expected for a normally distributed random variable. This

viewpoint, the random-walk hypothesis of stock price behavior, is reviewed

in Chapter IV. The time series of stock price changes is important be­

cause a common stock price relative is the sum of the dividend yield, and

the price change divided by the price at the start of the period, plus 1.0.

The naive prediction models, designed to predict future price relatives based on the behavior of past price relatives, are influenced by the na­

ture of stock price changes over time. The random-walk studies concen­

trate on short term price changes, so they do not preclude the possibility

of an upward drift in stock prices over annual periods.

The random walk literature is also reviewed in order to determine

the implications of this hypothesis of stock price behavior with respect

to the predictability of common stock price relatives using econometric models. Advocates of the random walk hypothesis not only accept that prediction may be possible, but they also argue that perfect markets, or nearly perfect markets, are necessary in order to explain random price movements. Adjustments to information about economic forces in the economy 20

are rapid and complete in highly efficient markets. Prediction of those

economic factors that influence stock price changes should lead to pre­

diction of stock price changes, and thus prediction of price relatives.

Four naive prediction models are developed to predict common stock

price relatives based on past price relatives. A constant model, a linear

trend model, and a constant exponent model are. tested. The constant terms

in these models are estimated by means of exponential smoothing. In ad­

dition to these three models, we use a simple arithmetic mean of the prior

ten price relatives in order to predict next year's price relative.

The final section of Chapter IV contains the development of the eco­

nomic prediction model, also referred to as the growth model which is

described on page 7. The value Line Investment Survey Method and several

empirical research studies are summarized, and their implications for

our model explained. The analysis explaining the inclusion of each

variable in the prediction model, and a statement of the model, conclude

Chapter IV.

The relative predictive performance of the naive prediction methods

and the proposed growth model is evaluated in Chapter V. The five in­

dustries selected for study are the beverage producers, cement producers,

integrated domestic oil producers, textile makers and producers of tire

and rubber goods. The financial data for the firms was provided by the

Binary-Type Annual Industrial Compustat magnetic tape library distributed by Standard s Poor's, Inc. After eliminating those companies for which

twenty years of financial data is not yet available, there are 61 companies

in the five industries to be included in the sample. Predictions are 21 made for the years 1958 through 1967, inclusive.

Summary of the Results

A capsule summary of the overall performance of the five prediction models is presented below.

Average Mean- Prediction Method Squared Error

Naive Models:

Constant model 0.142

Linear model 0.148

Constant exponent model 0.140

Mean of the prior ten price relatives 0.133

Economic model 0.175

The three naive time-series models performed well and the results are interesting in several respects. The constant exponent model is generally superior to the other two models in minimizing the mean-squared prediction error over the test period 1958 through 1967, as we had expected. The ranking of the naive models is significant at the 99.9% confidence level, and the Kendall Coefficient of Concordance was 0.64, which indicates a generally high degree of agreement among the rankings (see Table 7).

However, the difference between the constant model and the constant exponent model is minor, and it is not significant at the 95% confidence level when the two models are compared directly.

The problem of selecting an optimal exponential smoothing constant for each of the three naive prediction models, for each firm, proved to be an important issue. Each of the three naive prediction models was implemented in two ways. One method involves finding that smoothing con­

stant value which results in the smallest mean-squared prediction error

for the test period. The other method involves using a set of smoothing

constants, one for each year. For example, equation constants needed to predict 1959's price relative, for a particular company, are estimated by means of the exponential smoothing constant which results in the smallest mean-squared prediction error over the previous ten years, 1949 -

through 1958. For 1960's prediction, the smoothing constant which results in the smallest mean-squared prediction error for 1950 through 1959 is used to estimate the equation constants. For this method, the smoothing constant may change from year to year to year, so that there is a set of ten optimal smoothing constants associated with each price relative series.

The same smoothing constant tends to be used over time for each company, but there are many cases for which several different smoothing constants are used over the ten year test period.

The use of shifting smoothing constants is our simulation of the investment prediction process. The determination of an optimal smoothing constant, ex-post, was done in order to determine the maximum possible benefit of improved smoothing constant selection procedures. For the constant exponent model, the potential average reduction in the mean- squared prediction errors is 14%. The details of this analysis are contained in tables 12 and 13.

The Kolmogorov-Smimov test indicates that the prediction errors resulting from the constant exponent model are not normally distributed, although a visual inspection of a histogram of the prediction errors does 23 indicate significant non-normality. A comparison of the observed interval frequencies with those of a standardized, normally distributed random vari­ able indicates that there is a greater concentration of prediction errors about the mean error, and less concentration in the tails of the error distribution than is expected. As a result, there is no evidence that the variance of prediction errors is undefined. Tables 15 and 17 provide the details of the tests of normality.

The fourth naive prediction model, the use of' the mean of the prior price relatives to predict next year's price relative, performed better than the constant exponent model (see Table 41). As a result, it is the best of the four naive prediction methods.

The performance of the postulated growth model was interesting, but disappointing. The average mean-squared error from the growth model was about 20% greater than that of the constant exponent model using shifting smoothing constants. This difference is significant at the 99.9% confidence level for the Wilcoxen matched-pairs, signed-ranks test, and is significant an the 99.9% level for the t-test. We must reject the hypothesis that the growth model is better than the naive models.

In formulating the economic model, it is not possible to specify in advance how each variable should best be measured. By using a stepwise regression process, those variables which have the highest historical explanatory power, with respect to price relatives, are selected for use in the final prediction model. The growth rate of sales and the growth rate of operating earnings before depreciation are highly correlated (par­ tial correlation) with common stock price relatives, as is the uncertainty of operating income before depreciation. For some industries, two year 24 growth rates are more highly correlated with price relatives than six- year average growth rates. In other industries, the longer term average growth rates dominate. In all cases, one of the six measures of general economic activity enters the regression equations at about the fifth step. The general economic measures include annual and fourth quarter annualized growth rates for gross national product, gross disposable per­ sonal income, and personal expenditures on durable goods. A more complete description of the variables used in the prediction equations for each industry is presented in Chapter V.

The prediction errors resulting from use of the postulated growth model appeared to be approximately normal in distribution when reviewed in histogram form. However, the Kolmogorov-Smimov test indicates (Table

24) that these price relative prediction errors are not normally distri­ buted. A comparison of the observed interval frequencies with those of a standardized, normally distributed random variable indicates that there is a greater concentration of prediction errors about the mean error, and less concentration of large errors in the tails of the error distribution than is expected. These results are consistent with the analysis of the prediction errors from the constant exponent prediction model.

The major purpose of this research is the measurement of uncertainty.

The mean-squared prediction error is the criterion by which we evaluate uncertainty. We argue that predictions should be based on data available to investors at the time predictions are actually being made. Because uncertainty is measured in terms of prediction errors, that model which minimizes the mean-squared prediction error is the best prediction model. 25

Consistently, the best available measure of uncertainty is the mean-squared prediction error provided by the best prediction model.

If a new prediction model can be develpped which incorporates informa­ tion available to investors at the time predictions are made, and which results in smaller prediction errors, then that model should be used to measure uncertainty. Until such an improved model is developed, the mean-squared error of predictions based on the mean of prior price rel­ atives best measures uncertainty.

Hypothesis 2 is accepted^ The variance of a series of price relatives about the mean price relative for that series does tend to understate uncertainty as measured by the mean-squared prediction error, because of the difficulty of predicting the mean. All of our prediction models re­ sulted in an average mean-squared prediction error which was significantly larger than the average price relative variance, for the companies tested

(see Table 37).

The failure of the three naive prediction models to provide better predictions than the use of a mean of the prior ten price relatives leads us to two possible conclusions. One conclusion is that there is little or no information in the series of past price relatives which is not cap­ tured by the simple arithmetic mean. The other conclusion is that price relatives can be described by a mean-reverting process for the majority of companies. In our tests of the naive prediction models, the smooth­ ing constants tended to approach 1.0 for a few companies, which indicates that the process associated with those price relatives may not be mean- reverting. However, our over-all results indicate that price relatives tend to be mean reverting. We have further evidence that the variance may be a misleading measure of uncertainty- We found no linear first degree o< second degree

equation which can be used to relate the mean-squared prediction error

from a price relative to the variance estimate of that price relative.

In other words, the mean-squared error was not simply larger than the variance estimates by some constant amount or by some proportionality

factor. The importance of this finding is that the relative uncertainty

of two price relatives will vary, depending on which measure of un­

certainty is used. Ordinal rankings, as well as interval scale com­ parisons, may be influenced by the measure selected. For example, let

the variance of the price relatives of company B equal the variance of

the price relatives of company A. Their respective mean-squared prediction

errors may be equal, or the mean-squared error for B may be greater than,

or less than, that of A. Moreover, let the variance of B's price relatives

equal two times the variance of A's price relatives. Our findings indicate

that the mean-squared error of company B's price relatives may be three

times as large as the mean-squared error of A's price relatives.

Our inability to find a statistical relationship between the mean-

squared prediction errors, and the price relative variances for the

sixty-one companies in the sample has implications with respect to one's

ability to separate firms into risk classes. As summarized in Table 20 we found significant mean effects in the industry classifications of mean-squared prediction errors, even after adjustment by linear regression

for the price relative variances. The within-industry regression co­

efficient is clearly significant, but industry differences in the mean- 27 squared errors remain after adjustment for the regression. These results support the belief that using mean-squared prediction errors to define risk classes will result in a different classification than results from used price relative variance as the measure of uncertainty.

To summarize, the mean-squared error of predictions based on the arithmetic mean of the prior ten price relatives is believed to be a better measure of uncertainty than the price relative variance about the mean price relative for the prediction period. CHAPTER II

THE INVESTMENT PROCESS

Much has been written about the operation of economic systems and the

problem of investment alternative selection within the context of a compet­

itive market economy. In spite of the complexity involved, many writers

have been attempting to capture the essential elements of the processes by which people make investment decisions. Indeed, one cannot study the in­ vestment performance of individual common stocks, mutual funds, or other

forms of investment, if the goals of investors have not been determined be­

forehand.

It seems safe to say that the needs of investors vary greatly, and

that the ways in which one might model the investment process are numerous.

Our approach to this problem is to begin with a conceptual model of the

investment process which has its origins in accepted theories of rational

economic behavior. These theories are generally accepted by most re­ searchers in finance and economics, so they represent the current state of development in economic theory.

Individual Investment: A Postulated Process

The three classes of elements entailed in any purposive act "are (1)

the decision-maker, (2) a set of alternative actions, and (3) a set of g o a l s . "17 With respect to the investment process, the decision-maker is

17C. West Churchman, Prediction and Optimal Decision (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1961), p. 137.

28 the investor, who may be representing a large group of other people, or who may be investing his personal resources. The set of alternative courses of action consists of the set of all common stock portfolios that each investor might select. Portfolio selection involves decisions as to whether or not certain securities should be included in the portfolio, as well as decisions regarding the proportional dollar amount to be invested in each security to be included in the portfolio. The set of goals con­ sists of an operational specification of the investment results each inves­ tor wants to achieve.

The three elements of the purposive act of investment are made opera­ tional by describing the investment process, as we perceive it, in terms of six major elements: the formation of investor goals, the recognition of constraints on the investment process, the creation of an algorithm for selecting a portfolio of securities, the search for and description of ac­ ceptable securities, the application of the algorithm to the set of poten­ tial security investments, and the selection of an efficient set of securi­ ties. The fourth step, a search and description of securities, is the focus here, but that step cannot be discussed independently of tha three preceding ones.

Investor objective: step 1 .— Imputing a specific goal to an individ­ ual's behavior, while maintaining some ..relevance to the behavior of actual investors, is a difficult task. At the present time, we know very little about the collective values of man., and very little about the collective 18 values of the subset of man which we shall label common stock investors.

1 fi See Churchman, p. 3. A major portion of Churchman's book is de­ voted to an analysis of the methodology needed to ascertain personal values. 30 *

However, in order to measure performance, which is goal achievement, we must know more about the objectives of investors with respect to their in­ vestment activities.

We assume a goal for investors, which is based on accepted economic theories about man's behavior. In particular, w e ‘assume that investors maximize expected utility, where expected utility is a function of the ex­ pected rate of return on common stock investments, and the uncertainty of that expected return as measured by the variance of its probability density function. As is argued below, these assumptions made about investor goals are customary in studies of investment performance. Although widely used, these basic assumptions do limit the study and preclude generalization of the results obtained here to all investment activities, or to all common stock investors.

We begin by accepting the expected utility maxim as a proper descrip­ tion of rational behavior. In this context, the rational investor "acts as if he (1) attaches numbers (utilities) to each possible outcome and (2) 19 chooses that option (or strategy) with the largest expected value." Axiom- 20 atic presentations of the expected utility maxim are presented by Markowitz, and von Neuman, 21 while a more philosophical discussion is presented by

- 22 Churchman.

19 Michael C. Jensen, "Risk, The Pricing of Capital Assets, and the Evaluation of Investment Portfolios," The Journal of Business, Vol. XLII, No. 2 (April, 1969), pp. 167-247.

^ H a r r y M. Markowitz, Portfolio Selection; Efficient Diversification of Investments (New York: John Wiley and Sons, 1959).

21John von Neuman and Oscar Morgenstern. Theory of Games and Economic Behavior (Princeton, New Jersey: Princeton University Press, 1953). 31

Next, we assume that theinvestor maximizes a single objective func­ tion which is of the form

(2.0) U = f (C^, Wjj C2, W2; ...; Cn , Wn , E), where

U is utility,

C_£ is the dollar amount of consumption during year i

(i = 1> 2, •«•, n),

W^ is the unconsumed wealth remaining at the end of period i

(i = 1, 2, ..., n),

E is the net cash equivalent value of the investor's estate

at the time of death, and

n is the expected or probable maximum life of the individual

investor.

We ignore institutional investors to the extent that they do not have ob­ jective functions of the form of 2.0 or are investing for a set of indi­ vidual investors. Individual investors can choose mutual funds or trust managers based on their beliefs about the ability of these agents to maxi­ mize their (the investors') personal utility functions.

Equation 2.0 is formulated to emphasize the interdependence of con­ sumption and wealth. Increased current consumption must decrease wealth, which will usually decrease future increments in wealth, and thus decreases potential total lifetime consumption. Regardless of how the investor makes the choice between consumption and investment, he must maximize annual in­ vestment return in order to maximize total life-time consumption plus estate value, whether or not they are discounted values. In other words,

22 Churchman, Chapter 8. 32 to maximize equation 2.0, the investor must select the most profitable set of investments available, assuming that utility is proportional to the present value of future consumption expenditures and estate value. The ap­ parently divergent consumption and investment objectives of investors can be recognized in at least two ways. A multiple goal .structure can be for­ mulated, or the investor's noninvestment objectives can be treated as con­ straints on the investment process. The latter approach is adopted here because the more complex goal structure is beyond the scope of this paper.

Consumption, once decided upon, becomes a constraint because it limits the amount of money available for investment.

At this point some comment about the maximization assumption is use­ ful. It has been stated that "Most human decision-making, whether indi­ vidual or organizational, is concerned with the discovery and selection of satisfactory alternatives; only in exceptional cases is it concerned with 23 the discovery and selection of optimal alternatives." The assumed ob­ jective of an investor can be tentatively summarized as follows: given some dollar amount of cash to be invested, and a set of potential security investments, the investor will select a portfolio of securities which will maximize his utility, where his utility is a function of the expected re­ turn from the portfolio and the uncertainty associated with that return.

This assumed objective is not necessarily inconsistent with the observation that man seeks satisfactory alternatives, not optimal alternatives. For non-professional investors, the cost of searching for good investments may be very high, which would limit his search to those popular securities for

9 ^ James G. March and Herbert A. Simon, Organizations (New York: John Wiley and Sons, Inc., 1958), pp. 140-141. 33

which information is readily available. The same is true for the profes­

sional administrator of the investment funds of an organization, except

that the list of securities that are evaluated is much larger.

As stated, the wealth maximization assumption is an accepted represen­

tation of actual investor behavior. The assumption includes no normative

statements about how much money an investor should commit to investment in

securities in order to achieve some lifetime objective. We need only to

observe that people and organizations do invest in corporate securities.

More importantly, the knowledge that investors only consider some small

subset of all investment alternatives does not affect the assumption as

stated. We only assume that once some set of securities is included in the

investment process, the investor will prefer greater utility to less util­

ity. In other words, man may be a satisficer with respect to his search

for alternatives, because of perceived cost and time constraints, but given

a set of alternatives, there is no reason to suspect that he will not max- 24 imize utility.

The assumption that investors attempt to maximize expected utility

must be followed by a specification of the general form of a utility func­

tion. Since we have regarded current consumption as a constraint on in­

vestment, we can reduce equation 2.0 to the form

24According to Jensen, p. 172, Fama has shown that an investor will appear to behave as though he is maximizing E |. U (Ct» w t+j)j even though he is faced with the multi-period consumption-investment decisions indi­ cated in 2.0. Therefore, the assumption of period by period utility maxi­ mization seems justified. Eugene Fama, "Multi-Period Consumption - Invest­ ment Decisions," Report No. 6830 (Chicago: University of Chicago, Center for Mathematical Studies in Business and Economics, June, 1968). 34

(2.1) U = f (Wj.)j where W^ is defined as before. To avoid scale problems, 2.1 is restated in terms of the rate of return on beginning wealth, plus 1.0, which is equal to and is denoted R^. The reformulation is:

(2.2) U = f (Rt), where

R t ’

Restating these formulations in terms of expected values, we have:

(2.3) E [u (Rfc)] = the expected utility of the price relative, Rfc.

The investor is assumed to select investments so as to maximize E ^ U (Rj.)]

As before, we are faced with the need for information about investors that is not directly available. We need to know how investors react to uncertainty and how they interpret the meaning of uncertainty.

The usual method of incorporating uncertainty into an investment de­ cision is to assume that the rate of return or the price relative is a random variable for which possible values can be described in terms of a probability density function. If the probability density function is

Gaussian, then the first two moments of the density function can be used to specify all of its higher, non-zero moments. In such cases, it is ap­ propriate to assume that the utility function of the investor is a function of the mean and the standard deviation of the probability density function of the return from a particular security, or from any linear combination of 25 security returns (i.e., a portfolio).

25 Eugene Fama, "Risk, Return, and Equilibrium: Some Clarifying Comments," Journal of Finance (March, 1968), pp. 29-40; William Breen, "Homogeneous Risk Measures and the Construction of Composite Assets," Journal of Financial and Quantitative Analysis, Vol. Ill, No. 4 (December, 1968), pp. 405-413; and J. Tobin, "Liquidity Preference as Behavior Towards Risk," Review of Economic Studies (February, 1958), pp. 65-86. 35

The problem of non-gaussian distributions is discussed in Chapter III. At this point we will only note that extensive use has been made of- the mean, variance model, and that at least some empirical evidence supports its use 26 as a description of risk adverse investors. Many investors may not be risk adverse, or their desire for low risk may change with changes in 27 wealth levels. As a result, we are only concerned with that subset of all risk adverse investors for which a quadratic utility function is ade­ quate. The general form of such utility functions is

(2.4) U = R - ba, where R is the mean (expected) rate of return, b is a constant which reflects a particular investor's aversion to risk, and a is the standard deviation of R.

The measures of uncertainty discussed in this study are incomplete whenever the deviations of actual return from expected return are not rep­ resented by a Gaussian probability density function, and when the inves­ tor's utility function is not quadratic. In theory both conditions must hold for the under-specification to exist. As a practical matter, inves­ tors may not be influenced by small deviations from either condition.

Constraints on investment: step' 2 .— For purposes of this study, two constraints are imposed on the investment process. One constraint is the

See Markowitz; Fred D. Arditti, "Risk and the Required Return on Equity," Journal of Finance, Vol. XXII (March, 1967), pp. 19-36; and Kalman J. Cohen and Jerry A. Pogue, "An Empirical Evaluation of Alternative Portfolio Selection Models," Journal of Business, Vol. XXXX, No. 2 (April, 1967), pp. 166-193. Many of the sources referenced in the bibliography in­ corporate the variance or standard deviation as the measure of uncertainty. 27 Milton Friedman and Leonard J. Savage, "The Utility Analysis of Choices Involving Risk," The Journal of Political Economy, Vol. LVI, No. 4 (August, 1948). 36

limitation of investment periods to one year. This fixed investment period

constraint was made necessary by the decision to consider the problem of

prediction over time. Each prediction period within a series of prediction periods must include the same interval of time in order to be comparable.

The effect of increasing the investment time period is a separate issue, which should be tested separately.

By using finite investment periods, we recognize that investors do not

accumulate wealth over some very long period, but- invest for some purposes within their own expected life spans. At the present time, very little is known about the typical investment period duration for any investor type,

so the extent to which this assumption deviates from actual practice is un­ known.

The other constraint is that dividends are held until the end of the investment period or are consumed immediately. In either case, the annual return consists of the cash dividends received, plus the price change that took place during the year.

Evaluation models; step 3 .— Given a set of alternative investment opportunities, and given a well defined utility function, a selection pro­ cess can be defined so that utility can be maximized. Inputs to the selec­ tion process include information about the common stock, as well as infor­ mation about the utility function. If common shares can be described in ratio scale measures, and the utility function can be formalized, then a formal algorithm can be developed to perform the process of selecting an optimal portfolio from a set of investment opportunities. Four of the more refined methods of selecting portfolios are the state-preference approach, the relative strength method, the mean, variability approach, and the 37

dominance theorm.

In the state-preference approach, securities are resolved into distri­

butions of dated contingent claims to income, defined over the set of all 28 possible states of the world. Each state of the world is specified and a

probability assigned to that state of nature. In addition, each possible

amount of income from each investment must be specified for each state of

nature that has been specified. Hirschleifer argues that this approach in­

volves a more precise statement of uncertainty in that each event is as­

signed a probability.

The state-preference approach is not viewed as an empirically relevant

method for portfolio selection for various reasons. First, an algorithm

has not been developed for the multiple investment opportunity case, even

though Hirschleifer has provided an example involving a single commodity

and two mutually exclusive states of nature. Lacking a precise algorithm

for selecting a portfolio, it is difficult to determine exactly what the

information requirements of such an algorithm will be for the multiple in­

vestment case. Second, it is almost impossible to conceptualize the com­

plete specification of all possible states of nature, and all possible con­

tingent claims for each state of nature. With respect to security invest­ ments, the essence of uncertainty is not our inability to specify outcomes

in advance, but our inability to assign a meaningful probability to each

outcome. Hirschleifer has assumed away this essential element of uncer­

tainty in order to formulate his theory. As a result, his theory is of

28 J. Hirschleifer, "Investment Decision Under Uncertainty: Choice- Theoretic Approaches," The Quarterly Journal of Economics, LXXXIX, No. 4 (November, 1965), pp. 509-536. 38

little assistance in the actual portfolio selection process in an uncertain

world.

A second portfolio selection method is based on the relative strength 29 criterion of investment performance as described by Levy. In summary, it

is postulated that stocks performing well in recent market movements will

continue to perform well in the future. Past performance is compared by

ranking the percentage price changes of a set of stocks. The highest posi­

tive percentage increase in a period is given a rank of 1, while the

largest percentage price decrease is given a rank of N, where N stocks are

being ranked. This ranking process can be performed over any number of in­

vestment periods and over any investment period duration.

Levy did find that high performance stocks do tend to continue to per­

form well, and that low performance stocks do tend to continue a record of

poor performance. As is true in any such study, it is possible that his

results are related to the time periods over which the tests were made.

The most important limitation of the study was explained by Levy him­

self. No measure of uncertainty was used to determine if greater rates of

return are related to greater levels of uncertainty. Usual hypotheses

about the theoretical relationship of higher rates of return to uncertainty would suggest that high rates of return persist due to greater uncertainty,

and will continue so long as uncertainty is high. Because Sharpe has found

that mutual funds with higher rates of return do tend to have higher stan- 30 , dard deviations of annual return, Levy s results may be entirely

29 Robert A. Levy, "Relative Strength as a Criterion for Investment Selection," The Journal of Finance, Vol. XXII, No. 4 (December, 1967), pp. 595-610. 30 William F. Sharpe, "Risk-Aversion in the Stock Market: Some explained by the relative uncertainty of the stocks selected. High rate of return stocks will continue to command a high rate of return so long as the associated uncertainty of return is high. Low rates of return usually are associated with stability and low uncertainty. In summary, the relative strength criterion of investment performance has not been sufficiently validated, because Levy's findings can be explained in terms of traditional utility theory. Also, the relative strength criterion is counter to the random-walk theory of price behavior which is discussed more fully below.

The relative strength criterion of common stock selection is thus not a theoretical base upon which.to build a selection algorithm, from which we can impute the stock description input requirements.

Although the relative-strength method may not be an adequate criterion for investment selection, Levy's findings do have some significance for the process of price relative prediction. Of the three ratio-type measures of price performance which Levy tested, that price ratio which yields the most consistent intertemporal ranking of companies is the current week's ending stock price divided by the average ending price for the 27 weeks prior to and including the current week. Two hundred companies are ranked weekly for the 260 week period from October 24, 1960, through October i5, 1965.

Levy found that companies tend to maintain a relatively high or low price ratio over the entire test period. The implication of these results is that past price ratios are helpful in predicting future price ratios.

Levy's results have implications for price relative prediction because both price relatives, as defined above, and his price ratios are measures

Empirical Evidence," The Journal of Finance, Vol. XX, No. 3 (September, 1965), pp. 416-422. 40 of relative price increase. Upward price movements result in high price relatives and high price ratios. Similarly, downward price movements re­ sult in low price relatives and low price ratios. Therefore, Levy's find­ ings are expected to hold for price relatives, as well as for price ratios.

Although we do not test Levy's proposed method of stock selection, the naive prediction models which are evaluated in this study are based on the hypothesis that past price relatives are useful in predicting future price relatives.

The mean, variability approach to portfolio selection is the more 31 fully developed method currently being proposed. The expected utility of a common stock's return is assumed to be a function of the expected (mean) rate of return and the standard deviation of the rate of return. The exact trade-off between return and uncertainty is a function of each investor's utility function.

Operationally, the mean, variability approach requires information about the expected rate of return of each stock, the standard deviation of that rate of return, and the covariance between each stock's"rate of return and every other stock's rate of return. Theories about equilibrium condi­ tions in capital asset markets, as well as empirical studies of stock prices and rates of return, indicate that investors must accept higher

31 Some of the major works in this approach include: . H. Makower and J. Marschak, "Assets, Prices, and Monetary Theory," Economics N. S., Vol. V (1936), pp. 261-288; Harry M. Markowitz, Portfolio Selection: Efficient Diversification of Investments (New York: John Wiley and Sons, Inc., 1959); James Tobin, "Liquidity Preference as Behavior Toward Risk," Review of Eco­ nomic Studies, Vol. XXV (1958), pp. 65-86; and more recently, D. E. Farrar, The Investment Decision Under Uncertainty (Englewood Cliffs,. N.J.: Prentice- Hall, 1962). A recent contribution of ideas is provided by William F. Sharpe, "Capital Asset Prices: A Theory of Market Equilibrium under Condi­ tions of Risk," Journal of Finance, XIX (September, 1964), pp. 425-442. 32 levels of uncertainty in order to earn higher rates of return. Markets may temporarily deviate from equilibrium conditions, but the tendency is to move toward equilibrium. Any one investor may be able to find an invest­ ment which is expected to yield a higher rate of return, with no increase in uncertainty. However, large numbers of investors are not able to do so.

Using the mean, variability approach to portfolio selection insures that the level of uncertainty (as measured by the standard deviation) is mini­ mized for whatever rate of return is desired. A basic assumption of this method involves the use of the standard deviation of rate of return as a measure of uncertainty. For a Gaussian probability density function, the third moment about the mean and all higher moments are either equal to zero, or can be expressed in terms of the mean and variance. As a result, the mean and variance (standard deviation squared) completely describe the den­ sity function, and no other information is needed.

If the probability density function of the rate of return is non-

Gaussian, then one must know all of the moments, or at least all of the non­ zero moments of the probability density function in order to completely spe­ cify the probability density function of the rate of return. Samuelson has quite properly argued that one ought to consider the first four moments of the probability density function of each security's rate of return in any 33 portfolio selection model. He concentrates on the first four moments

■^ama, "Risk, Return ..., pp. 29-40; Sharpe, "Risk Aversion ..., pp. 418-419; and Sharpe, "Capital Asset ..., pp. 425-442.

^3?aul A. Samuelson, "General Proof that Diversification Pays," Journal of Financial and Quantitative Analysis, Vol. II, No. 1 (March, 1966), pp. 1-13. 42 *

because it is difficult to explain in non-mathematical terms why an inves­

tor should be interested in the tenth moment about the mean, whereas the

second, third and fourth moments — which represent variability, skewness

and kurtosis — can be explained in a layman’s terminology. Another major

factor for considering only four moments of a stock's rate of return proba­

bility density function is the limitation of data. Higher moments tend to

be unstable in repeated sampling experiments, which implies that they are 34 of little value in the portfolio selection problem.

Because we know very little about the form of investor's utility func­

tions, it is discouraging to argue that we need to know all n-moments of a

probability density function in order to maximize any investor's n-degree

utility function. The dominance theorm is presented as an alternative

method of ordering uncertain prospects, which requires relatively little 35 information about an investor s utility function for uncertainty. To

summarize, the probability density function g is said to dominate a second

probability density function f, if and only if, the cumulative density

function of g (denoted G(x)) is less than or equal to the cumulative den­

sity function of f (denoted F(x)), that is: G(x) < F(x) for all xel, the

strict inequality holding for at least one xel, where x is the value of a

random variable X, and I denotes the range of values taken on by X. In

34 Paul G. Hoel, Introduction to Mathematical Statistics (Second edition, New York: John Wiley and Sons, Inc., 1954), p. 48. 35 Josef Hadar and William R. Russell, "Preference Ordering and Stock- astic Dominance" (unpublished Working Paper No. 17, Case Western Reserve University, November, 1968); M. K. Richter, "Cardinal Utility, Portfolio Selection and Taxation," Review of Economic Studies, Vol. XXVII (1960), pp. 152-166; and J. P. Quirk and R. Saposnik, "Admissibility and Measur­ able Utility Functions," Review of Economic Studies, Vol. XXIX (1962), pp. 140-146. 43

the investment case, we are saying that one security is generally preferred

over another, if higher rates of return are more probable than lower rates

of return, relative to the second investment security.

In theory, the dominance theorm is more general and thus less restric­

tive than the mean, variance or moment approach because it holds for all

concave (monotonic) utility functions. Theories of risk aversion imply

that utility functions of investors are usually concave, and this issue can

be tested in the laboratory. There is little corresponding evidence to

suggest that we will ever be able to specify an investor's utility function

to the degree required by a strict interpretation of the mean, variance

theory of portfolio selection when extended to include the first four mo­ ments of the probability density function.

Nevertheless, the dominance theorm of stock selection is the more

troublesome approach in an empirical sense, because one must know what the probability density function of one security is in order to determine if it dominates another density function. At the present time, the solution to this problem is not clear.

Both the mean, variance approach and the dominance theorm require knowledge about the probability density function of each security's rate of return. Empirical methods of determining probability density functions in­ volve goodness-of-fit tests or use of sample estimates of the probability density function moments. In this study, sample data is examined and pre­ sented as historical evidence upon which an investor could form subjective estimates of probability density function moments. Due to the limited amount of data for each firm, only expected return and the variability of return are evaluated. To measure skewness and kurtosis with small sample 44 evidence would be a waste of time and effort because the results would be highly suspect.

Search for and description of acceptable securities: step 4 .— The process by which investors become aware of potential investment opportuni­ ties is called the search process. At some point in time, an investor be­ comes aware of a particular corporation's stock through a broker's tip, some financial news, or various other types of communications. Once aware of a particular stock, many courses of action are open to the investor.

The initial set of information elements transmitted to the investor may be sufficient for him to decide to buy the stock, or to reject it. If the initial set of information is not sufficient for the investor to make a decision, then more information must be collected.

The search process is a continuous one in the sense that some initial awareness of a security will lead to more and more information, until a de­ cision can be made. If the decision is to purchase the stock, then the stock's performance must be reviewed periodically. Current rejection of a security may lead to either permanent rejection or to another evaluation at a later date.

Because we have accepted the mean, variability method of selecting portfolios as a starting point, the primary items of information needed for stock selection are the expected price relative and the associated standard deviation of the price relative. We shall now turn our attention to the proper specification and measurement of these two items of information. CHAPTER III

UNCERTAINTY AND PREDICTION MODEL ERROR

\

Uncertainty is defined as the extent to which the future can be speci­ fied in the present. In this case, uncertainty is the extent to which a common stock's price relative can be predicted or specified one year in ad­ vance. We are concerned with an objective dimension of uncertainty, which is the historical predictability of a common stock's price relative over a series of one year periods. The statistic used to measure historical pre­ dictability is the mean-squared difference between the predicted annual price relatives and the actual price relatives subsequently observed over a period of ten years.

A brief review of some traditional notions of uncertainty is presented as a prelude to our own analysis of the concept of uncertainty and the ap­ propriate measure of uncertainty.

Uncertainty: Some General Notions

Investment is the process of exchanging a known amount of cash for the right to receive a greater amount of cash in the future. When the amount of cash to be received during a future period is known at the time of the exchange, the investment is said to take place under conditions of certain­ ty. When many different amounts of cash return are possible, investment is said to take place under conditions of uncertainty, because the actual rate of return that will be earned over some predefined period of time frequently

45 46 will not be equal to the rate of return that was expected at the start of

that investment period.

A dictionary definition of uncertainty is the "quality or state of be- 36 ing uncertain; lack of certainty; doubt..." Conversely, certainty is de­

fined as "that which is certain or sure; the truth; the fact; also a cer- 37 tain account. A certain or definite number or quantity." Such comments and definitions illustrate the vagueness of the concept of uncertainty which leads to a variety of possible measures of uncertainty.

Weston has said that "reality is a continuum of cases running from cer- 38 tainty to increasing degrees of uncertainty." For discussion purposes, authors have frequently divided this continuum into two or more categories of cases according to the quality of information we have about each case.

In this context, information quality is synonomous with the degree to which probabilities can be objectively specified. The objectivity of a discrete probability estimate is defined as the variance of the probability estimates about the true probability. When the true probability is unknown, the ob­ jectivity of a discrete probability estimate is defined as the variance of a set of estimates about the arithmetic mean of all such estimates. For continuous probability density functions, the objectivity of the probabili­ ty estimates is defined as the variance of the probability density function parameter estimates about the true parameters, or about the arithmetic mean

•^Webster's New International Dictionary of the English Language (2d ed.; Springfield, Massachusetts: G. and C. Merriman Company).

Fred Weston, "The Profit Concept and Theory: A Restatement," The Journal of Political Economy (April, 1934), pp. 154-155. 47

of all estimates of each parameter where the true parameters are unknown.

Knight for example, is frequently quoted for distinguishing between 39 risk and uncertainty. He uses the term risk to refer to those situations

for which the probabilities of alternative outcomes are known. In general, probabilities may be known because of some physical characteristics of the experiment, or by empirical calculation of the relative frequencies of al­

ternative outcomes for a large number of experiment replications. The physical characteristics of. a die imply that the probability of any number appearing on the top surface is 1/6 when the die is thrown, because each die has six faces and each face should have an equal chance of appearing on top if the die is square and its center of gravity is at its geometric cen­ ter. For those outcomes which do not have their probabilities implied by some physical characteristics of the experiment, as does the die, the rela- 40 tive frequency approach can be used. The relative frequency of an out­ come is the proportion of all trials on which each particular outcome oc­ curs. The important point is that the same experiment must be repeated a large number of times in order to determine the relative frequencies of the various possible outcomes. If each experiment cannot be repeated, then the relative frequency approach has less validity because the resulting rela­ tive frequencies relate at best to a set of similar, but not equivalent, experiments.

•^Frank H. Knight, Risk, Uncertainty, and Profit (Boston: Houghton Mifflin Company, 1921), pp. 19-21.

^®The experiment is the throwing of the die, and the event or outcome is the integer value of the top surface of the die. In Knight's terms, the outcomes of an experiment are said to be uncer­

tain if the probability of each of those outcomes is not known The un­

certainty case is usually associated with those events and activities for

which we have little experience. Without such experience, one may have

little basis for evaluating the probabilities of various possible outcomes,

and so the outcomes are regarded as uncertain. All of the possible outcomes

may not even be known, which creates another form of uncertainty.

Marshak divided the continuum of the cases running from certainty to 42 increasing degrees of uncertainty into four classes. These classes are:

1. Incomplete information, nonstochastic case,

2. Incomplete information, stochastic case,

3. Complete information, stochastic case, and

4. Complete information, nonstochastic case.

As used here, information is the ability to specify the alternative out­

comes. All possible outcomes are assumed known in the complete information

cases, while all of the outcomes cannot be specified in the incomplete in­

formation cases. The term stochastic implies that the probabilities of the defined outcomes are specified, or are somehow known. The probabilities of alternative outcomes are not known in the nonstochastic cases.

The distinction between risk and uncertainty made by Knight is impor­

tant to us because it is frequently quoted in economic studies of invest­ ment under conditions of uncertainty. Likewise, the more modern terminolo­ gy incorporated by Marshak is frequently used in conjunction with discus-

42 Jacob Marshak, "Role of Liquidity under Complete and Incomplete In­ formation," American Economic Review, Suppl. XXXIX (May, 1949), pp. 183-184. 49

sions of the investment decision under uncertainty. These frequently used

terms have special meaning to many people, so our use of the terms must be

made clear, and must be contrasted with traditional definitions.

Using Marshak's terminology, we have complete information with respect

to the possible rates of return that can be earned on investment in a com­

mon stock. All the possible rates of return can be specified in advance if

we are willing to aqcept the existence of an upper bound on market price

per share. Given an absolute upper bound on price, a rate of return can be

calculated for each possible combination of cash dividends and ending mar­

ket price because the number of possible combinations is finite. If no up­

per bound on ending market price existed, then the number of possible rates

of return would not be finite, and one could never specify all possible

rates of return. Even with an upper bound on price, the number of possible

rates of return is very large. Although it is very likely that investors will disagree about the probable maximum price of any common stock share,

agreement on a very high upper bound is plausible. As a practical matter,

then, we are able to specify the possible rates of return on investment in

any common stock, with few errors.

Specification of the probability of each rate of return outcome is

another matter. We argue that it is very difficult, if not impossible, to

specify in an objective manner the probability of each rate of return out­

come. The observation that the investment services, accountants, research­

ers in finance, and investors do not assign a probability measurement to

each possible rate of return outcome for each common stock serves as prima

facie evidence for this point of view. 50

If we assume that the probability of each possible rate of return can­ not be objectively specified, then what approach does one take? We shall suggest a specific approach to the uncertainty measurement problem after first discussing a more philosophic methodology question.

At least two approaches to the problem of uncertainty specification and measurement can be visualized. One approach is to describe and define a concept of uncertainty, and then construct a measure of uncertainty which is consistent with the concept. A second approach is to select a measure of uncertainty, and then define uncertainty in terms of that measure. The approaches are similar in that the end result is a concept of uncertainty and a measure of uncertainty which relates to the concept.

Neither approach can be judged correct or incorrect by logic alone.

The theoretical correctness of a measure of uncertainty is a matter of the / *5 collective judgement of the relevant scientific community. This situ­ ation prevails regardless of the method used to define and measure uncer­ tainty. Usefulness is another criterion by which we might evaluate the relative merits of different concepts and measures of uncertainty.

Neither the usefulness criterion nor the criterion of general accep­ tance by a scientific community are helpful in the initial evaluation of a proposed change in current practice or theory, which is our purpose. Our approach consists of three stages: (1) the definition of uncertainty in terms of predictions, (2) arguments in support of the particular definition adopted, and (3) the specification of a measure of uncertainty which is consistent with the definition proposed. Because our definition of uncer-

^ S e e Churchman, Chapters 4 and 5. tainty is developed within the context of a particular conceptualization of

the investment process, it is not being presented as the only concept of uncertainty which is relevant to the general process of investment.

Uncertainty and Prediction

The first step in any analysis of uncertainty is a specification or description of the subject of a prediction. The subject of a prediction is 44 the function which determines the description of an event. Consistently, an event is being predicted because it is uncertain, and that event must be

described in some manner so that we can verify its occurrence. In this

context the subject of prediction is the common stock price relative, which is defined below.

The price relative as a measure of return.— Earnings on investment can be measured in a variety of ways. Some of the more frequently used measures are the rate of return on investment and the price relative, which is a transformation of the rate of return on investment. The rate of re­ turn is the rate of increase in wealth over some period of time. The an­ nual rate of return represents the relationship of wealth at year end to wealth at the start of the year. For example, let

W± be the wealth that exists at the start of year t, for invest­ ment i, and let

W. be the wealth that exists at the end of year t, for investment i. i > t The rate of return on investment i over period t is denoted R OR^, and is defined such that:

44S. A. Ozga, Expectations in Economic Theory (Chicago: Aldine Pub­ lishing Company, 1965), Chapter 1. 52

(3.1) (1 + EORlt) - W1>t.

In the case of common stock investments, the wealth associated with invest­ ment i at the start of period t is N. „ , • P. t where N. , is the x,t-l i,t-l’ i,t-l number of shares of security i owned at the start of period t, and P i»t—1 is the price per share of security i at the start of period t, which is to say:

(3.2) W. . = N. . -P. , . i,t-l i,t-l i,t-l

Most common stocks pay cash dividends during each year and intermittently change the number of shares outstanding through stock splits and stock div­ idends. Cash dividends become a part of stockholder wealth and may be in­ corporated into the wealth calculation in a variety of ways.

One approach is to assume that dividends will be reinvested immediately upon receipt in fractional shares of the same security. The number of shares of security i owned at the end of period t is thus N. + F. , x , t x , t where F. is the decimal number of shares of security i that the investor is able to purchase during period t with his period t cash dividend pay­ ments. Another approach is to assume that cash dividends are held until year end and thus are reinvested in the same security at the start of the next investment period. A third approach is to assume that cash dividends are reinvested in whatever securities seem to be the best buy at the time the cash dividend is received.

The first approach of assuming immediate reinvestment upon receipt of cash is not used in this study because the price and quarterly dividend data necessary to do so are not readily available. The difficulty and cost of collecting this data does not seem to be justified in terms of the ex- pected affect on the results of this study, and the nature of the alterna­

tive assumption that is made. The third approach of assuming that cash

dividends are invested in the most profitable alternatives available at the

time of cash dividend receipt is probably the best representation of the manner in which portfolios are actually managed. However, the performance

of individual investments cannot be evaluated when cash flows are reinves­

ted in other securities. Because we are concerned with predicting the price relatives of individual securities, the approach of assuming rein­ vestment in other securities cannot be used in this study.

It is assumed that cash dividends are held until year end, or are used for consumption purposes. As a result of this assumption, end of period wealth, denoted W. , is defined as i,t’

(3.3) W. = N. . • (D. . + P. .), where D. . i s the dollar amount i,t i,t i,t i,t i,t of cash dividends per share paid during year t. As before, N. denotes the number of shares owned at the end of year t.

The term N. . in equation (3.2) is equal to the value of N. in 1 y C 1 1 j t equation (3.3) whenever there are no changes in the shares owned by the investor. The two events leading to a restatement of the number of shares owned, are stock dividends and stock splits. Simple increases in the num­ ber of shares held by any particular investor due to purchases do not af­ fect our rate of return calculations, but stock dividends and splits do af­ fect the relationship of the number of shares held at period start to the number of shares held at period end.

First, consider the case of stock dividends. If a 2% stock dividend is distributed during period t, then 54

(3.4) N. = 1..02 . N. j, assuming no other changes for security i X , t X ,t—I have taken place. Because this study is a historical analysis of stock prices, our adjustment must be in terms of current shares outstanding. So

for each share outstanding at the end of period t, there are 1/1.02 or

.9804 shares outstanding at the end of year t-1 on an equivalent investment basis. The year used as the base period is the most recent year used in

the study, or 1967 in this case. Each previous N. is adjusted so that l , t per share statistics represent a given percentage ownership in the company,

or a common investment base.

Stock splits were the second complicating factor in computing wealth

changes over time. Suppose there is an X for Y stock split, which means

each owner receives X shares of security i for each Y shares held before

the split. The relationships of shares owned at the beginning of the year

to shares owned at the ending of the period is expressed as

(3.5) N. = N. , (X/Y). Thus for each share of security i owned X y C X 5 t — I

at the end of period t, there were Y/X shares owned at the start of the period t, assuming the split took place during period t and there were no

other changes.

The terms N. . and N. are adjusted whenever there is a stock split X)t*l 1)t or a stock dividend. Hereafter, no explicit mention will be made of this I adjustment because it will always be made, and the adjustment accounts for

the fact that N. may differ from N. ,, for any i and t. X , t X)t"X Restating the rate of return formula, we have

W. - W. . (3.6) ROR. = ^ --- , which also can be restated as ’ Wi,t-1 55

w t (3.7) ROR. = 77-*------1, where (3.7) is the computational form. X,t i,t-l

The price relative for a security i over some period t is defined as

W i t (3.8) R. = t —, W. . > 0. Notice that R. = 1 + ROR. . i,t w_^ ^ >t i ,t

Due to the limited liability feature of corporations, stock market prices are non-negative, even in the case of liquidation. Because W. 1»t is almost always greater than zero, the price relative is almost always greater than zero. In contrast, the rate of return is negative when the price decline exceeds the value of dividends, which frequently happens when investment periods of one year are used. Price relatives are more conve­ nient to process because of this non-negativity feature. The basic measure of return on investment used in this study is the price relative as des­ cribed in (3.8).

The prediction aspect of uncertainty.— An investor can estimate the current dollar value of his common stock investments by referring to stock market prices or other evidence of the net cash price of each investment.

For each common stock, the valuation process usually involves multiplying the ending market price per share by the number of shares currently owned.

Total wealth is then determined by adding the dollar value of all the indi­ vidual common stock holdings.

Suppose the aforementioned investor would like to know something about the dollar value of his wealth at the end of the current calendar year.

Because stock prices and cash dividends change over time, the investor's wealth may assume many different dollar values by year end. Since many 56

values of his wealth are possible, the investor either must predict the

most likely ending value of his wealth, or he must describe the possible

wealth values in terms of a probability density function. He may predict

total wealth directly, or he may predict the ending dollar value of each

common stock and then combine the results to construct a prediction of the

dollar value of total wealth.

; In Chapter II we established that investors need to predict the future

value of wealth in order to decide upon current consumption levels. The

need to predict the dollar value of one's wealth at the end of one year is

thus built into our model of investor behavior.

The need to predict future stock prices and cash dividends is actually

greater than the above comments imply. If an investor wants to maximize

the wealth-consumption function described in 2.0, then he must invest in a

combination of the most profitable investments available. The most profit­

able investments are those for which next year's price relative will be

greatest (we have assumed a one-year investment horizon). Obviously, one

must formulate some predictions about each common stock's price relative

for next year in order to make a selection of the most profitable ones.

To summarize, an investor must predict future price relatives in order

to evaluate investment alternatives — especially if an attempt is being made to maximize or increase rate of return on investment.

To note that an investor must predict price relatives over some in­ vestment horizon is only a start. Next, we must consider the nature of

these predictions and the resulting prediction errors.

When only two outcomes are possible, predictions are either correct or

incorrect. The football predictor either picks the eventual winner of the 57

Super Bowl, or he does not. Likewise, a baseball fan either correctly

picks the winner of the World Series, or he does not.

For common stock investors the notion of correct versus incorrect pre­

dictions is of little relevance because price relatives are seldom predic­

ted in advance. Since each price relative can take.on many different val­ ues, it is unlikely that any investor will be able to predict which value will actually be earned over the next year. However, a skilled predictor may make predictions that are nearly correct in the sense that the predic­

ted price relatives are close to the price relatives subsequently earned.

It is this closeness between predicted price relatives and observed price relatives which we analyze in more detail.

Uncertainty Defined

In the investment context, uncertainty is defined as the extent to which price relatives cahftbe predicted one year in advance. Historical evidence and expectations about the probability density function of the price relative prediction errors provide the basis for a measure of the extent to which price relatives can be predicted.

The evaluation of predictions through an evaluation of the prediction 45 error probability density function is widely used and accepted.

This definition of uncertainty provides for the estimation of a prob­ ability density function from historical evidence, or from any other source.

Markowitz has described how a security analyst might crudely structure his beliefs about the probable rate of return that each common stock can be

45 James C. Naylor, "Some Comments on the Accuracy and the Validity of a Cue Variable," Journal of Mathematical Psychology, Vol. 4, No. 1 (February, 1962), pp. 154-161, and Henri Theil, Applied Economic Forecast­ ing (Chicago: Rand McNally and Company, 1966). 58 46 expected to earn. The beliefs of the analyst reflect the totality of his

experience, which includes any data about the firm which he has absorbed.

The bases of the analyst's expectations are varied and are not well defined.

We are interested in the historical evidence of uncertainty which is

provided by past prediction performance, where the predictions are made by

means of well defined models. Consistently, the remainder of this study is

concerned with the development and application of a measure of historical

uncertainty. The measure will be historical in the sense that it is de­

rived from historical data and relationships. Such a measure is considered

important because most estimates about the future start with interpreta­

tions of past performance and behavior. Our ability to draw any conclusions

about the confidence we have in our parameter estimates is unknown at pres­

ent.

A Measure of Uncertainty

Uncertainty is defined in terms of the probability density function of

prediction errors. Ideally, there would be no difference between the ex-

ante prediction of a particular price relative and the ex-post measure of

that price relative. As a practical matter, however, the predicted price

relative of any frequently traded common stock is apt to vary significantly

from the actual price relative that is earned.

The mean of a series of prediction errors should tend to zero if we

are to minimize prediction errors through constant improvement of the pre­

diction model. Given this objective to reduce prediction errors, the best measure of uncertainty is the second moment about zero of the prediction

lif\ Markowitz, Chapter 2. 59

47 errors, which is the mean-squared prediction error.

The mean-squared error is the arithmetic average of the differences

between the numerical value of an observed price relative and the numerical

value that was predicted for that same price relative. The formula is

(3.9) MSE = ln (R - R .. )2 /n, where J1 t—1 JJ-1'

it is the actual, observed price relative for common stock i

over the period t, 'V, R ^ t is the predicted numerical value of as made by pre­

diction model j as of the end of period t-1, and

n is the number of periods involved. In this case, n equals

10 (years).

In addition to being useful as a measure of total uncertainty, the MSE

statistic is amenable to division into two component parts. This division

provides a useful basis for analyzing the nature and source of prediction

errors in general. Because this separation of the MSE into component parts

can be expressed in terms of traditional statistical measures, we start with a discussion of prediction errors and the relevant statistical mea­

sures of these errors,

First, observed price relatives are compared to predicted price rela­

tives in order to obtain a series of differences, D.. . as indicated by Jit’

^ T h e mean-squared error is a measure commonly used in statistics. For example, see W. Allen Wallis and Harry V. Roberts, Statistics: A New Approach (New York: The Free Press, 1956), p. 448. The formulation of mean-squared error is also widely used in various contexts, such as in Yuri Ijiri, The Foundations of Accounting Measurement (Englewood Cliffs: Pren- tice-Hall, Inc., 1967), p. 142; and Miltin Friedman, The Interpolation of Time Series by Related Series (Technical Paper No. 16, New York: National Bureau of Economic Research, 1962), p. 14. 60

'b (3.10) D.. = R.^»- R..^, where j denotes the prediction model used, jit it jxt

i denotes the common stock for which price relatives are being predicted

and t denotes the year. Two rather common sample statistics, the mean and

the variance of the prediction errors, are defined as

(3.11) MN.. = In D . . /n = DD. .., and Ji Lt=1 Jlt: J’i

(3.12) VAR.. = l n (D - 5 )2 /n. JA t=l J

Using the relationship defined in 3.10, formula 3.9 can be put in the form

2 (3 ,13) MSE.. = yn D .. /n. Ji L t=l Jit

These two component parts of the MSE are the mean prediction error squared and the variance of prediction errors about their mean error. For- 48 mula 3.12 can be restated as

(3.i« vae^. - v 2 /- - [rM »jlt /»]2. which means equation 3.13 can be restated as,

(3.15) MSE.. = VAR.. + (MN..)2 . Ji Ji Ji

The mean prediction error, the variance of prediction errors, and the mean-squared error are operational measures of bias, precision and accura­ cy, respectively. Bias is the extent to which a prediction model or pro- 49 cess persistently over or under predicts. Excessive overprediction im- 'V plies that Rjit will exceed R^t more frequently than not, or that large

48rhe reason for using the biased estimate of o^2 as shown in 3.14 is explained in Appendix A^ Actually the right side of equation 3.15 is an unbiased estimate of o , or MSE... e ji ^Naylor, pp. 155-156. 61

prediction errors are more often associated with overprediction than with

underpredictions. Conversely, excessive underprediction implies that IL 'b will exceed more often than not, or that the larger prediction errors

made by the model are associated with underpredictions more often than with

overpredictions. Due to the formulation of the difference term (3.10) ex­

cessive overprediction results in a negative average error, and underpre­

diction results in a positive average error.

The term precision has been used by Cochran and Cox to denote the

closeness with which a measurement approaches the average of a long series

of measurements, made under similar conditions.^ In this study, the term

precision shall be used to denote the closeness with which a prediction

error approaches the average prediction error over the: time period during which the model was operated. The difference between the two uses of the

term precision is in the set of conditions under which the measurements or

predictions take place. In the context of measurement described by Cochran

and Cox, repeated measures are being taken of a particular characteristic

of a person or an object, ceteris paribus. In our prediction context, con­

ditions are changing and these changing conditions are only partially rec­

ognized in the prediction model. Therefore, one has a series of'single ob­

servations in the prediction case, whereas one has repeated observations within a single time period in the measurement case. The basic idea that

the measurement (or prediction) differs from the true value by some error

is common to both cases, however, so the term precision is adopted for use in our context of prediction.

G. Cochran and G. M. Cox, Experimental Designs (2d ed.; New York: John Wiley & Sons, Inc., 1957), p. 16. 62

As mentioned before, the operational measure of imprecision is the variance of prediction errors about the mean prediction error. Precision is thus unaffected by any bias that exists in the series of prediction errors. This implies that a prediction model can result in precise pre­ dictions which may or may not be biased predictions. Likewise, a predic­ tion model can result in relatively imprecise predictions which may or may not be biased predictions.

Accuracy is the term used to signify the closeness with which the pre­ dicted value of a price relative approaches the numerical value actually obtainedAccuracy is thus a measure of the extent to which prediction is achieved, since the ultimate goal of prediction is the minimization of differences between predicted price relatives and the actual price rela­ tives subsequently obtained. Note that accuracy can be determined without regard to the source of the predictions.

The accuracy of a set of predictions is reflected by the mean-squared error of the resulting prediction errors. More precisely, formula 3.13 is 52 a measure of the inaccuracy of a set of predictions. The term inaccuracy is used because the value of M S E ^ increases as inaccuracy increases, and this relationship is easier to comprehend than the inverse relationship that exists between the numerical value of and accuracy as defined.

As indicated in formula 3.15, bias and imprecision both contribute to

51 This definition follows a parallel definition of accuracy in mea­ surement by Cochran and Cox, op. cit., p. 16, and is the same definition used by Naylor, p. 156, in the context of the predictive performance of a cue variable.

■^Naylor is-responsible for this use of the term inaccuracy. Ijiri, p. 143, uses essentially the same formulation to define reliability in accounting measurements. 63 prediction errors. Because the mean-squared error is increased by the mean error, squared, it becomes quite apparent that finding unbiased predictions

is very important. Moreover, formula 3.15 provides an exact indication of

the trade-off that is involved between biased predictions and imprecise predictions. Bias should be reduced if and only if the can be re­ duced. Likewise, imprecision should be reduced if and only if the can be reduced. Whenever bias and imprecision can be reduced simultaneous­ ly, there is no trade-off and MSE.. is reduced, of course. . Ji The utility function.— A slightly different form of the usual quadra­ tic utility function is implied by the definition and measurement of uncer­ tainty adopted here. The utility function stated in 2.4 is restated as

A A (3.16) U = R - b oe, where R is the predicted price relative, b is a constant which reflects the particular investor's aversion to uncertainty, and is the variance of prediction errors about zero. MSE is the statis­ tic by which we estimate a ■ e

Price Relative Variance as a Measure of Uncertainty

The range and the standard deviation are two popular measures of un­ certainty. As a starting point, consider the range of the possible values of a price relative as a measure of uncertainty. Lange has argued that people "need not, and usually do not visualize an exact probability distri- 53 bution of possible prices." Further, people only consider a practical range, which excludes extreme values, he argues. Lange does not provide any theoretical or empirical evidence to support this hypothesis, but his

53 Oscar Lange, Price Flexibility and Employment (Bloomington, Indiana: The Principia Press, Inc., 1952), p. 29. 64

comments serve to indicate at least one notable proponent of the use of

range as a measure of uncertainty.

Suppose that a price relative A has ranged rather uniformly between

.90 and 1.50 over the last twenty years. Suppose that a second price rela­

tive, price relative B, has also ranged between .90 and 1.50, but that all

but two price observations are between .95 and 1.35. Clearly, the range of

price relative A equals the range of price relative B. Investment B is

preferred, however, because there is less variability in investment B, than

there is in investment A. The range is not sensitive to any differences

within the range, and thus is a less complete measure of uncertainty than

the standard deviation.

The standard deviation of return per dollar of investment is another 54 popular measure of uncertainty. It is the positive square root of the variance of a variable about its mean value, and thus is a measure of dis­

persion.

Under certain conditions, the standard deviation is a relatively com­

plete measure of uncertainty. If a price relative were properly described

as a random variable such that each price relative is independent of every other price relative, and the probability density function of the random

See Jacob Marshak, "Money and the Theory of Assets," Econometrica, VI (October, 1928), p. 320; H. Makower and Jacob Marshak, "Assets, Prices and the Monetary Theory," Economica, N. S., V (August, 1938), p. 272; Albert Gailord Hart, "Risk, Uncertainty, and the Unprofitability of Com­ pounding Probabilities," Studies in Mathematical Economics and Econometrics, 0. Lange, R. McIntyre, and T. Yntema, editors (Chicago: The University of Chicago Press, 1942), pp. 110-118; Donald E. Farrar, The Investment Deci­ sion Under Uncertainty (Englewood Cliffs: Prentice-Hall, Inc., 1962); H. Markowitz, Portfolio Selection (New York: John Wiley & Sons, Inc., 1959); P. A. Samuelson, "General Proof that Diversification Pays," Journal of Financial and Quantitative Analysis, II, No. 1 (March, 1966), pp. 1-13; and many others. 65

variable is symmetric, then the standard deviation is a relatively complete

measure o£ uncertainty.

Even under these conditions the variance understates the difficulty of

predicting price relatives because the true mean of the price relatives is

unknown. If price relatives are properly described by a random variable

with a stable probability density function, then the best prediction of

each year's price relative is the mean price relative, which is estimated

by means of sample data. Prediction errors will be caused by the variabil­

ity of the price relatives as well as by sampling errors in estimating the

mean value of the price relatives.

The problem created by use of sample estimates of the true mean may

not be too great. The standard deviation of the sample means is a function

of the number of observations and the variance of the price relatives

(o = a/ n ) . This implies that the variability of the price relatives, R and the sampling errors associated with estimating their mean, are both

proportional to the standard deviation of the price relative population.

As a result, the standard deviation of past price relatives may be an ade­

quate measure of uncertainty under the conditions stated.

There are criticisms of the standard deviation which are much more

basic than the simple issue of sampling error. Suppose price relatives are

not described by a stable probability density function. If a much more

complex stochastic process better describes the price relative data, then

the standard deviation of price relatives over some past period does not

represent uncertainty. It represents variability about a mean, but that

mean may have little relationship to data from other time periods.

I 66

The standard deviation of price relatives should understate uncertain­

ty under these circumstances because the time model of the stock price re­

latives is unknown, and because the mean price relative for the test period may not have been predictable. It is difficult to speculate as to the exact effect of using the standard deviation to represent the uncertainty of price relatives that are generated by an unknown stochastic model.

In contrast it is conceivable that the price relative variance could overstate uncertainty. In time series data, some trend frequently exists,

and that trend could be used to predict future price relatives. As a mea­ sure of variability about a mean, the standard deviation calculation ig­ nores any trend information which is contained in the price relative time series. The variability about the predictable trend may be less than the variability about the mean, which would imply that the standard deviation overstates uncertainty.

In addition to trend, the value of a price relative may be related to other variables. Using these relationships, one might be able to make pre­ dictions which are highly correlated with the price relative series. The differences between the price relatives and their mean may be greater than

the differences between the price relatives and the price relative predic­ tions based on other variables, so that the standard deviation would over­ state uncertainty.

In the next section we present more formal arguments for attempting to predict common stock price relatives in order to reduce uncertainty. 67

Analytical Arguments for Attempting To Predict Price Relatives

In general, prediction is useful if it serves to reduce uncertainty.

The manner in which prediction can lead to reduced uncertainty is dis­ cussed in this section.

A summary of the argument can be stated as follows: Let R be a ran- 2 dom variable, and let p be the mathematical expectation of R and a be the variance of R. It is assumed that the variance of a random variable is an adequate measure of its uncertainty, so prediction of actual obser­ vations of R is desirable if the conditional variance of R, given the prediction, is less than the variance of R.

The validity of this statement can be demonstrated by considering the relevant conditional probability arguments. Accordingly, let

Rit a rand°m variable which represents the ratio of the market

price of security i at the end of period t to the dollar amount

invested at the start of period t,

P2£t be the mathematical expectation of and

2 a „ . be the variance of R._. 2it it

Let X. be a random variable for which and R.„_ have the joint prob- lt xt it ability density function • Dropping the subscripts

for convenience, the conditional probability density function

of R, given that X=x, is

(3.17) f(r|x) = f (r,x)/f^(x), f^x) > 0, where f^x) is the marginal probability density function of X.'*’’

-^The following discussion can be found in many standard mathematical 68

By definition.

(3.18) E(r|x) = /^r-f(r|x) dr, and from 3.17

(3.19) E(r |x ) - £ 4r

The conditional expectation of R given X=x, is a function of x alone.

If it is a linear function of x, denoted by g(x), then it can be written

in the form g(x) = a+bx. Expression 3.18 can thus be restated as:

(3.20) E(R|x) = f_l dr = a+bx, or

(3.21) /“r-f(r,x)-dr = (a+bx)*f1(x)

Integrate both members of 3.21 on x to obtain

00 00 00 / rf„(r) dr = af fn(x) dx + b f x f-(x) dx, — 00 2 — 00 — CO 1 which by substitution of symbols, can be restated as

E(R) = a + b E(X), or as

(3.22) ^2 = a + by^.

Multiply the members of 3.21 by x and then integrate on x to obtain

f f x x f(r,x) drdx = af xf-(x) dx + bf x2 f1(x) dx, — 00— 00 ' 7 ' — 00 — 00 X which is by definition,

(3.23) E(XR) = a •E(X) + b-E(X2).

From the facts that E(X2) - [E(X)]2 = a^2 or E(X2) = a^2 + y^2, and

statistics texts. This particular development is from Robert V. Hogg and Allen T. Craig, Introduction to Mathematical Statistics (2d ed.; New York: The MacMillan Company, 1965), Section 2.2, pp. 54-65.

56Hogg, op.cit., p. 63. 69

E(XR) - y . ^ = E[ (X-y^) • (R-y2) ] - Pi2°la2’ exPression 3*23 can be restated as: 2 2 (3.24) P12CTlCT2 + m1P2 = aPl + b ^al + ui

The simultaneous solution of equations 3.22 and 3.24 yields

a 2 °2 b = an^ a = y2 - ~ so ^*20 can be restated as

°2 (3.25) E(R|x) = (y2 - P12 ~ *1^) + Pl2‘cr2/,C^1*X, which reduces to

°2 (3.26) E(r|x) = y2 + ~ * (x-y-^) > t*ie conditi°aal expectation of

R given X=x.

We shall now consider the conditional variance of R given that X=x, for the case where the conditional variance of R is given by

(3.27) E {[R - E (r|x)]2 | x} =

£ > [r - y2 - P12 (x-P^ ] 2 • f (r|x) *dr, or

f 2 t(r-y2) “ P12 (x-y^ ] 2 • f(r,x)*dr

f-L(x)

The variance is non-negative and at most a function of x alone, so f multiplication by f^(x) and integration of 3.27 on x will be non-negative.'

Recall that E[h(x)] = / h(x) f.(x)»dx by definition, so letting h(x) equal ™°° _L the right hand member of 3.27, we have

57Ibid., p. 64. 70

^(r-u2)-p12 (x-)J1)2 • f(x,r)dr j f^x) dx (3.28) E(h (x)) = CC

0 0 0 0 = f j a (a) J „ „ -Joh OOm OO

2

E [«-u2)2j -2p12^ E [tt-UjMR-Pj)] + Pi22 ^ 2 E [«-“l>2]

„ 2 2 0 2 2 2 2 = °2 2p12 o7 p12°la2 + P12 2 °1 1 al

2 „ 2 2 2 2

" °2 12 a2 p12 2

= c22 d - » 122) ; o

The expected value of the conditional variance of R, given X=x, is 2 2 equal to a2 (1-p ). Notice that the higher the correlation coefficient of

X and R, the smaller is the expected variance. Perfect correlation im­

plies perfect prediction and a correlation coefficient of 1. If p = 1, 2 p = 1 and the conditional variance of R is zero.

The above formulation is based on the assumption that the costs of

prediction are zero and that uncertainty should be reduced, ceteris pari­

bus. The postulation that some investors seek to maximize wealth implies

that if expected return is unchanged, then less uncertainty is preferred

to more uncertainty. Less uncertainty is preferred becasue the probability

that actual wealth will be within ± X% of expected wealth is increased as

the uncertainty is decreased.

* 71

When prediction is not cost free, the extent to which the investor will undertake prediction, or pay for predictions, depends on the utility he has for less uncertainty. The utility he has for less uncertainty is reflected in the marginal rate of substitution of more predictability

(less uncertainty) for expected return.

The utility functions of some investors can theoretically be ex­ pressed as"^ 2 (3.29) U(R) = y^ - Acr^ , where R is the financial rate of return,

U(R) is the utility of satisfaction derived from earning R,

y^ is the mathematical expectation of R,

2 is the variance of R, and

A reflects the curvature of the investor's utility of money

function.

From this illustrative utility function, we can demonstrate the effect of prediction costs, in a somewhate elementary manner. Expression

3.29 becomes

(3.30) U2 (R) = (y^C) - A(o12 - Aa^2) , or

U2 (R) = (y^-C) - A a ^ - d - A),

where C is a prediction cost per dollar of planned invest­

ment and 2 A is the proportional reduction in resulting from pre­

diction.

5%onald Eugene Farrar, The Investment Decision Under Uncertainty (Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1962), p. 15; and Milton Friedman and Leonard J. Savage, "The Utility Analysis of Choices Involving Risk," The Journal of Political Economy, Vol. LVI, No. 4 (August, 1948), pp. 279-304. 72

Prediction will take place if and only if ^ ( R ) > U(R). The condi- 2 2 tion ^ ( R ) > U(R) implies ^( R ) - U(R) = AXo^ - C. Because AXo^ - C 2 must be greater than zero, AXa^ > C must be satisfied for prediction to take place.

The discovery and specification of utility curves is beyond the scope of this project, so the determination of prediction costs is of little value. We can point out that as the dollar volume of investment increases, the prediction cost per dollar of investment decreases. This relationship helps to explain the reason for the extensive analysis staffs of large investors as opposed to the rather naive methods of individuals.

We have established that if price relatives are a linear function of a cue variable, then the conditional variance of the price relatives may be reduced. The analysis shall now be extended to show the relationship of predictions to the mean-squared prediction error, which is the measure of uncertainty adopted here. The mean-squared prediction error may be de­ composed as follows:

(3.31) I (R - R)2/n = (R - R)2 + (S - rS )2 + (1 - r2) S 2 where p A A

R is the mean price relative,

R is the mean predicted price relative,

Sp is the standard deviation of the price relative predictions,

is the standard deviation of the actual price relatives, and

r is the product-moment correlation coefficient of predicted

and actual price relatives.

Substitute the symbol R for a + bx, where x is the predictor or cue vari- 73 able discussed in equation 3.20. Under ideal conditions, predicted price relatives will be equal to actual price relatives, so that the only source of prediction error will be the lack of perfect correlation between x and

A R, (p-^2 < 1) • This statement can be verified by noting that if R = R, then

X A A R = R, and R - R = 0. The statement R = R is a special case of the state- SA SA ment R = b*R, for b* = 1. Because b* = r — , and b* “ 1, we have 1 = r — b b P P and S = rS., or S - rS. = 0. p A ’ p A The first two terms of 3.31 are thus zero, which leaves the term 2 2 S. (1 - r ), which is equivalent to equation 3.28. A We have now shown how the use of a prediction variable will lead to a reduction in the mean-squared prediction error where the linear relation­ ship between the predictor variable and the variable being predicted is known. The effect of biased predictions (R = a + bR) on total uncertainty is discussed earlier in this chapter and in Chapter V.

Summary

Uncertainty is defined as the extent to which price relatives can be predicted one year in advance. The extent to which price relatives can be predicted is reflected in the probability density function of the predic­ tion errors. The mean-squared prediction error is used to measure uncer­ tainty since it is the second moment about zero of the prediction errors.

The variance of past price relatives about a mean is not believed to be a good measure of uncertainty because the variance calculation ignores information contained in the price relative series, as well as ignoring the fact that a cue variable may exist for which the conditional variance of the price relatives may be less than the unconditional variance. More importantly, the use of a mean of past price relatives to predict future price relatives assumes a specific model of stock price relatives which may not fit the data.

The relationship of predictions to the price relative variance and to the mean-squared prediction error are discussed in detail. Having estab­ lished predictability as the aspect of uncertainty which we wish to study, we now turn to the problem of formulating predictions. CHAPTER IV

THE PREDICTIVENESS OF COMMON STOCK PRICES

AND PRICE RELATIVES

In spite of the large number of common stock selection methods, prac­ ticing investment analysts do seem to fall into two clusters*^, €ne cluster being the technical analysts who chart stock price movements in order to determine the proper time to buy a particular common stock, and the other cluster being those analysts who examine growth, earnings stability, price/ earnings ratios, and other measures of economic performance in order to evaluate intrinsic value. Intrinsic value is a concept of long-run value to which market price should converge.

The nature of technical analysis is discussed in the introduction to this chapter. In the second section, we review the random walk hypothesis, and then we analyze its implications for price relative prediction in sec­ tion three. The remainder of the chapter is concerned with the development of prediction models. Three naive models are presented as one approach to the formulation of price relative predictions. They are linear, first de­ gree equations for which the constants are estimated using exponential smoothing of past price relatives. A fourth naive method is the use of the mean of the prior ten price relatives as the prediction of the next price relative. An economic model which simulates The Value Line Investment Sur­ vey methodology is developed in the final section of this chapter.

75 76

Introduction

For many years attempts have been made to predict the future prices of individual common stocks, as well as future movements in the various common stock price averages which reflect general market trends. A superficial examination of the prediction problem might lead one to conclude that past prices or price changes should be of some value in predicting future prices and future rates of return. However, two conflicting theories about stock price movements have emerged over the years, and they are of fundamental importance whenever stock price predictions are being considered. Cootner went so far as to say "... it is hard to find a practitioner, no matter how sophisticated, who does not belie^Ie that by looking at the past history of prices one can learn something about their progressive behavior, while it is almost as difficult to find an academician who believes that such a 59 backward look is of any substantial value."

"Technical analysis" is the name given to the process of investigating past stock price movements in order to predict future price movements.

Through a careful examination of past common stock prices, a good analyst can presumably determine what future market movements will be. Because the analyst must be correct in his predictions more often than not, in order to attract a clientele, it is safe to say that a proponent of technical analy­ sis would argue that price increases or decreases can be predicted more often

59 Paul H. Cootner (editor), "Introduction," The Random Character of Stock Market Prices (revised edition; Cambridge: The M.I.T. Press, 1964), p. 2. 77

than is possible by using a chance model, or by making naive predictions.

Chartists and technical analysts do not argue that they can predict the mag­

nitudes of the future changes in common stock prices.

In order to state a theory of technical analysis, we must observe the

practice and note the writings of analysis practitioners because technical

analysis is an art which is practiced by many different types of people,

and not a field of scientific investigation. Because of the current state

of this art, however, the theoretical construct of technical analysis is

not in a generally accepted, testable form. The theory of technical analy­

sis that appears below is thus an imputed theory.

Underlying the use of technical analysis is the theory that common

stock prices tend to move in identifiable patterns, and that these patterns

will be repeated in future years. The various forces that affect stock

prices must therefore come into play in specific sequences and in magnitudes

comparable to previous periods. The significant idea is that there are cer­

tain stock price patterns which tend to be repeated.

One implication of the technical analysis approach is that meticulous

study of the historical series of a stock's prices and knowledge of the

current price movements of that stock can lead to above average trading pro­

fits. Another implication is that prediction models of future common stock

prices should incorporate information about the recent price movements of

the stock, or should include the predictions of a skilled analyst in order

to be complete models in an informational sense.

Due to the lack of objective, well defined estimates made by technical analysts, it is not possible to include such estimates in a formal prediction model. Moreover, some characteristics- of stock price behavior have been 78 tested and the conclusion drawn is that formal prediction models based on past prices are of little use.

The second, and conflicting theory of stock price behavior is that a common stock price series resembles a random walk. The level of a stock's price is not believed to be random, but the price changes from period to period are random and unpredictable. Any trends or complex patterns that appear in the series of past prices are due to a stochastic process, and are not apt to be repeated in the future. If the process generating com­ mon stock price changes is a random walk, then technical analysis is use­ less, and there is no need to search for the various patterns that techni­ cal analysts supposedly look for.

In the context of this project, the problem is to determine if know­ ledge of past prices is of any substantial benefit in predicting future price relatives (rate of return). If past prices or price changes are of little benefit in predicting future price relatives, then other means must be used to predict future price relatives. Conversely, if past prices or price changes are useful in the prediction of future price relatives, then any model used to predict future price relatives should make use of such knowledge. The method used to evaluate the potential prediction value of past stock prices or price changes is a review of the formal, empirical studies of stock price behavior. The results of this review are presented in the next section. 79

The Random Walk Hypothesis: A Review

Technical analysis persists in stock market circles even though some empirical evidence supports the hypothesis that changes in the price of a common stock behave as if they were generated by a simple chance model.

Roberts gives two reasons for the lack of a "widespread recognition among financial analysts that the patterns of technical analysis may be little, if any thing, more than a statistical artifact.One possible explana­ tion is that the usual methods of graphing stock prices gives a picture of successive 'levels' rather than of 'changes', and levels can give an arti­ ficial appearance of 'pattern' or 'trend'. A second is that chance behav­ ior itself produces 'patterns' that invite spurious interpretations.^ The patterns and trends which exist in historical market price data cannot be used to predict future market prices, unless these same patterns continue into the future. The evidence suggests they do not.

Two heuristic arguments in support of the random-walk hypothesis pro­ vide an intuitive introduction to the theory. "If the stock market behaved like an imperfect roulette wheel, people would notice the imperfections and 62 by acting on them, remove them." Information about market prices is gen­ erally available, so every investor has the opportunity to discover consis­ tent price behavior. General knowledge of the behavior would probably re­ sult in market activity which would change the pattern, and thus eliminate it.

^Harry V. Roberts, "Stock-Market 'Patterns' and Financial Analysis: Methodological Suggestions," Journal of Finance, Vol. XIV, No. 1 (March, 1959), p. 1.

^Roberts, p. 2. 62 Roberts, p. 7. 80

A second argument which is implicit in Roberts' comments is the more

fundamental one. If a simple mechanical device (chance model) can be cre­

ated which duplicates many of the characteristic features of stock-price

movements, then stock-price movements can be described in terms of that

chance model. The best model is that model which duplicates more charac­

teristic features of stock-price movements, or does a better job of dupli­

cating certain important characteristics. At the present time, no method

or set of criteria has been proposed which would provide a convincing basis

for selecting a particular model as being the best model of stock prices.

The .models of stock price behavior developed to date are not completely

satisfactory because different models seem to fit the empirical data, even

though they conflict in a rather basic way. The stock-price literature is

extensive, so only the better known of these models are discussed and con­

trasted.

The random-wallc hypothesis of stock price behavior was first proposed 63 by Louis Bachelier. His model was of the form

(4.0) p(t) - p(t-l) = e(t), t=l,2,..., "where p(t) denotes the price of a

security at time t and e(t) represents a Gaussian process of independent variables, that is, E j e(t)*e (t+T) != 0, T ^ 0, T integer, and for every

64- time tg, e(tg) has a normal distribution." Two basic ideas are involved

in the Gaussian behavior hypothesis of Bachelier. One is that bits of in-

63 L. J. B. A. Bachalier, "Theorie de la Speculation, Gauthier- Villars, Paris, 1900; translated in Paul H. Cootner (editor), The Random Character of Stock Market Prices (revised edition; Cambridge: The M.I.T. Press, 1964), Chapter 2.

^ S. James Press, "A Compound Events Model for Security Prices," The Journal of Business, XL, No. 3 (July, 1967), p. 318. formation are essentially independent over time. Good news does not tend to follow good news or bad news, with any pattern in other words. If the news reaching investors is serially independent, then price changes should be independent also. Secondly, if the news of information is received in a large number of small bits, spread uniformly over time, then by the central limit theorm, the distribution of stock price changes tends to its limiting distribution, the Gaussian or normal distribution. Because the work of

Bachelier was so advanced in 1900, very little testing of his theory ap­ peared until the 1920's and 1930's.

Working and Slutsky later found that a series generated by summing ran- 65 dom numbers is similar in appearance to an economic time series. * 66 These two works thus provided support for the earlier work of Bachelier, however, empirical work by Cowles and Jones cast some doubt on the independence of price changes. They found a first order serial correlation in the first dif- 67 ferences of some price 4-ndex series. In a presentation and subsequent pa­ per dealing with the analysis of economic time-series, Kendall also found some evidence of non-random behavior in cotton prices, but the rest of his work supported the random-walk hypothesis of Bachelier, even though that was •68 not the original purpose of this study. Kendall studied twenty-two time

65 H. Working, "A Random Difference Series for Use in the Analysis of Time Series," Journal of American Statistical Association, XXIX, (1934), pp. 11-24. 66 E. E.« Slutsky, "The Summation of Random Causes as the Source of Cyclic Processes," Econometrica, V, pp. 105-146 (1937), translation.

^ k . Cowles and H. Jones, "Some A Posteriori Probabilities in Stock Market Action," Econometrica, V (1937), pp. 180-294.

68M. G. Kendall, "The Analysis of Economic Time Series — Part I: Prices," Journal of the Royal Statistical Society, XCVI, Part I (1953), pp. 11-25. 82 series, and the series of cotton prices was the one series that appeared to be a notable exception to the pattern of random price changes.

Working independently of each other, Work i n g ^ and Alexandei?® discov­ ered that the use of an average of weekly price observations as a monthly price observation introduces first-order serial correlation in the first differences of a price series. This serial correlation is created even though the original data is a random walk. Cowles noted the validity of this argument and revised some of his earlier conclusions, because some of his indexes involved averaging of observations within his basic period. 71

This new information also explained Kendall's findings with respect to cotton prices because part of that series involved the use of averaged prices whereas the other indexes he studied did not.

An improvement in the basic model was proposed by various writers at about the same time. That improvement is the use of a logarithmic trans­ formation. The model is thus restated as

(4.1) logg p(t) - loge p (t-1) = e(t), where e(t) is distributed as in 4.0.

The logarithmic transformation is based on the observation of past stock price performance. The natural logarithms of price changes fit a normal distribution better than the price changes themselves do. M. F. M.

69 H. Working, "Note on the Correlation of First Differences of Aver­ ages in a Random Chain," Econometrica, XXVIII, No. 4 (October, 1960), pp. 916-918.

^Sidney S. Alexander, "Price Movements in Speculative Markets: Trends or Random Walks," Industrial Management Review, II, No. 2 (May, 1961), pp. 7-26.

^Alfred Cowles, "A Revision of Previous Conclusions Regarding Stock Price Behavior," Econometrica, XXXVIII, No. 4 (October, 1960), pp. 909-915. 83

Osborne attributed the empirical behavior of price logarithms to the Weber-

Fechner law. "The Weber-Fechner law states that equal ratios of physical

stimulus, for example, of sound frequency in vibrations per second, or of

light or sound intensity in watts per unit area correspond to equal inter­

vals of subjective sensation, such as pitch, brightness, or noise. The value

of a subjective sensation', like absolute position in a physical space, is

not measurable, but changes or differences in sensation are, since by experi­

ment they can be equated, and reproduced, thus fulfilling the criteria of 72 measurability." By hypothesizing the Weber-Fechner law, Osborne is saying

the absolute level of stock prices is of no significance because only changes

in prices can be measured by traders or investors. The loge of the ratio

p(t)/ p(t-l) is equal to loge p(t) - loge p(t-l), as in 4 .1.

The limited liability feature of investment in corporations and the

capital budgeting model provide some basis for the argument that investors

should a priori expect something like a log-normal distribution. Fisher has

argued that "traders know the distribution of price changes must be skewed.

The size of negative changes is restricted by the limited liability feature of the corporation. Price of a share of stock cannot fall below zero. In

the upward direction there is no such limit. At any momemt each firm, cur­ rently controlling only a small portion of the world's wealth, has some small chance of great (unbounded?) growth with a consequent rise in the price of its stock. Thus expectations must be skewed, having a lower, but no upper bound. Further, the most probable changes are small rather than ex­ treme .

72 M. F. M. Osborne, "Brownian Motion in the Stock Market," Operations Research, VII (March-April, 1959), p. 146. 84

Professor Fisher further suggests that such an expected distribution may not be strictly log-normal, but may simply be similar to a log-normal 73 distribution, i.e, skewed, and with its lower bound at zero."

The present value approach to the valuation of future income provides another reason why one might expect the changes in the logs of prices to be more appropriate for analysis than arithmetic first differences. A log­ normal distribution arises when some variant of a law of proportional ef­ fect is at work. The combination of the present value formula and a posi­ tive discount rate suggests that the discount rate itself is a source, perhaps the source, of the proportionality of price changes to prices, and the attendant normality of changes in the logs of prices.

Investors seek yields which are acceptable, and prices change in order to bring expected income and the desired yield into equilibrium. "Thus prices are subjected to numerous small multiplicative adjustments as traders estimate and reestimate yields. Even in the absence of uncertainty, price changes are proportional to price since the present value formula implies that prices change at r percent per year."^

In summary, the logarithmic transformation provides a model that fits the empirical data better.^

73 Arnold B. Moore, "Some Characteristics of Changes in Common Stock Prices," Cootner, The Random..., pp. 142-43.

^Moore, pp. 144-145.

^ F o r this quote and a more extensive discussion of the logarithmic transformation of prices, see Moore, pp. 142-145. 85

The independence assumption of the revised model is still subject to 76 question. Taussig has argued much earlier that market prices fluctuate within a range or penumbra. General consensus about underlying supply and demand factors prevents the movement of prices outside the penumbra, but lack of knowledge about current, specific market conditions results in a great deal of fluctuation within the penumbra.

Working^ provided a slightly different view of the same issue. He postulated that traders received information in a substantially independent series. The gradual interpretation and dissemination of information results in a series of price changes, however. If each trader cannot significantly influence the market, then price movements will consist of a series of very small movements.

Neither Taussig's analysis nor Working's analysis provide a very useful basis for predicting the behavior of stock prices over time. Price changes may be correlated within Taussig's penumbra, or they could tend to be inde­ pendent, depending on the time period selected. The correlation of price changes in Working's theory should depend on the time period studied and the relationship of the time period to the frequency of major bits of information.

A rapid flow of information might tend to reduce the probability of trends due to the existence of conflicting new items in various stages of dissemina­ tion to all traders.

76 F. W. Taussig, "Is. Market Price Determinate?", Quarterly Journal of Economics, XXXV (May, 1921), pp. 394-All.

^H. Working, "A Theory of Anticipatory Prices," American Economic Re­ view, XLVIII (May, 1958), pp. 189-199. 86

78 Moore found that weekly differences in logs are negatively correla­ ted, even though a price index he constructed was not. He attributed his findings to two factors; the standard deviation of the price relative of the security and the amount of dividend being paid. His findings involved a small sample of stocks (30) and he only suggested that his study provided an indication of possible sources of negative correlation. He did acknow­ ledge that Taussig and Working may have provided the reasons for a corre­ lation of price changes.

Cootner has argued that price changes are not independent, and that 79 they do not form a Gaussian or normal distribution. He postulates that there are two distinct types of investors. One group consists of people with occupations unrelated to the stock market. Costs of search are high for people in unrelated occupations, so this group of investors tends to accept current market prices and price movements as good measures of the value of a stock. A second group might be called the professionals. The professional group includes all of those investors who have access to in­ formation about the basic profitability factors of a corporation. The stock market is expected to behave as a random walk except in those cases in which prices tend to move away from some equilibrium level. If a stock becomes underpriced relative to its basic earning capacity, then the pro­ fessionals will begin to buy it and thus push the price up. If a stock becomes overpriced, the professionals will sell shares they acquired pre­ viously or will sell short, thus forcing the price downward. The cumula-

78 Moore, pp. 146-147.

^ P a u l H. Cootner, "Stock Prices: Random vs. Systematic Changes," Industrial Management Review, Vol. Ill, No. 2 (Spring, 1962), pp. 24-45. 87 tive action of professional investors thus creates a set of barriers beyond which prices will not tend to move. Even though Cootner admits a great deal of the movement of stock price change can be described by a random walk, there are some significant implications of his formulation. Most price changes will tend to be independent, but those price changes in the direction of a nearby barrier will frequently be followed by price rever­ sals. These price reversals at the barriers are used by Cootner to ex- 80 plain the slight negative autocorrelation found independently by Moore 81 82 and by himself. Larson and Alexander also provided some evidence that changes in futures prices are not independent.

A second significant element of Cootner's analysis is the statement that the distribution of price changes over short periods of time would be more leptokurtic than the normal distribution. The existence of barriers on prices implies that there should be more short or small price changes, and more large price changes than one expects from a normal distribution of price changes. Price movements toward a barrier are cut short, whereas they would not be in Bachelier's model. Likewise, large price changes be- . tween the barriers would exist more often than is consistent with a normal 83 distribution of price changes; Sample evidence reported by Fama supports

80 Moore, pp. 146-147. 81 Arnold B. Larson, "Measurement of a Random Process in Futures Prices," Food Research Institute Studies, I, No. 3 (November, 1960), pp. 313-324. 82 Sidney S. Alexander, "Price Movements in Speculative Markets: Trends or Random Walks," Industrial Management Review, II, No. 2 (May, 1961), pp. 7-26. 83 E. F. Fama, "The Behavior of Stock Market Prices," Journal of Business, XXXVIII, No. 1 (1965), pp. 34-105. 88 the hypothesis of a leptokurtic, long tailed distribution with non-zero mean for logged price changes.

Mandelbrot has proposed a model which differs in important ways from 84 that of Bachelier and that of Cootner. He argues that past research has over emphasized the approximate normal distribution of price changes, and ignored the obvious deviations from normality. 85 His suggested model is the difference equation

(4.2) logeP(t) - logep(t-l) = e*(t), where e*(t) is a memeber of the family of distributions called stable Pare­ tian.

Mandelbrot's model is a generalization of Bachelier's model because the Gaussian distribution is a special case of the general family of stable

Paretian distributions. This model has some appeal because it is more gener­ al than the Gaussian hypothesis, and because it explains some of the erratic behavior of successive price changes observed empirically. A major disadvan­ tage is that the variance of a stable Paretian distribution is not finite, except in the special case of a Gaussian distribution. Much of the portfolio literature is based on the existence of a finite variance, and all of the existing portfolio literature would need to be reformulated in terms of stable Paretian distribution parameters. Furthermore, many statistical tools used in econometric studies are based on the existence of finite variances,

®^Benoit Mandelbrot, "The Variation of Certain Speculative Prices," Journal of Business, XXXVI, No. 4 (October, 1963), pp. 394-419.

®^Kendall (op. cit.) found that weekly price changes of Chicago wheat and British common stocks were almost normally distributed. Moore (op. cit.) found that his sample of 30 common stocks from the New York Stock Exchange were also approximately normally distributed. 89 and may be of much less value than originally thought, if the variables be­ ing studied do not have finite variances.

Fama supports Mandelbrot's hypothesis in that he believes that

"... the stable Paretian hypothesis is more consistent with the data than 86 the Gaussian hypothesis." He examined the distributions of daily first differences of log price of each of thirty stocks included in the Dow-Jones

Industrial Average.

Mandelbrot's hypothesis has been attacked on the basis that it is only 87 an alternative model. Cootner criticizes Mandelbrot's casual methodology and his use of spot prices, as well as arguing that the evidence is not very compelling. Godfrey, Granger and Morgenstern examined some securities using spectral analysis and concluded that "no evidence was found in any of these series that the process by which they were generated behaved as if it pos­ sessed an infinite variance."^®

Press recently postulated a statistical model of stock price which has a finite variance and is consistent with the empirical data. "The model is similar to previous analyses of stock market price behavior in that logarith­ mic price changes are assumed to be independent (random walk models). How­ ever, this model differs from earlier work in that the logarithmic price changes are not assumed to follow some stable distribution (which might

86 Eugene F. Fama, "Mandelbrot and the Stable Paretian Hypothesis," Journal of Business, XXXVI, No. 4 (October, 1963), p. 428. 87 Paul H. Cootner, "Comments on the Variation of Certain Speculative Prices," in Paul H. Cootner (editor) The Random Character of Stock Market Prices (Cambridge: The M.I.T. Press, 1964), pp. 333-337. 88 M. D. Godfrey, C. W. J. Granger, and 0. Morgenstern, "The Random Walk Hypothesis of Stock Market Behavior," Kylos, XVII (1964), p. 13. possibly be normal). Instead, the logarithmic price changes are assumed to follow a distribution that is a Poisson mixture of normal distributions. It is shown that the analytical characteristics of such a distribution agree with what has been found empirically. That is, this distribution is in gen­ eral skewed, leptokurtic, more peaked at its mean than the distribution of a comparable normal variate, and has greater probability mass in its tails 89 than the distribution of a comparable normal variate." Press' model has been applied to ten common stocks, but no comprehensive testing of the rela­ tive merits of this model and those previously hypothesized has been made.

At this stage one can only argue that the Press model is consistent with the empirical evidence, and that the Mandelbrot model is consistent with some em­ pirical evidence. Both models represent the empirical data better than the

Gaussian model of equation 4.1.

In summary, no single model of the changes in stock market prices has been thoroughly tested and generally accepted. Existing models do imply that changes in price appear to be nearly independent. Slight negative autocorrelation may be induced by the effect of prices going ex-dividend, and by the limiting barriers on price created by the cumulative activities of professional traders. Distributions of logarithmic price changes tend to be leptokurtic and more long-tailed than the Gaussian or normal distri­ bution. Knowledge of past prices is of little expected benefit in the pre­ diction of price relatives.

®^Press, p. 317. 91

The Potentiality of Stock Price Prediction

Some writers have assumed that the random-walk hypothesis of stock price behavior implies that stock price changes are not predictable. This assumption is false. "When statisticians hypothesize that the course of a

stock's prices describes a random walk or Brownian motion, they do not imply

that a skilled student of the subject cannot forecast price changes. They merely imply that one cannot forecast the future based on past history

Moreover, if one examines the random-walk literature carefully, then it becomes apparent that predictions of intrinsic value are a basic part of the random-walk theory. Cootner states some heuristic arguments behind the 91 random walk theory as follows:

If any one group of investors was consistently better than average in forecasting stock prices, they would accumulate wealth and give their forecasts greater and greater weight. In the process they would bring the present price closer to the true value. Conversely, investors who were worse than average in forecasting ability would carry less and less weight. If this process worked well enough, the present price would reflect the best information about the future in the sense that the present price, plus normal profits, would be the best estimate of the future price.

Presumably, this process of rewarding good forecasters of intrinsic value, and eliminating poor forecasters, is a continuous process which gradually improves forecasts of intrinsic value.

Further evidence that predictions of intrinsic value are a fundamental part of the random-walk hypothesis of stock price behavior can be found in

90 Paul H. Cootner, "Introduction - Part II," in Paul H. Cootner (editor), The Random Character of Stock Market Prices (Cambridge: The M.I.T. Press, 1964), p. 80.

^^Cootner, "Introduction ...," p. 80.

9 92 the models of Cootner, Working, and Larson. The flexible barriers to stock price movements that Cootner envisions are based on the ideas professional 92 investors have about the intrinsic value of each security. Professional investors have the time and the resources to investigate the economic power of each company, and are able to formulate some expectations about the future dividends or earning capacity of the firm. These expectations are translated into a market price range beyond which the stock price will not be allowed to move without corrective action. Professionals would have no basis for formulating these ideas about the extent to which a stock is underpriced or overpriced if the stock's price were not closely related to some economic forces.

Working argues that current prices reflect the best projections made 93 by those involved in the market. Price changes arise due to revised ex­ pectations and not due to the haphazard movement of prices in the short run as suggested by Taussig. 94 The expectations discussed by Working are expec­ tations about basic supply and demand forces. Because truly new information emerges randomly, or in predictable trends, price movements tend to be ran­ dom in his model. "Working's model of price movement in an ideal market is 95 thus a random walk; that is, price is the cumulation of random movements."

92 Paul H. Cootner, "Stock Prices: Random vs. Systematic Changes," op. cit. 93 H. Working, "A Theory of Anticipatory Prices," American Economic Review, XLVIII (May, 1958), pp. 189-199. 94 F. W. Taussig, "Is Market Price Determinate?", Quarterly Journal of Economics, XXXV (May, 1961), pp. 394-411.

^ArnoId B. Larson, "Measurement of a Random Process in Future's Prices," Food Research Institute Studies, I, No. 3 (November, 1960), p. 316. 93

Working's model is tested by Larson and is found to be more consistent with the behavior of changes in corn futures prices than the model of

Taussig, which stressed the haphazard movement of prices within a penumbra.

The penumbra in Taussig's model is determined by supply and demand forces, but most of the short-run movement of prices is based on predicted price trends in the market, not fundamental economic forces.

Although there may be some information in a historical stock price series, such as slight autocorrelation, the information is generally insuf­ ficient to be of economic value in predicting future price changes.^ As a result, estimates of intrinsic value must be based on more fundamental eco­ nomic forces. The valuation models developed below are an attempt to incor­ porate economic forces in the prediction of common stock price relatives.

One should not conclude from the above analysis that past price rela­ tives are not useful in the prediction of future price relatives. If the relatives are mean-reverting, then past data provides an estimate of that mean. On the other hand, if the price relatives follow a random walk, then the most recent price relative is the best prediction available for next year's price relative. In both cases, some information about prior price relatives is absolutely necessary.

^ F o r a more complete discussion of this point see Robert A. Levy, "The Theory of Random Walks, A Survey of Findings," American Economist, Vol. II, No. 2 (Fall, 1967), pp. 34-48. 94

Naive Prediction Models

A price relative prediction model must reduce price relative uncer­ tainty if it is to be of any predictive value. Price relative uncertain­ ty is reduced whenever the conditional variance of a set of observed price relatives, given the respective predicted price relatives, is less than the variance of the observed price relatives without regard to the predictions.

This concept of uncertainty reduction is discussed in detail in section six of Chapter III. However, for any small sample of predictions, the rela­ tionship of actual and predicted price relatives will deviate from that equality relationship which is expected at the time the predictions are being made. For such samples, total uncertainty consists of two elements.

One is the conditional variance of the price relatives, given the set of price relative predictions, while the second is the portion of the predic­ tion errors which is created by the lack of knowledge about the true linear relationship between the actual and the predicted price relatives. When combined, these two components of uncertainty are equal to the mean-squared prediction error, which is the sum of the squared differences between pre­ dicted price relatives and the corresponding observed price relatives, di­ vided by the number of differences.

The information contained in a past series of common stock price relatives may be used in\arious ways to make predictions about future price relatives. At one extreme, one might use the most recent price relative as the prediction of next year's price relative. At the other extreme, one might use an average of the price relatives observed for many years in the past as a prediction of next year's price relative. The 95 proposed naive prediction models involve various weightings of past price relatives that include these two extremes, as well as thirty-nine grada­ tions in between. Prediction models using only past data in the same series are called naive models because they incorporate no other economic variables in the formulation of predictions. In this case, specific eco­ nomic forces and the relationships of these forces to common stock price relatives are ignored by the naive prediction models. However, it should be noted that the general class of naive prediction models includes those involving sophisticated mathematical techniques which cannot be regarded as simple, or trivial. , n 97

The naive models are comparable with the growth model method in terms of cost to operate, assuming that the basic data is available in useable form. They are also comparable in the sense that they are well-defined, objective methods which can be duplicated by others. Such would not be the case if the predictions were made by individuals using more intuitive methods

Three of the four models of stock price tested for price relative pre­ diction performance are a constant model, a linear trend model and a con­ stant exponent model. The constant terms in each equation are estimated by means of exponential smoothing. The optimal smoothing constant, or set of constants, is allowed to change from company to company. More details of the smoothing constant selection method are presented in Chapter V.

The three models of stock price tested, along with their respective

^Charles F. Roos, "Survey of Economic Forecasting Techniques," Econometrica, Vol. XXIII, No. 4 (October, 1955), pp. 363-395. 96 smoothing statistics, coefficient estimates (equation constants) and price relative prediction models are summarized in equations 4.3 through 4.18:

Constant model;

(4.3) Model of the process: R$t = A^gt

(4.4) Process observed with noise: R^fc = + e^

(4.5) Smoothing statistic: + ^i i t -1

(4.6) Estimate of coefficient:

A (4.7) Prediction: Rit+1 = Ai0t

Linear Model;

(4.8) Model of the process: R$ = + A^^fcT

(4.9) Process observed with noise: R ^ = A^Qfc + A^-^T + e^t

(4.10) Smoothing statistics: = aRit + ^ H t - 1

(4.11) S2 .t - aSU t + B S ^

(4.12) Estimate of coefficients: A ^ ^ = 2 S ^ t -

Ailt - “/S(SU t - s2it>

(4.13) Prediction: Rit+1 = AiQt + Allt for T = 1

Constant exponent model; Ai0t (4.14) Model of the process: R* = e

Afot + (4.15) Process observed with noise: R_^t = e

(4.16) Smoothing statistic: = aloge R^t +

(4.17) Estimate of coefficient: A^q = S ^ t 97

A -n (4.18) Prediction: = e 1

R*t is the true price relative of security i (i = 1, 2, ..., 61) for

period t (t = 0, 1, 20)

R^ is the price relative observed for security i (i = 1, 2, ..., 61)

for period t (t = 0, 1, 20)

A R^ is a prediction of R^t

Sfit* are smoothing statistics as defined

are equation coefficients (constants) for security i

(i = 1, 2, 61) for period t (t = 1, 2, ..., 20)

e. is an error or disturbance term which is normally distributed it with zero mean, constant variance and is independent of past

disturbance terms and past price relatives.

The initial conditions for the constant model and the linear trend model are determined in the same manner. The initial value for S., _ (t = 1, lit 2, ..., 5) is equal to the arithmetic mean of the first five price rela­ tives observed during the years 1948 through 1952 (t= 1, 2, ..., 5). For 5 any t (t= 1, 2, ..., 5), = j£l where j is the time subscript in the summation index. The initial value of is assumed to be zero 2it for 1948 (t = 1) only. The smoothing statistic for the constant exponent model is determined in a similar manner, except that the data is in loga- 5 rithm form, or S.._ = log R. ,/5 for t = 1, 2, ..., 5. lit j=l e ij The annual price relative is decomposed into the dividend yield ele­ ment and the price change element in order to formulate some expectations about the value of the smoothing constant, a. First, consider a price 98

relative such that:

(4.19) Rlt - (D.t + Plt)/Ple.x where

is the price relative for security i (i = 1, 2, 61)

and t is period t (t = 1, 2, 20)

D is the cash dividend paid during period t for security i

is the equivalent price per share for security i for

period t .

The expression in 4.19 can be separated into two parts such that

D P (4.20) R = ^ + ^ — , where t-1 t-1

is the annual price relative for year t

is the amount of cash dividends paid during year t

P is the common stock price at the end of year t assuming

adjustment for all stock dividends or stock splits.

The company subscript, i, has been omitted for convenience and clarity.

If the expected price change is zero, we can write the equation

(4.21) E (P - Pfc i) = 0, and since P^_ ^ is known at the start of year t,

(4.22) E (Pfc) - P = 0 or

(4.23) E (Pfc) = Pfc in which case

P. (4.24) E 1 'E (PJ = 1 ?t-l Pt-1 1

If the expected price change is some constant percentage, representing the upward drift in common stock prices, then 99

(4.25) E (Pt - Pt_1) = wPt_1

(4.26) E (Pfc) = Pt_1 + wPfc_1

(1 + w) P t-1 (4.27) E = 1 + w, t-1 t-1 when w is the constant percentage of upward drift.

Let Y be a random variable with mean My which is the dividend yield on common stocks. We are assuming that dividend yield is mean reverting in the short run even though shifts in mean yield may occur on occasion.

Since E (pt;/I>t j_) = My, we can saY that

(4.28) E (D /Pfc p j_) = My + 1 + w, the mean dividend yield plus a constant, 1 + w.

If 4.28 represents the behavior of price relatives, then one would expect a low smoothing constant because the price relative series will fluctuate about an average equal to My + 1 + w.

In contrast, suppose that the expected change in dividend yields is zero. The price relative can be decomposed as follows

D D , P P' t t-1 t t-1 (4.29) R.-V- l P , P „ t-1 . t-2P t-1 , P t-2 ^ M. D ; Dt t-1 + E It t-1 (4.30) E (Rt - Rt_1) = E ; y P P t- 1 t-2t-1 t-2 t-1 t-2t-1 but the first term on the right side of 4.30 is zero by assumption, so

P t-1 (4.31) E (Rt - Rt_1) = 1 + w - p « 0 . t-:2

The nature of the price relative depends on the nature of the dividend yield. The assumption that annual price changes are zero has not been validated, but the assumption has been validated with respect to short­ term price movements (see review of random walk). If we are free to as­ sume that annual price changes tend to zero, or some constant percentage of drift, then the nature of dividend yield becomes the important issue.

If the expected dividend yield is zero, then the expected price relative is If the expected dividend yield is equal to the mean of a stable distribution (My), then the expected price relative is equal to My + 1 + w.

Model No. 1 or Model No. 3 should outperform Model No. 2 whenever dividend yield is mean-reverting and there is some constant proportional drift in the changes in stock prices. The expected smoothing constants for such a process should be less than 0.3 for a and greater than 0.7

(1 - a) for 3. Because the dividend yield is non-negative, and the expec­ ted price change is zero or slightly positive for most firms, the expected price relative for most firms will be greater than or equal to the pre­ vious year's dividend yield plus 1.0. Individual price relative observa­ tions frequently drop below 1 .0 , but the expected price relative is some­ what above 1 .0 .

Smoothing constants are selected for Model No. 1, for each firm, in two ways. One method is to use Values between zero and one in increments of .025 for a, on all twenty price relative observations; then calculate the mean-squared prediction error for each value of a; and then adopt the smoothing constant which resulted in the lowest mean-squared prediction error for the test period from 1948 through 1967. For each company there exists an optimal smoothing constant, ten predictions resulting from use 101 of the optimal a, and ten prediction errors. The second approach is to use forty-one smoothing constants from 0.0 through 1.0 in increments of

0.025 for years one through ten (1948 through 1957). That smoothing con­ stant which resulted in the smallest mean-squared prediction error over the base period 1948 through 1957 is used to estimate Awhich is the prediction of the price relative for 1958. That smoothing constant which results in the smallest mean-squared prediction error for the period 1949 through 1958 is used to estimate A^q , which is now used to predict the price relative for 1959. This process must be repeated for the remaining predictions on a year to year basis so that each annual prediction is based on the statistic provided by the best smoothing constant for the prior ten years. When the process is completed, there are ten price relative pre­ dictions and ten price relative prediction errors as was the case for the first method. However, there may be as many as ten different smoothing constants used to make those predictions. The usual case is to have one or two different smoothing constants for the entire ten years of predic­ tions .

Model No. 3, which is the constant exponent model, is presented in equations 4.14 and 4.15. This model is based on the hypothesis that the annual rate of return varies about the company's average rate of return, which is determined by the growth rate of earnings and dividends, and by the level of uncertainty associated with that company. The expression for the geometric average rate of return is:

(4.32) (1 + r)n = (1 + r ^ U + r2)...(l + rn) where r is the average rate of return and r^, r^j are the rates 102 of return for the respective periods 1, 2, n. If some rate of re­ turn is associated with a particular corporation's uncertainty level and growth rate, then for any t (t < n), E(1 + r ) = 1 + r, because E(rfc) = r so long as the growth rate and uncertainty level do not shift. The rate of return for any year, plus 1 .0 , is equal to the price relative of that year. Let the logarithm of the annual price relative be symbolized by

Rt> and the average price relative be symbolized by R. The averaging method used to approximate R is:

(4.33) R = aR^ + a(l - + 01 (i " a^ ^ t-2 * ’ ’ ’ ” 01

In this way the more recent price relatives are given more weight than they would receive from a simple, unweighted average of the price relatives dur­ ing the period from 1948 up to the year for which a prediction is' being made. The estimate of R is denoted R. Model No. 3 averages the logarithms of past price relatives in order to approximate the weighted geometric average relative, whereas Model No. 1 is a weighted arithmetic average of past price relatives.

As in the case of Model No. 1, two methods are used to select the op­ timum smoothing constant, or set of constants, for each firm. One method involves finding the best single smoothing constant for the years 1948 through 1967. The other method involves the use of that smoothing constant which minimizes prediction errors for the prior ten years to estimate the statistics used to predict the next year's price relative. These are the same two approaches described more fully with respect to Model No. 1.

Model No. 2, the linear trend model, is based on empirical evidence about stock price changes and rates of return earned on common stocks in 103

prior periods. Farrar was forced to simulate investor expectations in

his study of uncertainty, and in doing so he tested some naive prediction 98 models. In predicting growth stocks' annual price changes, he found that

the continuation of the previous year's arithmetic change in price is more

likely than the prediction of no price change, or the prediction of a con­ stant rate of change. In other words, predicting a continuation of last year's arithmetic price change results in smaller prediction errors than

results from the prediction of a zero price change or from the prediction

that last year's rate of price change will continue for another year. Most of the companies in the sample being used in this study are not growth stocks, but Farrar's findings are potentially relevant for some of the

firms that are growing at above average rates.

Translated into price relative terms, the predicted price change is positive for growth stocks. Although Farrar did not test for an average amount of annual price change, it is reasonable to think in terms of an average price change because the expected price changes are positive for growth stocks. Suppose that a growth stock does experience a constant annual increase in stock price over the years. The price change element of the price relative will be declining because the relatively constant amount of price increase is being divided by larger and larger share prices. If dividend yield is increasing, then it is possible that in­ creases in the dividend yield offset the declining ratio of price increase, so that the price relative is constant. If dividend yield is constant, then the total price relative will decline over time because the price

98 Donald E. Farrar, The Investment Decision Under Uncertainty (Engle­ wood Cliffs: Prentice-Hall Inc., 1962), p. 57. 104 change ratio declines. Under these conditions, one would expect to find a trend in the price relatives.

An examination of general movements in the rates of return earned on a broad sample of common stocks also suggests that a linear trend model 99 may be relevant for limited periods of time. For example, there was an increasing trend in the cumulative rate of return from 1945 to 1956. The fact that the cumulative rate of return was sharply increasing implies that the annual rates of return were also increasing. There is little evidence of trend in subsequent years, but the earlier trends suggest that linear trend models may be appropriate in predicting price relative trends. The evidence is not strong.

The naive prediction models described in equations 4.3 through 4.18 are appropriate models of common stock price relatives under varying cir­ cumstances. First, if the price relative series of a common stock does contain information, such as trend, then this information will be incorpo­ rated into the price relative predictions of the linear trend model. Under different circumstances, the price relatives may follow a random walk. The most recent price relative should.be the best prediction of next year's price relative if the relatives do follow a random walk. Either the con­ stant model or the constant exponent model can readily adapt to the random walk case because the smoothing constant (a) can vary from 0.0 through 1 .0 .

However, if the price relatives are best described by a stable prob­ ability density function, then the mean of the past price relatives should

^Lawrence Fisher and James H. Lorie, "Rates of Return on Investments in Common Stock: The Year-by-Year Record, 1926-65," The Journal of Business, Vol. XXXX, No. 3 (July, 1968), pp. 1-26. 105

provide the best prediction of next year's price relative.

Accordingly, a fourth method of predicting future price relatives is

the use of the mean of the prior ten price relatives to predict the next

price relative. The prediction model may be described as:

(4.34) R.- = £n R. . /n, where j=l * L J

R^t is the common stock price relative for company i for

period t,

R. is the prediction of R. , xt Xt *

n is the number of observations (10 in this case).

Superior price relative prediction performance of this model will serve as

evidence that the price relatives can be represented by a stable probabili­

ty density function.

In summary, four naive methods of predicting price relatives are for­ mulated. Three models incorporate equation constants estimated by means

of exponential smoothing, while the fourth method incorporates a simple

average of the prior ten price relatives. The resulting prediction errors

are evaluated in order to determine the best overall naive prediction model.

The results of these prediction performance tests are discussed in Chapter

V in conjunction with the evaluation of the economic growth model's perfor­ mance .

An Economic Model of Stock Value

The thrust of the random-walk hypothesis is that technical analysis

is not useful in buying common stocks. Even though brokers and analysts

still claim that technical analysis can lead to more timely purchases, 106

academic research seems to reflect the general agreement that technical

analysis is not useful. The empirical evidence in support of the random walk hypothesis is extensive and technically sophisticated, but it is

somewhat naive in that the statistical tests performed do not test for

the complex patterns envisioned by the technical analyst. Nevertheless, academic researchers seem content to dismiss the notion of technical ana­

lysts.

A somewhat different situation exists with respect to the determina­

tion of the intrinsic value of a common stock. Intrinsic value is an es­

timate of the true value of a common stock, which depends on the long-run profitability of the corporation. The determination of intrinsic value not only is important in the selection of the best common stock invest­ ments for long investment periods, but it also helps indicate which stocks seem to be underpriced or overpriced at the current time.

Both academic researchers and financial analysts have devoted a great deal of time and effort to the determination of common stock values. Among practitioners, the Value Line Investment Survey is one of the most widely known approaches to the determination of intrinsic value. The Value Line method is objective enough to be of interest to academic researchers.

On the academic side, extensive research has been devoted to the de­ velopment of normative, as well as positive, common stock valuation models.

In addition, various studies have attempted to determine the predictive power of accounting measures of performance. The normative models are de­ signed to indicate the relationship of common stock values to other econo­ mic variables, for a given set of assumptions. Empirical models involve 107

testing hypothetical models of stock price in order to determine which

theory fits the data best. Both the empirical (positive) and the normative

valuation models are designed to isolate and define those variables which

influence stock prices.

The VLIS approach and the models constructed and tested by academi­

cians share a common assumption — that stock prices and stock price

changes are predictable. Unless this assumption is made, then their ex­

tensive activities are not purposeful.

The steps included in our development of an economic model for pre­

dicting common stock price relatives are (1) a statement of the inherent

limitations of stock valuation models, (2) a brief review of some findings

from other empirical studies, (3) an explanation of the Value Line method,

(4) arguments in support of including certain variables in the model, and

(5) specification of the model.

Limitations of stock valuation models.— All common stock valuation models suffer from the same problem — uncertainty. Our inability to pre­

dict aspects of the future makes price prediction very difficult, regard­ less of the nature of the resource being priced.Predicting the price of the commodity steel is as difficult as predicting the net earnings to be reported by United States Steel, Inc., in other words. Nevertheless, improved predictions should theoretically lead to better decisions in an economic sense, and so the prediction process is worthy of study.

Prediction models based on theories of economic behavior are generally

^Slany of these general comments on stock valuation model construc­ tion are from William J. Baumol, "Problems in the Construction of Stock Valuation Models," paper read at the Spring Seminar of the Institute for Quantitative Research in Finance, April 25, 1968. 108

considered more fruitful because there is a greater probability that theo­

retically defined relationships will continue in the future, whereas spuri­

ous ones will not. Baumol has clearly stated the issue, which is:

The ad hoc models may do better as forecasters. If one is lucky and happens to hit on a mathematical relationship that cap­ tures for the moment the psychology of the investing public, good predictions may emerge. But there is no reason to believe that the ad hoc model that would have done well in 1960 will perform satisfactorily in 1967. We have no grounds on which to expect that the psychological climate will remain stationary over time, and perhaps only by the wildest coincidence will a mathematical relationship derived by ad hoc processes long follow successfully the vagaries of investor models. This means that even the rela­ tively successful among ad hoc models are likely to fail us when we need them most. It is the turning points in the behavior of the market that we want most help in predicting, and it is pre- cisely there that the ad hoc models will most probably go wrong.

With these limitations and prediction problems in mind, we turn to some of the findings of others, and a review of the Value Line Invest­ ment Survey Method.

Empirical evidence.— There is an extensive literature of finance and accounting in which the effect of financial variables on stock prices, or stock price changes, is tested. Many articles have been concerned with the usefulness of financial ratios in predicting the failure of corpora­ tions. Horrigan suggested that the usefulness of financial ratios may 102 have been overlooked in recent years. Horrigan had some success in predicting the long-term credit standing of corporations with respect to 103 bond ratings. Beaver has had some success in predicting failure using

^^Baumol, "Problems ... ." 102 James 0. Horrigan, "Some Empirical Bases of Financial Ratio Anal­ ysis," Accounting Review, Vol. XL, No. 3 (July, 1965), pp. 558-568. 103 James 0. Horrigan, "The Determination of Long-Term Credit Standing with Financial Ratios," Empirical Research in Accounting: Selected Studies, 109

104 ratios of accounting variables, but he also found that market price de­ clines may serve as more timely predictions of failure.These findings may be the result of investor predictions of failure due to early trends in the ratios as well as other information. The evidence presented in

Beaver's study suggests that accounting variables may be anticipated by the market, and thus are of little value in formulating price relative predictions.

There is evidence, however, that stock prices do move jointly with accounting numbers, and that different accounting measures do influence in- vestment«. decisions.j • • t Roper,, 106 „ Horngren, 107 Cerf, . 108 and . „Burns 109 . have used . the questionnaire or interview approach to determine the extent to which accounting data is useful in various types of decisions. Bruns,

1966, pp. 44-62. A supplement to Journal of Accounting Research, Vol. 4 (1966).

^^William Beaver, "Financial Ratios as Predictors of Failure," Empirical Research in Accounting, 1966, pp. 71-111. A supplement to Journal of Accounting Research, Vol. 4 (1966).

^^William Beaver, "Market Prices, Financial Ratios, and the Predic­ tion of Failure," Journal of Accounting Research, Vol. VI, No. 2 (Autumn, 1968), pp. 179-192. 106 Elmo Roper, A Report on What Information People Want about Policies and Financial Conditions of Corporations, Vols. I and II (New York: Con­ trollers Institute Foundation, Inc., 1948.

^^Charles T. Horngren, "Disclosure: 1957," Accounting Review, Vol. XXXII, No. 4 (October, 1957), pp. 598-604. 108 Alan R. Cerf, Corporate Reporting and Investment Decisions (Berkeley, California: Institute of Business and Economic Research, 1961).

Research in progress. 110 William J. Bruns, Jr., "Inventory Valuation and Management Deci­ sions, " The_^£counting_Revoew, Vol. XL, No. 2 (April, 1965), pp. 345-357. 110

111 112 Dyckman, and Jensen have used simulated data to determine if students or financial analysts make different operating decisions or different in­ vestment choices given accounting data in controlled experiments. Their results were mixed, and at most one can say there is sometimes a difference 113 in decisions under certain experimental conditions. Greenball used a simulation to determine the relative performance of different measures of income with respect to criteria such as bias and predictability.

Many empirical tests of the association of stock prices and account- 114 ing variables have been performed. Gordon tests various models of stock price on a cross-sectional basis in order to find that model which best represents the stock valuation process. Dividends, growth, earnings in­ stability and even leverage were found to be significant factors in the valuation of common shares. Gordon's results were very good in terms of the relationship between the linear regression coefficients and their stan- 115 dard.errors. Staubus has tested various measures of income by using

111 Thomas R. Dyckman, "The Effects of Alternative Accounting Tech­ niques on Certain Management Decisions," Journal of Accounting Research, Vol. II, No. 1 (Spring, 1964), pp. 91-107; and "On the Investment Deci­ sion," The Accounting Review, Vol. XXXIX, No. 2 (April, 1964), pp. 285-295. 112 Robert E. Jensen, "An Experimental Design for Study of Effects of Accounting Variations in Decision Making," Journal of Accounting Research, Vol. 4, No. 2 (Autumn, 1966), pp. 224-238. 113 Melvin N. Greenball, "The Accuracy of Different Methods of Account­ ing for Earnings — A Simulation Approach," Journal of Accounting Research, Vol. XI, No. 1 (Spring, 1968), pp. 114-129.

^\lyron J. Gordon, The Investment, Financing and Valuation of the Corporation (Homewood, Illinois: Richard D. Irwin, Inc., 1962). 115 George J. Staubus, "Testing Inventory Accounting," Accounting Review, Vol. XLIII, No. 3 (July, 1968), pp. 413-424; "Statistical Evidence of the Value of Depreciation Accounting," Abacus (August, 1967), pp. 3-22; and "Alternative Asset Flow Concepts," Accounting Review, Vol. XLI, No. 3 (July, 1966), pp. 397-412. Ill

alternative inventory and depreciation methods in order to find that in­

come measure which is more highly correlated with discounted stock prices.

The particular accounting method chosen does seem to influence the correla­

tion between the accounting income measure used and common stock prices.

Benston uses various averaging techniques to determine if one method of averaging past annual rates of increase in earnings results in a sig- H 6 nificantly increased partial correlation with stock prices. His re­ sults were generally poor, which is some evidence that predictions may be difficult. Benston does not make predictions, and his is a cross-sectional study. Nevertheless, he does use price relatives instead of prices, and he is concerned with annual rates of change instead of the raw variables.

Benston eliminates general market effects by means of a simple linear regression equation of the price relatives on Fisher's link relative, which is an average of the price relatives for securities listed on the N.Y.S.E.

He is analyzing the residuals from this linear regression. Industry mean- effects are isolated by means of dummy variables. Part of the reason for

Benston's poor results may be the fact he has eliminated some of the inter­ action effects by using simple linear regression to eliminate market ef­ fects, of which each company is a part.

The implications of these studies with respect to intertemporal pre­ diction formulation are not clear. In many cases, it is not clear to what extent the results obtained are dependent on the manner in which the study is formulated, or on the variables selected. As a result, many conflicting

11/' George J. Benston, "Published Corporate Accounting Data and Stock Prices," Empirical Research in Accounting: Selected Studies, 1967, pp. 1-54. A supplement to the Journal of Accounting Research, Vol. V. 112

conclusions may be drawn from similar studies. The conclusions drawn here are based on intuitive information instead of hard facts. One conclusion is that cross-sectional studies frequently ignore trend information, and therefore are somewhat limited in explaining the current price or price relative for a company because price changes are a function of changed ex- 117 pectations. The level of earnings is not so important as its deviation from expectations. A second conclusion is that variables that are impor­ tant in distinguishing between firms may not be important in explaining intertemporal price changes for any one firm. Dividend yield may help ex­ plain interfirm differences, but any particular firm's dividend may be relatively constant over time, and thus is of little predictive value. A third conclusion is that one cannot separate out industry and economy-wide forces in the prediction process because these factors must be predicted, as well as the interaction effect of general and specific economic forces.

Based on these conclusions, we cannot predict very accurately the possi­ bility of making annual price relative predictions on a firm by firm basis.

There are technical problems created by small sample sizes, but here again it is easy to be pessimistic or optimistic without empirical foundation for that attitude.

The Value Line Investment Survey.— Early in his description of the

VLIS approach, Bernhard notes that the present value of a common stock is equal to the sum of all future dividends of the stock, discounted by the 118 current interest rate. Although he notes that this statement is true,

117 See William H. Beaver, "The Information Content of Annual Earnings Announcements," Empirical Research in Accounting: Selected Studies, 1968, a supplement to the Journal of Accounting Research, Vol. VI.

■^Bernhard, p. 39. 113 he also considers it unfathomable because nobody knows what future divi­ dends are!

Our preliminary attempts to find a time series of the projections made by Value Line were not successful, so we adopted their general ap­ proach to valuation. We are not reconstructing Value Line predictions because we do not measure variables in the same manner, and because we can­ not duplicate the professional judgment factors which are embodied in some of the VLIS projections. However, similar variables and similar methods are used in our prediction model.

The Value Line approach involves ranking stocks with respect to (1) quality grade, (2) price appreciation potential over the next three to five years, and (3) probable market performance over the next twelve months.

These three factors are ranked separately, so each investor must decide which factors are relevant to him, and how they should be weighted.

Quality grade depends on growth and stability. Growth is measured in terms of percentage changes in earnings per share and dividends per share over one-year investment periods, three-year periods, and five-year periods. All possible percentage changes are divided into nine intervals and a code number from -4 to +4 is assigned to each interval. The growth of earnings or dividends falls into an interval for each time period con­ sidered and a code is assigned accordingly. The total of the code numbers for dividends is added to the total for earnings, multiplied by two, to create a composite growth index for earnings and dividends.

Stability is measured in terms of price stability. The price stabil­ ity ratio is an average annual price range over the previous eleven years, 114

119 adjusted for a secular trend factor.

Based on correlation studies, the VLIS analysts have concluded that stability should be weighted three times as much as growth in creating a quality index. After calculating the quality index, they rank more than eight hundred companies, and then place each company into one of nine quality grades. Judgment may be exercised to the extent of moving a stock up or down one grade, or even two.

Price appreciation potential estimates are based on a hypothesis about the general economic environment three to five years in the future. Each corporation's sales are then predicted based on the relationship of that company's sales to national income statistics in the past, and based on the probable performance of the industry in the hypothesized economic en­ vironment. Estimates of earnings and dividends are based on projections of gross margin rates and the past relationship between working capital and sales. Price estimates are made by using the average dividend yield rate and average price/earnings ratio associated with this firm in past years.

By comparing current price to the projected price, the Value Line analysts have a measure of price appreciation potential which can be ranked across companies.

Estimates of probable market price performance are based on the rela­ tionship of the moving-average of each stock's price over the prior 52 weeks, to the Value Line rating of intrinsic value. The intrinsic value rating is based on estimates of earnings and dividends in the next twelve months. The relationship among each corporation's intrinsic value, earn-

119 Bernhard, p. 46.

/ 115 ings per share, and dividends per share is based on the past relationship among average price per share, average earnings per share, and average di­ vidends per share, for a group of companies.

An empirically testable model of stock price.— The Value Line In­ vestment Survey method and the results of empirical research studies combine to provide insights and information about those factors which influence stock price relatives. An analysis of these insights and findings forms the theoretical basis for the formulation of an economic prediction model.

Economic theorists have argued that highly profitable investments

(r "s' k) represent a form of compensation for risk bearing, a return cre­ ated by monopoly position, or a reward for entrepreneurship and innova- 120 tion. According to the risk compensation theory of profit, higher re­ turns must be associated with higher risks so that expected returns, after deducting possible losses on some investments, will be comparable to the returns available on low risk investments. The well-accepted theory that above average rates of return are required by investors whenever risk is 121 above average, is supported by empirical studies of common stock returns. To conclude, both economic theory and empirical evidence support the hy­ pothesis that high price relatives should be associated with high uncer­ tainty levels, and low price relatives should be associated with low levels of uncertainty.

120 J. Fred Weston, "The Profit Concept and Theory: A Restatement," Journal of Political Economy (April, 1954), pp. 152-170.

■^^Sharpe, "Risk Aversion...," pp. 152-170; and Fama, "Risk Return ...," pp. 29-40. 116

Although the theoretical relationship of uncertainty to common stock price relative levels is clear, the proper method of measuring uncertainty is not obvious. Value Line has concentrated on price stability as a mea­ sure of risk, where their price stability indes is an average price range over the past eleven years, after adjustment for secular trend.

Another candidate for the measure of uncertainty is a measure of 122 earnings stability, such as that used by Gordon. In his cross sectional study, Gordon found that the stability of net earnings, with interest ex­ pense added back, is significantly correlated with common stock price.

The stability of earnings seems to be the more important measure of risk with respect to long-term investments, because investors can with­ stand temporary declines in price — so long as they are temporary. How­ ever, we are assuming fixed investment periods for which temporary price fluctuations will be important if they occur at year end. Investors prob­ ably are more sensitive to price stability than to operating income sta­ bility whenever their investment horizons are less than a year or two.

Rather than choose one measure of uncertainty over another, on an a-priori basis, the step-wise multiple regression algorithm will provide evidence as to which measure is the more highly correlated with past common stock relatives, on a company by company basis.

High rates of growth also create an element of uncertainty. Recall that growth takes place when investments are available for which the cor­ poration can earn a rate of return in excess of that rate required by in-

122Gordon, The Investment..., Chapter 12. 117 vestors. Very high rates of return are associated with risky investments, and with monopolistic returns. We have already noted that high risk pro­ jects are associated with high rates of return. But consider the high rates of return made available through marketing innovations, lower costs, improved products, and other circumstances that put one's product in a preferred or monopoly position. These circumstances tend to be short-lived, or at least they are not expected to be perpetual in nature. Uncertainty arises because one cannot always predict the duration of these'high return investments.

The prediction of growth rates is very important because past growth rates tend to be extrapolated into the future. When projected prices and dividends of high growth companies are discounted back to the present at market discount rates, the resulting stock price is a reflection of inves­ tor expectations about growth, and not about current dividend yields or normal price/earnings ratios. Therefore, any unexpected indications that a company's high growth rate is beginning to slow down should result in a significant price decrease. Even though a corporation's growth has been nearly constant over past years, an unexpected, permanent decrease in its growth rate will result in a substantial price drop. 123 Bernhard, Gordon, and W ippem have used growth variables to measure quality grade, to explain common stock price levels, and to explain earn­ ings/price ratios, respectively. Consistently, we expect that high growth rates will be associated with high price relatives, and that low growth

• 123 Bernhard, p. 41; Gordon, The Investment..., Chapter 12; and Ronald F. Wippern, "Financial Structure and the Value of the Firm," The Journal of Finance, Vol. XXI, No. 4 (December, 1966), pp. 615-633. 118

rates will be associated with low price relatives, ceteris paribus. One

would expect some interaction effects between growth rates and price sta­

bility, but high growth rates, which are also stable, may have an impor­

tant influence on common stock price relatives even in those cases where

stock prices have been stable too. Conversely, common stock prices and

price relatives may not be stable in those cases where growth rates are

low and stable. In summary, the inclusion of growth rates in the predic­

tion model is based on the VLIS method, and is supported by other empiri­

cal studies.

The rate of sales growth and the rate of growth of operating earnings

are the two measures of growth used in our model. Sales growth is used

because shifts in demand for a corporation's product frequently cause an

increase in sales without an immediate increase in profits. As capacity

is increased, and the corporation is better suited to handle the increased

sales volume, profits can be expected to increase. In other words, in­

creases in the rate of growth of sales are expected to precede increases

in the rate of growth of earnings,

Growth rates in operating income are used in the model because earn­

ings growth may not respond to sales growth in some companies. For these

cases, the better measure of long-run growth prospects is the rate of growth

in operating income. Operating income is used instead of net income be­

cause fewer transitory factors affect operating income. The less stable net income series is not expected to produce measures of growth that are

stable enough to be useful in the prediction of common stock price rela­

tives . 119

The availability of investment opoortunities on which a corporation can earn an above average rate of return is dependent on factors exogenous to the firm as well as factors endogenous to the firm. One very important exogenous factor may be described simply as the general condition of the economy as is reflected by employment levels, corporate profits, produc­ tion levels, and other similar measures. When business is good through­ out the economy, more people are employed, which increases total expendi­ tures on consumer goods and services. As the production of these goods and services approaches capacity, managers look to increased investment to cut costs and to increase capacity. As a result, groups of corpora­ tions, such as those producing machine tools and those constructing other productive facilities, undergo a rapid increase in activity and profit­ ability when the economy is in a rapid expansion phase. Likewise, reve­ nues of the transportation industry are higher during periods of high economic activity, and are lower during economic slow downs. Some indus­ tries, such as food processing, are less dependent on general economic conditions for growth and profits.

Value Line ranks corporations with respect to price appreciation potential over the next three to five years. Their starting point is a hypothesis about the economic environment in three to five years. In this case we only duplicate the Value Line method to the extent of pre­ dicting general economic activity one year in advance, since we have assumed a one-year investment horizon. The Value Line prediction of price appreciation could not be exactly duplicated here because so many 120 professional judgments and estimates are involved in predictions of sales and earnings potential.

We use the annual business forecasts of the Prudential Insurance Com­ pany of America in the prediction model. Many factors influence general business conditions in a rather complex manner. Fiscal policy, monetary policy, war labor disruptions, technological change and many more intan­ gible factors are among the determinants of economic expansion. All rel­ evant factors are supposedly included in the forecasts made for Prudential.

In addition to quality grade and price appreciation potential, the

Value Line Investment Survey ranks common stocks with respect to expected price performance over the next year. Our problem differs somewhat since we are attempting to predict future price relatives, not future prices.

One measure of possible market performance over the next year is the difference between the most recent price relative and the average of the price relatives for the five years prior to the year for which predic­ tions are being made. If last year's price relatives were abnormally low, then one might expect the stock's price to rebound this year, resulting in a high price relative. Conversely, an abnormally high price relative in one year may be followed by a low price relative the next year, due to an adjustment in the stock's price. The measure of expected price relative performance is analogous to the Value Line measure of the extent to which a stock is overpriced or underpriced.

The economic model of stock price is of the form:

B1 B B B, B B, (4.35) Rt = BQGt 0 ^ 1St^ D ^ 1Ut_1Vt.1, where 121

Rt is the annual price relative for period t (t =■ 1948,

1949, ...» 1967) for any security,

is a constant (i = 0 , 1 , ..., 6),

Gt is the growth rate of gross national product from the

year t-1 to the year t,

0^ ^ is one plus the average historical geometric growth

rate in operating income per share,

^ is one plus the average historical geometric growth

rate in sales per share,

D , is the difference between the price relative for year t-1 t-1 and the average price relative for the five one-

year periods ending with year t-1 ,

Ut_^ is the variance of the changes in operating income

over the five one-year periods ending with t-1 ,

V ^ is the variance of price relatives over the five one-

year periods ending with year t-1 .

A detailed description of the manner in which equation 4.35 is applied to companies in the five test industries is presented in Chapter I, while the prediction results are presented in Chapter V and summarized in Chapter

I.

Preliminary tests indicate that the multiplicative form of the model presented in 4.35 fits the data better than an additive model. This finding is consistent with many of the cross-sectional studies relating common stock prices to various independent variables. CHAPTER V

ANALYSIS AND INTERPRETATION OF THE

PREDICTION MODELS’ PERFORMANCE

The predictive performance of the four naive prediction models and the

economic growth model is reported in this chapter. Based on the mean-

squared prediction error criterion, the constant exponent model was the best of the three naive time-series models. The constant exponent model performed significantly better than the linear trend model, but only slight­ ly better than the constant model.

Prediction based on the mean of the prior ten price relatives, that is naive model number four, were better than those of the constant exponent model. These results lead us to the conclusion that price relatives tend to be mean-reverting, and that the time series of past price relatives con­ tains little or no information not contained in the arithmetic mean of that series.

The economic growth model failed to provide better predictions than the best naive method, thus leading to the rejection of hypothesis 1. The weaknesses of the economic growth model include insufficient data to achieve stable prediction equation coefficients, and an insufficient emphasis on anticipatory data among the independent variables.

In general, the variance of the price relatives for each security for the test period 1958 through 1967 is significantly smaller than the mean-

122 123

squared prediction errors associated with the best prediction model. Hy­ pothesis 2 is accepted on the basis of these findings.

Performance of the Exponential Smoothing Models

The three naive time-series models described in equations 4.7, 4.13, and 4.18, are

A (5.0) Model Is R.. = A. it 10

(5.1) Model 2: R.. = A. + A.- xt io xl

A. *LO (5.2) Model 3: R >4. = e where xt

A R^ is the predicted price relative of security i for

period t,

A^ , A ^ are prediction equation constants that are

estimated by means of exponential smoothing, as

described in Chapter IV.

Because A^q and A ^ are estimated by means of exponential smoothing,

the performance of each prediction model is influenced by the selection of

the smoothing constant, a. The a priori expectation is that the smoothing

constant will tend to be slightly more that 0 .0 , thus, weighting the past

observations heavily. In contrast, if the smoothing constant is equal to

1 .0 , then only the most recent price relative is used to predict the next

price relative.

Two methods of selecting optimal smoothing constants for each company are used in this study. One method is to predict price relatives for 1958

through 1967 by means of the prediction models 4.7, 4.13 and 4.18. Each 124 model Is tested forty-one times per year because the equation constants

A, and A,, are estimated with the forty-one smoothing constant values io il a = 0.0, 0.025, 0.05, 0.075, ...» 1.0. The best smoothing constant is that constant which results in the lowest mean-squared prediction error for 1958 through 1967, inclusive, in this method, the smoothing constant is said to be fixed because it is applied to the entire period over which predictions are being made.

The second method involves the selection of a smoothing constant on a year-by-year basis. As before, values of the smoothing constant are al­ lowed to range from 0.0 through 1.0 in increments of 0.025 while making predictions for the ten year period prior to the period for which a predic­ tion is being made. That smoothing constant which results in the smallest mean-squared prediction error over the prior ten years is used to estimate

A^q and A ^ , and thus the price relative of the next period. The values of

A^q and A ^ used to make predictions in 1958 are estimated by means of the smoothing constant resulting in the smallest mean-squared prediction error for the year 1948 through 1957, inclusive. In 1959, the value of A^q and

Aji used to make the price relative prediction will be those estimates which are made by the smoothing constant which resulted in the smallest mean-squared prediction error over the previous ten years, or 1949 through

1958 in this case. This method may result in ten different smoothing con­ stants being used for the ten predictions made during the test period, 1958 through 1967. The smoothing constants are said to be shifting in this method, because the smoothing constant used for any one company may change from period to period depending on past results. 125

Ranking with respect to mean-squared error.— Two expectations are

being tested In the evaluation of the three models of stock price rela­

tives. One expected result is that the constant models (1 and 3) will out­

perform the linear trend model. The basis of this expectation is the fact

that no significant trend is expected in price relatives, even though a

trend may exist in dividends and the stock price. Price relatives are com­

posed of the dividend element and the price appreciation element. Based on

the assumption of a constant proportional increase in stock price, the ex­

pected price relative is a constant, and does not increase over time (see

equation 4.28). If there is no upward drift in prices, then the expected

price relative is still a constant, and does not increase over time (see

equation 4.31).

Irving Fisher and John Burr Williams have written what are now con­

sidered classic works in the theory of interest and the theory of invest- 124 ment value. Their methods of representing the profitability of an in­ vestment by the annual rate of return which equates a stream of cash pay­ ments to a current price is still well-accepted today as evidenced by the

fact that many current authors of finance textbooks find it useful to asso­

ciate one average discount rate (geometric rate) with each given level of 125 uncertainty. As discussed with respect to the economic model, the dis­ count rate would remain the same until uncertainty changes, or until there

^■^Irving Fisher, Theory of Interest (New York: MacMillan, 1907); and John Burr Williams, The Theory of Investment Value (Amsterdam: North- Holland Publishing Company, 1964). 125 See Ezra Solomon, The Theory of Financial Management (New York: Columbia University Press, 1964), and Eugene M. Lerner and Willard T. Carleton, A Theory of Financial Analysis (New York: Harcourt, Brace and World, Inc., 1966). 126 is a shift in profitability of investments throughout the economic area.

An average rate of return is determined according to the following expression:

(5 .3) (1 + k)n - (1 + k-jHl + k2)(l + k3)...(l + kn), where k± is the rate of return for period i (i = 1 , 2, ..., n),

n is the number of periods,

k is the geometric average of the k^'s, and

(1 + k^) is the price relative for period i.

An average price relative is determined in the following manner.

(5.4) Rn = R2, ..., Rn » where

Rn = (1 + k)n , and

R.^ = (1 + k^), for i = 1 , 2 n.

(5.5) n log R = log R^ + log R 2 + ... + log Rr

log R1 + log R? + ... + log R (5.6) R = Antilogarithm of ------—

Naive model number three uses the calculation method described in 5.6, with alternative weighting schemes tested. Equal weights are implied by expres­ sion 5.6.

Based on traditional theory, and the arguments provided in equations

5.3 through 5.6, we expect that one geometric average rate of return is associated with each price relative, which implies naive nodel number three will provide the better predictions.

Past performance of the three prediction models using shifting smooth­ ing constants is summarized in Tables One, Two and Three. Past prediction 127 performance for the same three models using the optimal fixed smoothing constant for each model, and for each company, is summarized in Tables Four,

Five and Six. Using two methods of selecting smoothing constants for each of three prediction models results in six sets of predictions, and thus six measures of the mean-squared prediction error for each company.

The comparative performance of the three naive prediction models and the two methods of selecting smoothing constants is based on the ranking of the six measures of the mean-squared prediction error that result from the six sets of predictions made for each company. The model that performs best has the smallest total ranking over the sixty-one companies, because the ranking is from smallest to largest.

Model No. 3 performed best for the sixty-one sample companies over the ten year period 1958 through 1967. This model outperformed the other two models when an optimal fixed smoothing constant was used for each company and it also outperformed the other two models when shifting smoothing con­ stants were used.

The overall ranking was significant at the 99.9% level of confidence as indicated by the chi-square statistic of 123.76 reported in Table 7.

The Kendall Coefficient of Concordance is 0.43 out of a possible 1.0, which is also significant at a level of confidence well above 99.9%. We can con­ clude that the sample rankings are not due to chance, but are due to a real difference in the performance of the three prediction models. One could expect to observe similar findings in other test samples.

Additional evidence of the general preference for model three is gained from a review of the model rankings on an industry by industry basis. As indicated in Table 11, Model No. 3 was the best model in all but 128

one industry, textile producers. The industry rankings were all signifi­

cant at the 99.9% level of confidence except the Tire and Rubber Goods In­

dustry which is significant at the 99% level. An examination of the in­

dividual textile producers does not reveal any conclusive explanations for

the unusual results found in this industry. Dividend patterns seem to be

more irregular among the textile producers than among the cement producers,

for which Model No. 3 is clearly the best model. Dividend pattern does not

seem to be very conclusive evidence, however, because Model No. 3 performed

better for at least two textile producers with very irregular dividend pay­ ment records. Another possible factor is the variability of the price re­

latives being predicted. The average price relative variance for the tex­

tile industry was .1367 while it was .1131 for the cement industry. These

variances seem too similar to conclude that the difference in the vari­

ability of the price relatives in the two industries is the basis for the

difference in prediction model performance. Because the ranking in the

textile industry is clearly significant at the 99.9% confidence level, we

cannot ignore the evidence and suggest that the superior performance of

Model No. 1 in predicting textile industry stock price relatives is merely

a chance event. In view of the evidence, it is concluded that individual

industries may differ sufficiently to imply that models of stock price or models of stock price relatives must be unique to each industry. Similar

industries might result in similar stock price relative models, but two different models may be necessary to best represent the stock price rela­ tives of two dissimilar industries. More evidence on the effects of in­ dustry classifications is presented below.

The cement producer’s industry results are also of interest. Model 129

No. 3 was uniformly the best prediction model for all cement producers and

the overall agreement as measured by the Kendall Coefficient of Concordance

is 70%, which is high for this type of measure. These results suggest a

high degree of homogeneity in this industry, which is unmatched by any

other industry in this study.

The mean-squared prediction error can be decomposed into three parts;

constant bias, proportional bias and validity. The nature of this decom­

position is reviewed here prior to a ranking of the three prediction models with respect to each error element.

The goal of prediction model formulation is to attain predictions such

that

A (5 .7) Rit = Rit + e±t, where

R^t is the price relative of security i for period t,

A R^t is the predicted price relative of security i for

period t made as of the end of period t-1 , and

e^t is a normally distributed random variable with zero

mean, the smallest variance possible, and for which

cov (e^t» e^fc j) =0.0, j < t. Other desirable con-

A ditions are cov (e^t, **0.0 and cov (e^t» ^t^ =

0.0.

It is a goal because if e^t is at a minimum, then any other prediction model will result in greater prediction errors. In reality, predictions are related to actual price relatives by means of the equation

A (5.8) R^t = a + b R^_ + e , where all terms are defined in 5.7, 130

except the constants a and b. The constant of regression, a, arises be­

cause the average predicted relative is unequal to the average observed

relative, and the term b arises because the regression slope deviates from

one. 126 The mean-squared prediction error may be decomposed as follows:

(5.9) I(R - R)2/n = (R - R)2 + (Sp - rSA )2 + (1 - r2)SA2, where

R is the mean price relative,

R is the mean predicted price relative,

S is the standard deviation of the price relative predictions, P

S A is the standard deviation of the actual price relatives A

r is the product-moment correlation coefficient of predic­

ted and actual price relatives.

The first two terms on the right side of equation 5.9 are equal to zero if

a and b in 5.8 are equal to zero and one, respectively. The portion of the

“ _ 2 mean-squared error measured by (R - R) is labeled constant bias, and the 2 term (S - rS.) is due to proportional bias. The remaining portion of the p A 2 2 mean-squared error is equal to (1 - r )SA . This part of the squared-error

can be reduced by finding a prediction which is more highly correlated with 2 2 actual price relatives. For perfect correlation r = 1 and 1 - r = 0 .

This part of the mean-squared error is referred to as the correlation or

validity portion.

The relative performance of the prediction models with respect to

"^Henri Theil, p. 33. 131

constant bias are summarized in Table 8 . One Important constraint must be

emphasized, however. In all cases, the smoothing parameters were selected

on the minimum mean-squared error criterion. As a result, it is possible

that constant bias may be eliminated for any particular prediction model, but only by increasing the mean-squared error, which is undesirable. This

constraint is necessary, but it must be recognized in evaluating the rela­

tive performance of the three prediction models with respect to constant

bias, proportional bias, and validity.

Use of Model No. 3 results in the smallest constant bias, relative to

that of Models No. 1 and 2, for the sixty-one companies tested. The rank­

ings and the Kendall Coefficient of Concordance are both significant at the

99.9% confidence level. Although statistically significant, the Coeffi­

cient of Concordance is very low, indicating a low level of agreement among

the rankings. Model No. 3, using a fixed smoothing constant for each com­

pany, is uniformly the best prediction model with respect to constant bias,

except for the textile industry. It has previously been argued that the

• *\ textile industry may have some characteristics which suggest that a differ­

ent model should be used for this industry.

Bankings at the industry level indicate that the results for beverage producers, textile producers and tire and rubber goods producers were quite mixed, and not statistically significant.

In summary, Model No. 3 resulted in the smallest amounts of constant bias for.all but the textile industry. The other model rankings were mixed and provided little additional information.

Using Model No. 2 minimizes the amount of proportional bias of the predictions for sixty-one companies sampled, as is reported in Table 9. 132

The ranking across all companies, and the Kendall Coefficient of Concor­

dance, are significant at the 99.9% confidence level, but the amount of

agreement is only 0.18 of a possible 1.0. Proportional bias results from

a failure to pick up trends in the stock price relatives.

These results imply that some trends may exist in common stock price

relatives, but not to the extent that a trend model such as Model No. 2

actually outperforms the constant models (1 and 3) with respect to total

prediction errors.

The industry rankings indicate somewhat conflicting judgments. Of the

five industry classifications, Model No. 2 performed best in three classi­

fications and Model No. 3 performed best in two classifications. Model

No. 3 performed best in the Cement Industry and the results were clearly

significant at the 99.9% confidence level. For the Tire and Rubber Goods

Industry, Model No. 3 performed best, but the results were not statisti­

cally significant at the 95% confidence level, so the ranking may have been

caused by chance factors alone. Model No. 2 performed best for the other

three industry groups and the results were significant at the 95% confi­

dence level or higher.

In summary, the generally superior performance of the linear trend model in reducing proportional bias indicates that some short runs or

trends probably exist in the price relatives. However, the superior per­

formance of the constant models with respect to mean-squared error indi­

cates that such trends are not sufficiently long or pronounced to be of value in predicting future price relatives.

Using Model No. 1, with shifting smoothing constants, results in the least amount of correlation error when compared to Models No. 2 and 3. 133

Although this ranking and the Kendall Coefficient of Concordance are sig­ nificant at the 99% confidence level, the degree of agreement is so low

that it is of little use.

Except for the cement industry, none of the industry rankings are

significant at the 95% confidence level. Given this low level of agreement at the industry level, few conclusions can be drawn.

One result of interest is the generally sqperior performance of the shifting smoothing constants on an overall basis and on an industry by in­ dustry basis. In the three previous comparisons, the fixed smoothing con­ stants outperformed shifting smoothing constants on a model by model basis.

These results suggest that the predictions formulated by using shifting smoothing constants are more highly correlated with actual price relatives than the predictions formulated by using fixed smoothing constants. How­ ever, due to the resulting constant and proportional bias, the use of shifting smoothing constants is not preferred to fixed smoothing constants in minimizing mean-squared prediction error.

The influence of smoothing constant selection on the mean-squared prediction error.— Prior to the testing of the prediction models using equation constants estimated by means of exponential smoothing, the two methods of selecting smoothing constants were expected to yield very simi­ lar results. This expectation was based on the belief that after the ini­ tial ten year period 1948 through 1957, a particular smoothing constant would perform best thereafter. If this condition were met, then the opti­ mal smoothing constant for the prior ten years should be about equal to the smoothing constant for the entire prediction period. In other words, we expect stability in the process which generates the price relative 134

series, which means the best smoothing constant for any ten year period within the 1948 through 1967 period will be equal to, or nearly equal to,

the best smoothing constant for the entire period.

The test results indicate that the method used to select smoothing

constants for each of the three prediction models has a significant in­

fluence on the resulting measure of uncertainty, the mean-squared predic­

tion error. Using the mean-squared prediction error resulting from the use

of shifting smoothing constants as the base measure, uncertainty can be

reduced by 47.9%, 65.9% and 58.6% for Models No. 1, 2, and 3 respectively.

In addition to being large, these differences are statistically signifi­

cant at the 99.9% confidence level and are not considered to be the result

of chance.

The 58.6% reduction in the average mean-squared error associated with

Model No. 3 is especially important because it is the best overall predic­

tion model. The difference is even more significant when expressed in

terms of the average mean-squared error resulting from the use of a fixed smoothing constant for Model No. 3. The uncertainty associated with the use of shifting smoothing constants is more than 240% of the uncertainty associated with the use of a fixed smoothing constant for each company.

As reported in Table 13, similar results were found when comparisons are made of mean absolute errors.

The calculation of the prediction errors associated with the single best smoothing constant for the period 1948 through 1967 provides informa­

tion about the best model of stock price relative movements over that pe­ riod, as well as indicating the maximum prediction efficiency associated with the smoothing models. The resulting mean-squared prediction errors 135

are a standard by which we can evaluate our efforts to select an optimal smoothing constant on a year by year basis.

Model No. 3 was judged the best naive prediction model after the series of years for which predictions were being made. An investor would not have known which naive model was going to perform best, and so we have eliminated an element of the uncertainty which an investor would have faced over the years 1958 through 1967. However, Model No. 3 is a simple model and it was expected to perform well as compared to Models No. 1 and

No. 2. Model No. 3 was not much better than Model No. 1, so the results are relatively insensitive to the choice between these two models.

Predictions based on prior means.— If the price relatives are mean- reverting, then a sample estimate of the mean is the best prediction one has. Accordingly, the fourth naive method of predicting is

10 (5.10) R. = I R. J 9 xt ^ it-j

R^t is the price relative prediction for company i

(i = 1, 2, ..., 61) and year t (t = 1958, ..., 1967)

The results are different than those of the constant exponent model, thus supporting the hypothesis that the price relatives are mean-reverting

(Table 41). However, the difference was not statistically significant.

The average mean-squared prediction method is significantly greater than the average variance (Table 37) thus supporting hypothesis 2, which is that the variance does not completely reflect total uncertainty.

Performance of the Growth Model

The general form of the postulated growth model of stock price rela- 136

tives is presented in equation 4.35. In order to implement the model, specific measures of each variable must be made for each company. For example, corporate growth rate may be expressed as an average annual rate of sales growth, operating earnings growth, or cash flow growth, determined over the last two years, three years, or any period believed best.

A generalized stepwise regression program is used to determine which methods of measuring each variable are best. That measure which has the highest partial correlation with the past price relatives is the best mea­ sure available for prediction purposes.

The empirical models tested.— The actual variables selected in the stepwise regression tests for 1948 through 1957 varied from company to com­ pany and from industry to industry. To select the set of variables for each company on an individualized basis was not possible because of the cost and difficulty of doing so. Individual runs of the regression tests would require much more computer time, and thus more cost. Also, the data handling problems would be much greater for individualized processing.

In order to compensate for intercompany differences, more variables were included in the prediction equation and the elements of the variable set were allowed to vary from industry to industry. To illustrate, in the beverage industry, for many companies the best measure of groxfth was the average growth rate in operating income plus depreciation, and for other companies the best measure was the two-year growth rate in operating in­ come plus depreciation. Forcing both variables into the prediction equa­ tion insures that the most significant variables for all of the companies are included in the equation. If a variable has little relevance for any particular company, then the coefficient for that variable should not vary 137

significantly from zero, and it will not affect the predictions very much.

Six variables were selected for each industry classification based on

the frequency and order in which these variables entered the stepwise re­

gression equation for the years 1948 through 1957, for each company in the

industry. Once an industry variable set was selected, it remained fixed

throughout the test period 1958 through 1967.

For the beverage industry, the most significant variables are the

difference between the last annual price relative and the average price

relative for the past five years, the standard deviation of the price rela­

tives over the past five years, the variance of changes in operating income about the average change for the last five years, the average growth rate

in operating earnings over the past six years, the annual growth rate ex­ perienced in the past two years in operating income plus depreciation, and

the rate of growth of gross national product expressed in current dollars.

In the formulation o f (4.35) it was expected that the most recent price relative would be partially correlated with the current relative. The empirical tests indicate that the difference between the most recent price relative and the average price relative is more significantly related to the current price relative than the most recent price relative alone. This finding is consistent with the earlier finding that the smoothing constants tended to be very small for the naive prediction models. The conclusion drawn is that the price relatives tend to be mean-reverting for many com­ panies. This statement cannot be generalized to all companies because for several companies the optimum smoothing constant was equal to 1 .0 , indicat­ ing a random walk in the price relatives, and thus no mean-reversion. The same set of variables was most significant for the tire and rubber goods 138 industry, except that the growth rate of annualized fourth-quarter gross national product was used instead of annual gross national product.

The petroleum and cement industries exhibited some similarity with respect to the variables found the most highly correlated with the price relatives. The common variables were the difference between the last an­ nual price relative and the average price relative for the past five years, the average proportional price range over the past two years, the variance of changes in operating income about the mean change for the past five years, and the average growth rate in operating income plus depreciation for the past six years. The remaining two variables used in the petroleum industry are the average growth rate in net sales over the prior two years and the annual growth rate of gross national personal income over the pre­ vious year, expressed in current dollars. For the cement industry, the other two variables are the average growth rate in net sales over the prior six years and the growth rate of annualized fourth quarter gross national personal income. These are actually different measures of the same basic variables, so that these two industries are very similar. Average propor­ tional price range is a measure of uncertainty which is equal to the price range, divided by the beginning price in order to make the measure insen­ sitive to the absolute level or the share price.

The textile industry was unique in some respects, which is consistent with earlier findings with regard to the naive prediction models. The most significant variables on an industry basis are the most recent price rela­ tives, the relative price range over the prior five years, the variance of changes in operating income about the average change for the prior five years, the average growth rate in operating earnings plus depreciation for 139

the past six years, the average growth rate in net sales over the past two

years, and the growth rate in annualized fourth quarter gross national pro­

duct. This is the only industry for which the most recent price relative

was significantly related to the current price relative, in all others,

it was the difference between the most recent price relative and the aver­

age price relative for the prior five years. These results are not con­

sistent with the fact that the optimal smoothing constant for most of the

textile companies was 0 .0 , thus using the average price relatives of an

earlier period to predict current price relatives as opposed to using last

year's price relative to predict the current year's relative. At the pre­

sent time we do not have an adequate explanation of these somewhat con­

flicting results for the textile industry.

Bounded optimism and pessimism.— Preliminary trials of the postulated

growth model indicated that occasionally an extreme prediction was being

made; one that would exceed all reasonable expectations by a knowledgeable

investor. What might be called a programmed judgment factor was added to

the price relative predictions in order to prevent these extreme predic­

tions .

The programmed judgment factor took the form of an upper and lower bound for each prediction. The first stage of the bounding process in­

volves finding the maximum and the minimum price relatives over the prior

five years for each common stock. A visual examination of these upper and

lower bounds indicated that a wide range of values might appear during any

five year period, and that the range of these limits was still very great.

The second stage of the bounding process is designed to further reduce

the extreme values that may be predicted. An arbitrary maximum upper 140 bound of 0.80 were established after a visual examination of the price rela­ tives for the sixty-one companies during 1948 through 1957. Price rela­ tives of 1.50 were not unusual, but they seldom occurred in consecutive periods or within a five year period. The same is true of very low price relatives.

The net result of these two parts of the bounding process is the set­ ting of upper and lower limits on the predicted price relatives. The upper limit is equal to the maximum price relative for that company over the prior five years, or 1.50, whichever is smaller. The lower bound is equal to the minimum price relative observed for that company over the prior five years, or 0.80, whichever is larger. These programmed judgment factors were important with respect to the predictions of six companies. The pre­ diction equation coefficients for these six companies had large standard errors and tended to fluctuate greatly from year to year.

Comparative predictive performance.— The relative mean-squared errors and absolute errors are compared for sixty-one companies to determine if the postulated growth model resulted in smaller prediction errors than re­ sulted from use of Model No. 3, with shifting smoothing constants. The average results for these two models can be seen in Table 32, and Table 33.

Tests of significance are summarized in Table 38.

It is safe to conclude that Model No. 3 resulted in smaller prediction errors than resulted from the growth model. The average mean-squared error for the growth model was 0,034, or about 20%, larger than resulted from

Model No. 3. These results were significant at the 99.9% confidence level for the Wilcoxen matched-pairs, signed-ranks test, and are also significant at the 99% confidence level for the t-test. The cement industry results 141 were clearly mixed with insignificant differences. However, for the other industries and for all companies combined, the null hypothesis of no dif­ ferences is rejected, and the results accepted as evidence of the superior performance of Model No. 3.

The average absolute error resulting from use of the growth model was

0.0432 or about 14% larger than the average absolute error resulting from the use of Model No. 3. As was true for the mean-squared error, these results are significant at the 99.9% confidence level for the Wilcoxen matched-pairs, signed-ranks test, and are also significant at the 99% confidence level for the parametric t-test.

The growth model prediction errors contained less constant bias than the smoothing model but greater prediction error due to proportional bias and low correlation. Eliminating proportional bias may be the most promising correction factor available to reduce the growth model predic­ tion errors.

Linear correction factors, the distribution characteristics of the prediction errors, and possible methods of improvement are discussed in the subsequent three sections.

Optimal linear correction parameters.— One additional method of re­ viewing the prediction results is to examine the linear correction factors associated with the set of actual price relatives and the set of pre­ dicted price relatives. The results of the determination of the correction constants are summarized in Table 29. Apparently, the correlation was low between the actual price relatives and the predicted price relatives. 142

This observation is supported by the low value of b (see Table 29)

and the large value of a. Ideally, a = 0 and b = 1. The results of the

simple analysis of variance summarized in Table 27 are similar. The with­

in group regression is not significant at the 95% confidence level, even

though the group effects are significant. These two tests suggest that

the prediction performance of the growth model is very poor.

However, when each prediction was adjusted for the difference

between the average price relative and the average predicted price rela­

tive, the relationship of each predicted relative to the relevant actual

price relative was quite good (see Table 28). The regression of R^t on

A R^t was forced through the origin. The regression coefficient was 0.96149

(b = 1.0 is ideal) and the squared multiple correlation coefficient was

0.8861 out of a possible 1.0, which is relatively good. If the amount

of constant bias (a^) is relatively stable over time, then there is some promise that the use of linear correction factors may be used to improve

the prediction results of subsequent studies.

Prediction error distribution tests.— The nature of the distributions which represent the behavior of economic variables have long been of in­ terest and have received much attention in recent research. Benoit 127 Mandlebrot has renewed this interest because he has argued that price changes for various economic time series tend to be more leptokurtic

(long-tailed) than would be expected for the Gaussian distribution (normal).

127 Benoit Mandlebrot, "The Variation of Some Other Speculative Prices," The Journal of Business, Vol. XL, No. 4 (October, 1967), pp. 393-413. 143

Various authors have explained the apparant non-normal behavior of stock price changes in various ways. Mandlebrot has argued that price changes are represented by the general class of distributions called Stable

Paretian, of which the Gaussian Distribution is a special case. A most significant feature of this class of distributions is that of an un­ defined variance. Regression and many other statistical models are based on the existance of a finite variance measure, which only exists for one special case of the Paretian Distribution, which is the Gaussian Distri­ bution.

The distribution of price relatives and the distribution of price relative prediction errors are both studied to determine if these vari­ ables can be described by the Gaussian Distribution, and if not, how they deviate from the Gaussian Distribution. The prediction errors which result from the use of Model No. 3, using shifting smoothing constants, are not normally distributed according to the Kolmogorov-Smirnov test, as explained in Table 15. When plotted, the errors appear to be nearly bell-shaped, but that is not the case.

The most interesting aspect of this test is the nature of the devia­ tion from normality. It is not the tails that are more heavily weighted than the normal or Gaussian Distribution, it is the concentration at the peak that is greater than expected. These results are very different from those expected because of Mandlebrot*s work. Because there is little evidence of a very large or undefined prediction error variance, the use of mean-squared error instead of average absolute errors is preferred 144 because of the utility function assumed.

Similar results were found for the predicted price relatives made by Model No. 3 using shifting smoothing constants.

As reported in Table 14, the predicted price relatives tended to be more concentrated about the mean predicted price relative than one would expect for a normally distributed random variable. As a result, the variance of the predicted price relatives tends to have a finite variance, and there is no evidence that the statistical models used are not applicable to this data.

Similar results are obtained for the prediction errors that result from the use of Model No. 3 with a fixed smoothing constant for each company. The distribution is more heavily concentrated about the mean prediction error than would be expected for a normally distributed random variable (see Table 17). The predicted price relatives for

Model No. 3, using a fixed smoothing constant for each company, are similarly distributed (see Table 16).

The errors associated with price relative predictions made by the growth model for the sixty-one companies for 1958 through 1967 were not normally distributed. The errors were more concentrated about the mean error than one would expect for a normal distribution, as was also the case for the prediction errors from Model No. 3 (see Table 23).

The predicted price relatives for all sixty-one companies from 1958 through 1967 were less concentrated about the mean predicted relative than one would expect for a normally distributed random variable. These 145 results do not parallel the results obtained for prediction Model No. 3.

Greater weight is included in the tails of this distribution because the predictions are bounded, so that there are numerous predictions equal to

0.80 but none less than 0.80, and numerous predictions equal to 1.50, but none are larger. The distribution of these predictions is thus nearly uniform, with more weight at the ends than one would expect for a uniform distribution.

The 610 price relatives observed over the ten year test period were tested for normality. The relatives deviate significantly from normality in at least two ways. First, the price relatives are more heavily con­ centrated about the mean price relative then one would expect if the price relatives were normally distributed. Second, the actual price relatives are skewed toward the larger relatives.

The finding that prediction errors show no sign of excessively large variances is significant, and it justifies the use of mean-squared pre­ diction error and variance as measures of uncertainty.

Implications for Measuring Uncertainty

The mean-squared prediction error for a sample of predictions and the variance of price relatives about the mean sample relative are two surrogate measures of uncertainty that have been discussed in this study.

Using mean-squared error as the more logical measure of uncertainty, the variance of price relatives, for each corporation, is evaluated in terms of its appropriateness as a measure of the uncertainty inherent in price relative predictions. 146

Components of uncertainty.— The investor faces two problems when pre­ dicting price relatives and evaluating the uncertainty of those predictions.

The movement of price relatives over time can be represented by various stochastic models. One element of uncertainty facing any investor is the purely random element of a price relative's changes. This element of un­ certainty is inherent in the market process and cannot be eliminated.

The usual measure of uncertainty associated with this component of total uncertainty is the variance, or some equivalent but more elaborate measure, when the process is very complex. The second element of uncertainty is determining which stochastic process best fits the time series of prices or price relatives. Many models of stock price changes have been pos­ tulated and tested without conclusive results. The same results are to be expected for the study of price relatives. Likewise, there are many theoretically sound price relative prediction models, which vary in their effectiveness in predicting price relatives. Total uncertainty is thus a combination of the stochastic nature of a price relative, and man's inability to determine or fabricate the best possible model of stock price relative behavior, or the best possible predictions of the stock price relatives.

Four naive models of stock price relative behavior over time, and a growth model based on economic forces, are used to formulate predictions in this study. The resulting prediction errors are caused by the stochas­ tic nature of the pricing process as well as by the errors that result from incorrectly formulating the model. The variance of price relatives 147 about the mean price relative for some sequence of periods tends to under­ state the uncertainty associated with the formulation of predictions be­ cause the mean, or expected value, is known, for a particular sample, after the fact, and only then.

Comparisons of these measures of uncertainty are summarized in

Table 37. The average mean-squared prediction error resulting from use of the growth model is more than 33% greater than the price relative variance, using the former measure as the base value. When expressed in terms of the variance, the average mean-squared error is about 51% greater than the average variance. Less extreme, but similar results are found for Model No. 3, when the optimal smoothing constants are sel­ ected on a yearly basis. The average mean-squared error for Model No. 3 is 17.2% greater than the variance. Expressed in terms of the smaller base, the average mean-squared error is 121% of the average variance.

Unless better prediction models can be formulated, or better intuitive predictions can be made, these results indicate that the variance under­ states the amount of uncertainty inherent in price relative prediction.

It may be possible to find better models of the behavior of stock price relatives, and it may be possible to find much better predictions of price relatives. Progress may be made to the point that prediction errors are less than deviations from the mean, but the present evidence suggests that the variance estimate understates the level of uncertainty associated with price relative prediction.

Variance as related to mean-squared error.— If the variance under- 148

states true uncertainty by some constant amount or proportion, it is a

relatively simple matter to calculate the variance, and then apply some

correction factor in order to better estimate the level of uncertainty

associated with the price relative being predicted. Moreover, if any

functional relationship exists between the variance and the true, but unknown, uncertainty, then that measure of.true uncertainty can be es­

timated by means of the variance estimate and the known relationship between total uncertainty and the variance.

At the present time, we have two surrogates for true uncertainty, one of which is the mean-squared prediction error of Model No. 3 using shifting smoothing constants. Using this mean-squared prediction error as one available surrogate for true uncertainty, we tested various relation­ ships to determine if mean-squared error could be estimated by means of the variance. These attempts were relatively unsuccessful. Linear regression models failed to produce any model in which the variations in variance measures were useful in explaining more than 20% of the variation in mean-squared error for the sixty-one companies. The residuals were linearly correlated with the value of the mean-squared error so that the smaller values of mean-squared error were being overstated by the model and larger values were being understated. We do not argue that no such relationship exists, we only argue that the relationship is not obvious, and that it is unknown at present.

In summary, no relationship was found between the variance of a company's price relatives over the ten year test period and the mean-squared 149 prediction errors for that period. The tentative conclusion is that no such functional relationship exists, for this measure of true uncertainty.

Risk class discrimination.— As discussed in Chapter II, the selec­ tion of investments involves a measure of uncertainty. Unless it can be shown that alternative measures of uncertainty are linear functions of one another, it is likely that the selection of uncertainty measures will influence the portfolio selected from a given set of opportunities.

Because the squared correlation coefficient is invariant under linear transformation, it is our belief that a linear function of the variance would result in the selection of the same portfolio as would result from the use of the variance as the measure of uncertainty. As mentioned above, no such linear transformation of the variance was found, assuming that the mean-squared error is a good measure of true uncertainty.

Moreover, Modigliani and Miller have discussed the decomposition of investments into different risk classes, which requires only ordinal . 128 measurement of uncertainty. There is some evidence that even the selection of risk classes using the variance as the classification cri­ terion will lead to different results than would be obtained if the mean-squared error were the classification criterion.

Evidence that use of the variance to measure uncertainty may result in one classification, while use of the mean-squared error may result in another classification, is provided by the simple analysis-of-variance

■^^Modigliani, Franco, and Miller, M. H. "The Cost of Capital, Cor­ poration Finance, and the Theory of Investment," American Economic Review, XLVIII (June, 1958), pp. 261-279. 150

test summarized in Tables 19, 20, 21, and 22. The mean-squared error is

the dependent variable, while the variance is the independent variable,

thus providing sixty-one paired observations. Industry mean-effects

were significant at the 99% confidence level for both the variance and

the mean-squared error, which indicates that the average mean-squared

error differed from industry to industry, ds did the average variance.

The variance is then used as a control variable to determine if the mean-effects for the mean-squared errors can be explained by differences

in the variances. Referring to Table 19, the mean effects for the ob­

served measures of the mean-squared errors can be explained by differences

in the variances. Referring to Table 19, the mean effects for the ob­ served measures of the mean-squared error cannot be explained by dif­

ferences in the variance, so the null hypothesis must be rejected at the

99% confidence level. The implication of these findings is that the variance may not be a sufficiently complete measure of uncertainty for purposes of defining risk classes and assigning companies to those risk classes. At this point we cannot make any normative statements, we can only raise the problem and suggest that different classifications prob­ ably will result. Mathematical proofs of these statements and empirical tests of their validity are beyond the scope of this project.

Potential Improvements in the Growth Model Predictions

The prediction performance of the growth model is disappointing.

Most authors do not test their valuation models by actually making price relative predictions, so the performance of our model cannot be compared 151 to the performance of the other models that appear in the literature.

Nevertheless, the model is not useful in its present form. There are several approaches that one might use to improve the price relative pre­ dictions of an economic model. Most of these techniques are apparent in retrospect, even though they were not obvious at the start of the study.

Sample size increases.— The large standard errors of the regression coefficients undoubtedly contribute significantly to the average mean- squared prediction error resulting from use of the growth model. Increasing the sample size may reduce the standard error of each coefficient and thus reduce year-to-year fluctuations in these coefficients.

One method of increasing sample sizes is to combine the observations for all companies that are in one industry classification, thus obtain­ ing one set of equation coefficients per industry per year. Within indus­ try differences will still be significant because growth rates for the various companies will still differ, as will the measure of uncertainty.

For homogeneous industries this method should work well, but it probably is of little value in the case of conglomerates, and other loosely defined industry groups.

A second method of increasing sample size is to increase the number of years of data in the formulation of the regression model coefficients.

Two problems are created by the use of more data years, and these may de­ crease the performance of the growth model. One is the probable instability of relationships over long periods of time. If the relationships of indi­ vidual variables to the price relatives and to each other change over time, 152

then adding more observations may have a detrimental affect on the standard

errors of the coefficients. If the relationships are stable, then one

might expect the standard errors to decline, and prediction performance

to increase. It is difficult to predict the success of this method of

increasing sample size.

Variable measurement changes.— One of the more apparent opportunities

to improve the results of the approach used in this study is the continued

search for better measures of the variables used in the prediction model,

and the inclusion of some industry variables in the model. The particular measure of sales growth rate used in this study, or that of the uncertainty

of operating earnings, are not optimal in any true sense of the word. Much more research must be done in sales predictions, growth rate prediction

and other related measures before we can confidently state that the best measures of each variable have been included in the prediction model.

This type of searching never ends, but should result in moderately im­ proved results.

The more significant improvement of prediction results probably will be the result of including an industry activity variable in the prediction equation. For those industries in which annual industry predictions are made, in quantified form, the reduction in uncertainty should be much more than can be expected when such data or predictions are lacking. Here again, the more homogeneous industry classifications are the more desirable starting points for such efforts. The collection and processing of industry 153 information is a costly and time-consuming process, and the incremental cost may not be justified economically, even though predictions may improve.

Learning factors.— The most interesting source of potential im­ provement in the performance of the postulated growth model is that of learning. First, through an examination of price relative changes we may be able to set more scientifically determined upper and lower limits on the predicted price relatives. For example, if it is determined that price relative increases of 0.6 or more have a 0.5 probability, we establish the rule that the predicted price relative may not exceed last period's relative plus 0.6. Any larger change is considered highly unlikely, and thus should not be predicted. In this case we are incorporating know­ ledge about the distribution of price relative changes.

The inclusion of past prediction errors in the regression model and thus in the prediction model is a more direct application of the concept of learning. In this case we are admitting that the regression model does not fit the data quite as well as is postulated, because the errors are of no predictive value in theory. However, as a practical matter, the regression coefficients are based on data and relationships from the previous ten years, and they may not adjust to changing conditions as fast as is desired. Some additional information may be provided in prior prediction errors, and this method is an attempt to capture some of that information for prediction purposes.

A second method of incorporating the relationship of past price re­ lative predictions to past price relatives is to formulate predictions 154

as was done in this study, and then adjust these predictions for the

historical amount of bias associated with prior use of the growth model

predictions. The optimal linear correction of predictions was discussed

above, and this linear correction factor can be applied to each prediction,

as it is being made, in order to formulate an adjusted prediction. The

constants in the correction factor are estimated each year from the his­

torical relationship of price relatives and predicted price relatives.

More esoteric approaches may exist, but the latter two approaches

mentioned are straightforward methods of incorporating information about

past prediction errors in current predictions. The true test of any one

of the suggested improvements is its ability to reduce expected uncertainty.

Extension of theory.— Our economic model of stock price relatives

includes measures of growth and stability, a prediction of the economic

environment next year, and an estimate of the probable short-run change

in the price relative. These are the same factors evaluated by the Value

Line Investment Survey, even though the variables are measured differently.

If use or usefulness is any criteria of the empirical validity of a con- * * » struct, then our approach to price relative prediction has empirical validity because investors and stock brokers purchase the Value Line pro­ jections. However, the value line approach lacks the derivation and the

analysis that should be characteristic of a formal model of stock price

relatives.

A more complete theory of the economic forces causing price relatives

to vary with changes in expectations, changes in the supply of money, or 155 other forces, may lead to improved price relative predictions. Given the heterogeneous nature of investors in common stock, there is absolutely no assurance that a more formal conceptual model of stock price relatives will result in better predictions. 156

APPENDIX A

A series of predicted price relatives can be compared with the series of those price relatives subsequently observed in order to calculate a series of prediction errors. Let R represent the common stock price JL U> A relative for security i during period t, and R^ represent the predicted value of The prediction error, denoted is the difference be­ tween the actual price relative and its predicted value where

(A-l) dlt-Rlt-Rlt

The i subscript will be dropped for convenience since we are discussing a general measure of uncertainty, not the specific value of a statistic for a particular company. For any n-year period, the mean prediction error is given by n (A-2) d -trx dt/n, and the variance of the prediction errors is given by

(A-3) S ^ « . J 1 (dt - d ) 2/(n-l)

Suppose the prediction errors associated with the price relatives of a particular security can be described by a random variable D, where the 2 true, but unknown, mean difference is and the expected variance is o^.

The mean-squared rror is defined as

(A-4) a2 - E(D-O)2 - E(D2) e

By definition, 157

(A-5) a2 = E(D - yD)2 = E(D2 - 2DyD + y2)

(A-6) cr2 = E(D2) - y2

Note that:

(A-7) a2 + y2 = E(D2) - y2 + y2

(A-8) a2 + y2 = E(D2) = a2

The relationship defined in A-8 is true by definition of the terms. 2 2 2 A sample estimate of a , denoted S , is an unbiased estimate of a r e e e 2 2 if and only if E(Sg) = ae * also true that E(A + B) = E(A) + E(B)

for two random variables A and B. Our problem is to find an unbiased es- 2 timate of a . e We know that

(A-9) E(d - yQ)2 = o2/n

By manipulating the terms of A-9, we have that

(A-10) E(d2) = 02/n + y2

129 Another standard relationship in mathematical statistics is

(A-ll) E(S2) = o2

2 130 Define the biased estimate of as

(A-12) s2 = (n - 1) S2/n, so that d d

(A-13) E(ns2/(n - 1)) = E(S2) = a2

129 Hogg, p. 39.

130Hoel, p. 198. 158

2 2 The sample estimate of 0g is denoted Se> and can be decomposed as follows:

CA-14) s2 - trx d2/n - ( dt/„)2 + j 1 (dt - d)2 n

n (d. - d)2 d 2 _ n d because E------E - 2d E- — + d , t=l n n t-i n

n (d-d)2 n d 2 __ and thus E------= E. — - d t=l n t=l n

Because the expected value of a sum is equal to the sum of the expected values,

(A-15) E(S2) = E(d2) + E(s2)

(A-16) E(S2) = a2/n + y2 + E[(n - 1) S2/n]

(A-17) E(S2) = a2/n + y2 + (1 - £> E(S2)

(A-18) E(S2) = a2/n + y2 + a2 - a2/n

2 2 2 2 (A-19) E(Se) = aD + yjj» which is equal to a so we have shown that

2 2 2 2 (A-20) E(S ) = o , so S is an unbiased estimate of a . e e e e 2 It should be noted that adding the unbiased estimate of to the square of the unbiased estimate of y^ does not provide an unbiased estimate 2 2 of (yQ + oD), which is to say

(A-21) E(d2 + S2) * a2 + y2

The left side of expression A-21 does equal

(A-22) E(d + S2) ■ cr2/n + y2 + a2 which is biased by the amount a2/n. 159

APPENDIX B

Let R^t be the price relative for security i (i = 1, 2 ..... 61) for time period t (t = 1948, 1949, ..., 1967). Further, assume that the price relatives of each company can be represented by a random variable

- ]_oi R^, with a sample mean value of R^. The difference between each price relative and R is itself represented by a random variable, X. Let

(B—1) xt = Rt - R

The x^'s form a purely random series; one of random deviations from a norm.^^ The expected value of X is zero, because E(R - R) =0. Similar­ ly, let

(B-2) yfc = xfc+1 - xt, where

(8-5) *t+1 - xt = (Rt+1 - R) -

The expected value of X Is zero, so the expected value of Y is zero also.

The standard from of the correlation coefficient is

E(xt - x)(yfc - y) (B-4) r = £g-g------T"s , where x y

r is a correlation coefficient,

xfc is the value of the random variable X,

x is equal to Ext/n,

^^The i subscript is dropped for convenience since it adds little to the subsequent discussion. Also, all summations are from t=l to t=n.

"^^Holbrook Working, "A Random-Difference Series for the Use in the Analysis of Time Series," Journal of the American Statistical Association, Vol. XXXIX, No. 1 (January, 1934). 2 - Sx is equal to E(xfc - x)/n

y is a value of the random variable Y,

y is equal to Ey^/n, and 2 - 2 Sy is equal to ^(y^ “ y) /n.

If x = 0, and y = 0, then B-4 becomes

E¥ t (B-5) r = ng g 3 S x y

Recalling that y = xfc+j - xt> we can write B-5 as

Ex (x . - x ) (B-6> r = - M i — - * for x y

(B-7) Sx = ^ ( x t - 0)2/n = -v/Ex^/n

(B-8) Sy =.N/S[(xt+1 - xt) - 0]2/n

(B—9) nS S = n A x2 (e x2 , + Ex2 - 2Ex„_l,x \ x y _ t I t+1 t t+1 11 n n

However, xfc and tend to be uncorrelated, so the most probable value of 2 2 2 Exfc+jXt is 0. Also the expected value of 2xfc+1 = naR , as does Exfc. Substi­ tuting these probable relationships, we have that the expected value of r can be stated as: which can be reduced to 162

TABLE t

Na IVE PREDICTION MODEL NOi t

PREDICTION RESULTS POR THE TEARS 1951 T ROUQH 1967 USIN9 THE BEST SHOOTHINO CONSTANT POR THE TEN EARS PRECEEDINO THE TEAR POR WHICH A PREDICTION IS B NS MADE SOURCE PREDICTION ERRORS BSOLUTE INOUSTRT COMPANT var iance MEAN.SaUAflEO co n s t a n t PROPOR ONAL OH AVERAOE CODE eODE' es t imate ERROR BIAS BlA CORR LATION ER 20BE 49030 o;os6s 0,1474 0,0867 0. 531 0,3 2oS* 229S00 0.035* 0.0420 0,0696 0,33 0, 968 0,1 2082 255100 0.0612 0,0739 0,0291 0,23 0, 387 0,2 2082 321418 0.0697 0,0997 0,1292 0139 0, 878 0,2 2002 348600 011984 0,1789 0,1737 0102 0, 967 0,2 0,4 2082 997100 0.2210 0,3678 0,2960 0 f 33 0, 676 20 8 6 162600 071920 0,1616 0,0016 0, 199 0.2 20 8 6 162900 07 0904 0,0662 0,0969 0,91H? 0, 283 0,2 20 6 6 218930 0,2093 0,2923 0,1936 0,25 0, 920 0,4 20 8 6 976429 070666 0,0980 0,0009 0, 669 0.2 20 8 6 631490 010931 0,0846 0,0939 0,46 0. 613 0,2 2200 12200 070962 0,1090 0,0358 0,39 0, 709 0,2 2200 73900 070466 0,0926 0,2000 0, 0, 000 0,2 2200 149460 073196 0,3900 0,0482 0,33 0, 169 0,9 2200 179600 0.1019 0,1499 0,2015 0,36 0, 359 0,3 2200 203600 0.1249 0,1148 0,0211 6. o, 789 0.2 2200 322600 0.1079 0,1947 0,0107 0,59 0, 959 0,3 2200 374790 0.3216 0,4628 0,0039 0,37 0. 230 0,9 2200 438900 071049 0,1104 0,0207 0,23 0, 437 0,2 2200 694900 0.1190 0,1801 0,0164 2»*2 Ot 614 0,3 2912 93377 0 70164 0,0187 0,0174 0,38 0, 999 0.1 2912 192800 070234 0,0289 0,0899 0,24 0, 690 0.1 2912 167700 0 70316 0,0379 0,2016 0,04 0, 919 0,1 2912 312697 071249 0,1360 0,0996 0,08 0, 149 0,3 2912 407900 070939 0,0993 0,1221 0, 779 0,2 2912 492180 070281 0,0349 0,0067 0,69S'4. 0. 927 0,1 2912 913700 070219' . 0,0330 0,0739 0,21 0, 097 0,1 2912 606900 0 i 0960 0,0769 0,0693 HI 0, 679 0,2 2912 696900 070179 0,0313 0,2991 S'S? 0, 680 0.1 2912 699209 070492 0,0636 0,0299 0,34 0, 241 0,20 163 c 164

TABLE 2

n a i v e p r e d i c t i o n m od el n o ; 2

PREDICTION RESULTS POP THE YEARS 1«98 THROUGH 1967 USING THE BEST SMOOTHING CONSTANT FOR THE TEN YEARS RRECEEDINS THE YEAR FOR WHICH A PREDICTION IS BEINQ MADE

p e r c e n t a o e SOURCE OF PREDICTION RRQRS VDUSTRY COMPANY VARIANCE meanasquared co n s t a n t PROPORTIONAL LO AVERAGE BSOUUTE CODE CODE ESTIMATE ERROR BIAS BIAS CORREL ION ER R 20S2 49030 0i0863 0.2394 0,0248 0,7497 0 22 0.4 20«2 229900 0.0392 0.0341 0,1228 0, 0 87 S.l 20*2 299100 0 '0 8ll 0.1024 0,0009 0,4690 0 93 0.2 20*2 321419 0.0497 0.1068 0,1098 0,4319 0 46 0.2 2082 348800 0.1984 0.1747 0,1699 0,0291 0 80 0,2 2082 997100 0.2210 0.4084 0,2929 0,3981 0 34 0.4 2086 162600 0.1920 0.1729 0,0010 0,3490 0 69 0,2 2086 162900 0)0904 0.0900 0,0178 0,9764 0 40 0,2 2086 218930 0.2093 0,2891 0,1124 0,2840 0 60 0,4 2086 976429 0.0466 0.0847 0,0143 0,9193 0 46 0,2 2086 631490 0.0931 0.0712 0,0436 0,3189 0 43 0.2 2200 12200 0.0962 0.1067 0,0247 0,3836 0 99 0,2 2200 73900 0.0468 0,0926 0,2000 0, 0 80 0.2 2200 149460 0.3198 0.40390,0900 0,3303 0 61 0.9 2200 179600 0.1019 0.1996 0,2014 0,3600 0 41 0,3 2200 203600 0.1249 0,1148 0.0211 0. 0 97 0.2 2200 322600 0.1079 0,2149 0,0038 0,6289 0 36 0,3 2200 374790 0.3218 0,4743 0,0089 0,3812 0 60 0.9 2200 436900 0.1049 0,1130 0,0181 0,2499 0 73 0,2 2200 694900 0.1190 0,1994 0,0332 0,4920 0 47 0.3 2912 93377 0.0184 0,0217 0,1219 0,4272 0 49 0,1 2912 192800 0.0234 0,0293 0,0223 0,3649 0 61 0,1 2912 187700 0.0316 0,0366 0,1979 0,0726 0 76 0.1 2912 312697 0.1249 0,1371 0,0742 0,1042 0 81 0,3 2912 407900 0.0939 0,0993 0,1221 0, 0 67 0.2 2912 492180 0.0281 0,0399 0,0260 0.6212 0 39 0.1 2912 983700 0.0289 0,0336 0,0919 0,2421 0 70 0.1 2912 606900 0 i 0960 0,0820 0,0478 0,4109 • 0 94 0,2 2912 696900 0.0179 0,0323 0,2400 0,3396 0 42 0,1 2912 699209 0;0492. 0,0677 0,0022 0,4260 0 96 0.2 165

TABLE 2— CONTINUED

pe r c e n t a g e SOURCE or PREDICTION ERRORS INDUSTRY COMPANY v a r iance MEa NLSSUARED c o n s t a n t proportional low a v e Ra o e abs olute CODE CODE ' e s t imate ERROR BIAS BIAS correlation ERROR 2912 0*2000 0.04*9 0.0*73 0,0049 0,2724 0,7227 0,1*23 2912 0*3*00 0.0472 0.0932 0.0111 0,3640 0,6249 0.1*93 2912 0*0300 0.0201 0.030* 0,0001 0.4471 0.9920 0.146* 2912 000700 0.04*1 0.0421 0,0134 0, 0,9*66 0,1066 2912 099600 0.0291 0.040* 0,0690 0,2*00 0.0422 0,1793 2912 701220 o;o4*9 0.0994 0,0023 0,3626 0.6391 0,2076 2912 736700 0.0492 0.0476 0,1434 0, 0,094* 0.1730 291S 334*00 0.0244 0,0246 0,109* 0. 0.0942 0.1*10 2913 40*0(3 . 0.0262 0,0296 0, 0,2022 0.7970 0.1394 2913 *06200 0.0342 0,0396 0,2343 0,009* 0.7*00 0.1990 2913 0*6900 0.0394 0,0910 0,1000 0,3426 0.3966 0,1*39 2913 711202 0.0310 0.0344 0,2103 0,0099 0.7003 0,1699 3000 100700 0.4027 0,9021 0,0001 0,6200 0,3719 0,6401 3000 209000 0.2093 0,2199 0,0144 0,4943 0.9313 0.3694 3000 266300 . 0.0927 0,0723 0,0019 0,2642 0,6*39 0,2033 0.2914 3000 317900 0.0*97 0,0947 0,0260 0,9131 0,4601 3000 310100 0.062* 0,0094 0,2012 0,1904 0,6403 0.2360 3000 449700 0.9144 0,6096 0,0099 0,6030 0.3919 0,9929 3000 4*7100 0.0*2* 0,0000 0,0990 0,6302 0.3141 0,6233 3000 73*691 0.0649 0,0099 0,0007 0,9930 0,4063 0.230* 3241 19000 0.1114 0,1394 0,0930 0,3972 0,9491 0.3214 3241 19900 0.2446 0,3727 0,0100 0,4270 0.9622 0,4209 3241 304000 0.1410 0,1*92 0,0074 0,2720 0,640* 0,3926 3241 312*00 0.0000 0.0910 0,21*3 0, 0,7037 0,2942 3241 371200 0,0*09 0,0070 0,0490 0,0692 0,0*90 0.2343 3241 401126 0.1042 0,1399 0,1*49 0,1696 0,6*40 0.3370 3241 424300 0.0*02 0,1294 0,1232 0,6410 . 0,2390 0,2993 3241 434*00 0.0347 0,0432 0,2392 0,0974 0,6*74 0,1694 3241 4*6600 0.0*22 0,0910 0,0124 0,1039 0.0041 0,2330 3241 409210 0.0*1* 0,1004 0,1777 0,1229 0,6990 0,2*22 3241 971000 0.3900 0,3190 0,0004 0. 0,9996 0.4102 166

TABLE 3 NAIVE PREDICTION MODEL NOi 3

PREDICTION RESULTS TOR THE TEARS 1998 THR0U6H 1967 USINO THE BEST SMOOTHING CONSTANT FOR THE TEN YEARS PRECEEDINO THE YEAR FOR WHICH A PREDICTION IS BE|NS MADE

p e r c e n t a g e SOURCE OF PREDICTION ERRORS INDUSTRY COMPANY VARIANCE MEAN«SGUARED CONSTANT IOPORTIONAL LOW AVERAGE ABSOLUTE CODE CODE ESTIMATE ERROR BIAS BIAS CDRRE^ATIDN ERROR 20*2 49030 OiO**3 0.144* 0,1239 0,9174 0,3372 20»2 229900 0.0392 0,0496 0,10*9 0,3424 0.9481 0,1*34 20*2 299100 0.0612 0,0673 0,0022 0,2066 0.7*10 0,234* 2062 321419 0 i0*97 0.1061 0.1449 0,3974 0.4977 0,2963 2062 346600 0.1964 0.1626 0,1992 0,0264 0,7784 0.2679 2062 997100 0.2210 0,4099 0,3998 0,2924 0.3*78 0,4*06 2080 1*2600 0.1920 0,1631 0,0112 0,2587 0,7300 0,254* 2060 1*2900 o;o9o* 0,0901 0,0759 0,5037 0,420* 0.2479 2060 216930 0.2093 0,2972 0,1719 0,2436 0,9849 0,4091 2060 976429 o;o66* 0,1021 0,0006 0,9366 0,4629 0,2774 2060 631490 0.0931 0,0971 0,0597 0,4759 0,4644 0.2603 2200 12200 0,0*62 0,1060 0,0530 0,3797 0,9673 0,2307 2200 73900 0,0466 0,0997 0,2449 0 , 0,7551 0,2089 2200 1494*0 0.3199 0,36*2 0,0730 0,2799 0.6*75 0,4905 2200 179*00 0.1019 0.1906 0,2121 0,3999 0.4319 0.3213 2200 203*00 0,124* 0,1169 :,0390 0 , 0,9*90 0.2*32 2200 322600 0,1079 0.1979 0,0176 0,5939 0,3489 0,3637 2200 374790 0.3218 0,4772 0,0101 0,3838 0,6061 0,9234 2200 430*00 0.104* 0,1042 0,0061 0,2033 0,7*06 0.2929 2200 *94900 0.1190 0,171* 0,0609 0,9793 0.3998 0,3512 2912 93377 0.01*4 0 ,01*1 0,0174 0,4060 0,9766 0.1142 2912 192800 0.0234 0.0303 0,0570 0,3472 0,9998 0,1077 2912 167700 0.031* 0,034* 0,1616 0,0344 0,8040 0 .1*01 2912 112*97 .,124* 0,142* 0,1207 0,0910 0,7682 0,326* 2912 407*00 0,093* 0,0649 0,2476 0,0106 0,7418 0 ,21*6 2912 492160 0,0261 0,033* 0,0038 0,7488 0,2*74 0,139* 2912 963700 0,026* 0,0314 0,0437 0,2343 0,7220 0 ,1*01 2912 60*900 0.09*0 0,07*2 0,0769 0,3975 0,5696 0,2044 2912 *96900 0,017* 0,0281 0,2224 0,2978 0,9198 0,146* 2912 *99209 0,0492 0,0*26 0,0134 0,3483 0,6383 0,2003 167

TABLE 3.-CONTINUED

INDUSTRV COMEANT Variance mean -squared c o n s t a n t RROPORTIONAL LOW a v e r a b e BSOLUTE CODS cooe estimate ERROR. BIAS BIAS CORRELATION ER 2912 442400 0.0499 0.0942 0 0041 0,2434 0.730J 0.1 2912 I11400 0.0472 0.0934 0 0149 0.34H 0.4437 0.1 2912 114300 0.0211 0.0217 0 0004 0,4091 0.9191 0,1 2912 4I4TOO 0.0441 0.0432 0 0314 0, 0,9414 0,1 2912 499400 o;0291 0.0327 0 0392 0,1490 0.7937 0.1 2912 701220 0i04!9 0.0919 0 0099 0,2794 0.7115 0,2 2912 734700 0.0492 0.0419 0 1411 0, 0.1312 0,1 2913 334400 0.0244 0.0234 0 0420 0, 0,(310 0,1 2913 414019 0.0241 0.0329 0 0299 0,2439 0,7104 0,1 2913 414200 0.0142 0.0319 0 1949 0,0191 0,7177 0,1 2913 414900 010394 0.0490 0 1373 0,2749 0.9191 0.1 2913 7U2I2 0.0310 0.0137 0 1739 0,0101 0,1194 0,1 3000 144700 0.4127 0.4173 0 1017 0, 0,1913 0,9 3000 209010 0.2093 0.1194 0 0249 0,4707 0,9024 0,3 3000 244300 0.0927 0,0471 0 1219 0,1749 0.4992 0,1 3000 317900 0.0497 0.0927 0 0021 0,9917 0.4319 0,2 3000 311100 0.0424 0.0710 0 1192 0,0941 0.7144 0.2 3000 449700 0 i 9144 0.9413 0 0144 0,7019 0.2H9 0.9 3000 417100 <3,31*9 0.1111 0 0414 0,4173 0.3143 0.4 3000 731491 0.0449 0.0141 0 0004 0,9747 0,4229 0.2 3241 19000 0.1114 0.1414 0 0211 0,3494 0,4249 0,3 3241 19900 0,2444 0.1079 0 0070 0,3090 0,4140 0,4 3241 304000 0.1411 0.1721 0 0493 0,2719 0.4127 0,3 3241 312100 0.0100 0.0179 0 1079 0,0147 0.1094 0,2 3241 371200 0.0419 0,0117 0 0391 0,0290 0,9397 0,2 3241 401124 0,1042 0,1274 0 0194 0,1921 0.7174 0,3 3241 424300 0.0902 0,1309 0 0299 0,1119 0,1424 0,3 3241 414100 0.0147 0,0412 0 2310 0,0093 0,7997 0,1 3241 444400 0.0922 0.0199 0 0041 0,2297 0,7441 0,2 3241 419210 0.0119 0,0191 0 1749 0,0047 ' 0,1114 0,2 3241 971000 0,3901 0,3117 0 0, 1.0000 0,4 168

TABLE 4 NAIVE PREDICTION MODEL NO, 1 PREDICTION RESULTS FOR THE TEARS 1*98 THROUGH 1947 USING THE SMOOTHING CONSTANTS INDICATED PERCENTAGE SOURCE OF PREDICTION ERRORS DUSTRV COMPANY v a r iance MEAN-SQUARED CONSTa PROPORTIONAL LOW AVERAGE ABSOLUTE SMOOTHING CODE CODE est imate ERROR • IAS BIAS CORRELATION ERROR CONSTANT 20*2 49030 0 0443 0,0424 0,0024 0.2179 0,77*7 0,1*49 0,0*0 20*2 229900 0 0392 0,0341 0,1228 0, 0,8772 0,1611 0, 2082 299100 0 0412 0,0978 0,0392 .0,0201 0,9447 0,2034 0,090 2082 321419 0 0497 0,0818 0,1474 0,2717 0,9804 0,2264 0,190 2082 348800 0 1984 0,1410 0,1142 0, 0,8898 0,2488 0, 2082 997100 0 2210 0,2874 0,3080 0, 0,6*20 0,4449 0, 2084 142400 0 1920 0,1977 0,0043 0,3998 0,439* 0,2420 0,129 2084 142900 0 0904 0,0784 0,1394 0,42*7 0,4349 0,2214 0,290 2084 218930 0 2093 0,2480 0,1399 0,2477 0,6168 0,3792 0,179 2084 974429 0 0444 0,0774 0,0194 0,4197 0,3646 0.2467 0,179 2084 431490 0 0931 0,0499 0,1393 0,1992 0,6699 0,2266 0,229 2200 12200 0 0942 0,0949 0,0442 0,3790 0;S76B 0,221* 0,090 2200 73900 0 0448 0,0929 0,1494 0,4980 0,3766 0.2070 0,029 2200 149440 0 3198 0,2878 0,0001 0, 0.9*99 0,4980 0, 2200 179400 0 1019 0,1138 0,1979 0, 0.8029 0,2*98 0, 2200 203400 0 1249 0,1148 0,0211 0, 0.9789 0,2914 0, 2200 322400 0 1079 0,1199 0,1493 0, 0,8347 0.2846 0, 2200 374790 0 3218 0,3904 0,0838 0,4487 0,4476 0,49*7 0,190 2200 438900 0 1049 0,1004 0,0142 0,2087 0.7791 0,2821 0,090 2200 494900 0 1190 0,1184 0,0084 0,3949 0,9*6* 0,26*6 0,100 2«12 93377 0 0144 0,0173 0,0010 0,3721 0i426* 0,1109 0,179 2912 192800 0 0234 0,0298 0,0297 0,3431 0,4312 0,1060 0,179 2912 187700 0 0314 0,0338 0,0480 0,1419 0,4106 0,1924 0,179 2912 312497 0 1248 0,1009 0,0447 0,0243 0,9291 0,2798 0,429 2912 407900 0 0939 0,0919 0,0414 0,1132 0,8493 0,1*39 0,090 2912 492180 0 0281 0,0249 0,0124 0,2294 0,7980 0,1332 0,090 2912 983700 0 0289 0,0291 0,0428 0,1102 0,8470 0,1348 0,079 2912 404900 0 0940 0,0994 0,1911 0, 0,8489 0.170* 0, 2912 494900 0 0179 0,0213 0,0241 0,2904 0,7296 0,130* 0,300 2912 499209 0 0492 0,0989 0,0109 0,3284 0,6611 0,2013 0,290 169 - u *- »-• K ►(Eo o o o o o - o o o O O O O O O O O O O O I ooec a o z K t> ooooooooooeooooooooaoooooeooooo a< O K « • n « U.** « o JWh««»»NM««hhcvo>K«ho9isoh»c»0nho0» zwmooo«c ® o n nht«D«onoh«» ® o o ® w o Ul l D O . O D • • •

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TABLE S

NAIVE p r e d i c t i o n m o d e l NO, 2

PREDICTION RESULTS FOR THE YEARS 1*88 THROUGH 19 7 USING THE SHOOTHINO CONSTANTS INDICATED PERCENTAOE SOURCE or PREDICTION RRORS INDUSTRY COMPANY v a r i a n c e MEAN-SOUARBD CONSTANT PROPORTIONAL LOW AVERAGE ABSOLUTE SMOOTHING CODE eooE ESTIMATE ERROR BIAS BIAS CORRELA ION ERROR CONSTANT 20S2 *•030 0,0*83 0,0634 0,0074 0,2189 0 773 0 2000 0,029 2082 22*900 0,0392 0,0361 0,1228 0, 0 *77 0 1*11 0, 20*2 299100 0,0812 0.0977 0,0340 0,0178 0 948 0 2031 0,029 20*2 321419 0,0897 0,0*12 0,1398 '0,2609 0 603 0 2269 0,079 2062 348800 0.19*4 0,1610 0,1142 0, 0 *89 0 2488 0, 20S2 997100' 0,2210 0,2*74 0,3080 0 , 0 6*2 0 4449 0, 20*8 182800 0.1*20 0,19*0 0,0031 0,4710 0 929 0 23*4 0,090 2 086 1*2*00 0,0904 0,0797 0,062* 0,4737 0 463 0 2193 0,129 2 0 8* 218930 0,2093 0,2629 0,1326 0,2202 0 *47 0 3680 0,079 20S* 978429 0,0886 0,0778 0,0048 0,6408 0 394 0 2489 0,079 2 088 831490 0,0931 0,0660 0,1093 0,1839 0 710 0 2193 0,100 2200 12200 0.0*82 0,0*67 0,0392 0,3679 0 993 0 2227 0,029 2206 73900 0,046* 0,0926 0,2000 0, 0 800 0 2096 0, 2200 14*480 0.319* 0,287* 0,0001 0 , 0 **9 0 4980 0, 2200 179800 0,1019 0,1138 0,1*79 0, 0 802 0 2*98 0, 2200 203800 0.124* 0,1148 0,0211 0. 0 *78 0 2916 0, 2200 322800 0.1079 0,1199 0,1693 0, 0 834 0 2*46 0, 2200 374790 0.321* 0,3933 0,1237 0.9219 0 394 0 4*71 0,090 2200 438*00 0.104* 0,1004 0,0144 0,1888 0 7*6 0 2827 0,029 2200 8*4900 0.11*0 0,1169 0.0003 0,2262 0 773 0 27*9 0,090 2*12 93377 0,0184 0,0183 0,0168 0,3669 0 616 0 1160 0,079 2912 192*00 0.0234 0,0294 0,0109 0,2738 a 719 0 1022 0,079 2*12 187700 0.0316 0,0327 0,0419 0,0*99 0 (62 0 14*2 0,079 2912 312697 0.124* 0,1062 0,0942 0,0743 0 *71 0 2860 0,390 2912 407*00 0.093* 0,0914 0,0341 0,1044 0 861 0 1*26 0,029 2912 492180 0.02*1 0,02*9 0,0088 0,1948 0 7*6 0 1321 0,029 2912 983700 0.02** 0,02*2 0,0186 0,1360 0 849 0 133* 0,090 2912 808900 0.0960 0,09*4 0,1911 0 , 0 848 0 170* 0, 2912 898900 0.017* 0,021* 0,0148 0,281* 0 703 0 1309 0,129 2912 899209 0.04*2 0,0609 0,0001 0,3399 0 6*4 0 2011 0,100 171

TABLE 5— C0NTINUEB

PERCENTASE SOURCE op p r e d i c t i o n e r r o r s NDUSTRV COMPANY VARIANCE MEANiSOUARED CONSTANT PROPORTIONAL a v e r a q e abs olute SMCOTHIt CODE CODE es t imate ERROR BIAS BIAS CORN ATION ERROR C0NSTAN1 2911 442S00 0.0499 0.0912 0,0133 0,2249 0, 0.1921 0.090 2911 443400 0.04720.0499 0,0021 0,3214 0, 0.1774 0,029 2912 414300 0.0241 0.0243 0,0342 0, 0. 0,1362 0, 2912 414900 0.0441 0.0421 0,0134 0, 0. 0,1466 0, 499400 0.0291 0.0249 0,1304 0,0429 0, 0,1424 0,2902911 2911 701220 0.04S9 0.0444 0,0499 0,1979 0, 0.2004 0,029 2911 734700 0.0492 0.0474 0,1494 0, 0, 0,1730 0. 2913 334400 0.0244 0.0244 0,1094 0, 0, 0,1414 0, 2911 444019 0.0242 0.0244 0,0373 0,2444 0, 0,1943 0,079 2911 444200 0.0342 0.034S 0,0494 0,1400 o. 0,1416 0,079 2911 414900 0.0394 0.0401 0,0440 0.2144 0. 0,1744 0,100 2911 711292 0.0310 0.0319 0,0497 0,0994 0. 0,1634 0,079 3000 148700 0.4427 0.4940 0,0014 0,1693 0. 0,9931 0,029 3000 209010 0.2093 0.1494 0,0034 0, 0. 0,3479 0, 3000 244300 0.0927 0.0944 0,0329 0,1624 0, 0.1994 0,100 3000 317900 0.0497 0.0922 0,0010 0,3177 0. 0,2234 0,100 3000 318100 0.0424 0,0790 0,0904 0,1692 0. 0.2243 0,100 3000 449700 0.9144 0,4414 0,0304 0, 0, 0.9464 0, 3000 447100 0.4429 0,4394 0,0934 0, 0. 0.6130 0. 3000 734491 0.0449 0,0424 0,0300 0,1143 0, 0.1912 0,029 3241 19000 0.1114 0,1244 0,0940 0,1169 0. 0,3019 0,090 3241 19900 0.2444 0,2944 0,0393 0,2917 0. 0,4209 0,079 324} 304000 0.1414 0,1937 0,0721 0,0997 0, 0.3699 0,079 3241 312000 0.0400 0,0794 0,0492 0,0429 0, 0,2422 0,129 3241 371200 0.0449 0,0420 0,0247 0, 0, 0,2202 0, 3241 401124 0.1042 0,1142 0,0322 0,1493 0. 0,2976 0,100 3241 424300 0.0902 0,0744 0,0021 0,9042 0. 0,2332 0,179 3241 434400 0.0347 0,0404 0,2242 0, 0. 0,1442 0, 3241 444400 0.0922 0,0439 0,0044 0, 0. 0,2230 324i 449210 0.0919 0,0490 0,1349 0, 0, 0.2364 0,0°' 3241 971000 0.3904 0,3194 0,0004 0. 0, 0.4102 0, 172 . I H -I-:.. .1. -I-:.. H I . 1 ;l i* ;l

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TABLE 7

m i c e r e l a t i v e PREDICTION PERFORMANCE WITH RESPECT TO MEAN-SQUARED e r r o r FOR SIXTY ONE COMPANIES OVER THE TEN YE»R PERIOD 1959 THROUGH 1967 USING THREE EXPONENTIAL SMOOTHING '■30ELS HITH BOTH SHIFTING SMOOTHING CONSTANTS AND FIXED s m o o t h i n g CONSTANTS

MODEL NO, 1 MODELnuDEU NO,i'0, t 2 MODELnuucw NO,' i ]9 INDUSTRY COMPANY BEST OVERALL BEST ALPHA b e s t o v e r a l l BEST ALPHA BEST OVERALL BEST ALPHA CODE CODE ERROR RANK ERROR RANK ERROR r a n k ERROR RANK ERROR RANK ERROR RANK 2002 4903Q 0 1475 5.0 0.0626 2.0 0.2353 6,0 0,0634 3,0 0.1446 4,0 0,0601 1,0 2002 229500 0 0420 5.0 0,0360 2.0 0.0360 2.0 0,0360 2.0 0.0498 6,0 0.0389 4,0 2002 255100 0 0735 5.0 0,0570 3,0 0.1023 6.0 0.0577 2.0 0.0672 4.0 0,0552 1.0 2002 321415 0 0997 4,0 0.0810 2.0 0.1068 6,0 0,0812 1.0 0.1062 5,0 0,0861 3,0 2002 340000 0 1705 5.0 0,1610 1.5 0.1766 4,0 0,1610 1.5 0.1828 6.0 0.1682 3,0 2002 557100 0 3879 4,0 0.2074 1.5 0.4085 5.0 0,2874 1.5 0.4099 6.0 0.3198 3,0 2000 162600 0 1616 4.0 0.1577 1,0 0.1730 6,0 0,1590 2.0 0.1631 5,0 0,1614 3,0 2000 162900 0 0802 4.0 0,0784 2.0 0,0899 5,0 0,0757 1,0 0.0901 6.0 0,0816 3,0 2000 216930 0 2924 5.0 0,2680 2.0 0.2881 4,0 0.2625 1.0 0.2972 6,0 0,2763 3,0 2000 576425 0 0979 5.0 0,0776 1.5 0.0866 4.0 0,0776 1,5 0.1021 6,0 0,0800 3,0 2000 431450 0 0846 5.0 0.0695 2.0 0.0712 3.0 0,0660 1.0 0.0870 6.0 0,0726 4,0 2200 12200 0 1051 4.0 0.0969 2.0 0.1067 6.0 0,0967 1.0 0.1059 5,0 0.0991 3,0 2200 73500 0 0526 2.5 0,0526 2,5 0.0524 2.5 0,0526 2.5 0.0558 6,0 0,0556 5,0 2200 149460 0 3901 5.0 0,2878 1.5 0,4039 6,0 0,2876 1.5 0.3892 4,0 0,2895 3,0 2200 175600 0 1495 4.0 0.1138 1.5 0.1555 6,0 0.1138 1.5 0.1508 5,0 0.1155 3,0 2200 203600 0 1148 2.5 0.1148 2,5 0,1148 2.5 0.1148 2,5 0,1189 5.5 0,1189 5.5 2200 322600 0 1947 4.0 0,1160 1.5 0.2144 6,0 0.1160 1.5 0.1975 5,0 0,1208 3,0 2200 374750 0 4628 4.0 0,3906 1.0 0,4743 5,0 0,3933 2.0 0.4772 6,0 0,4092 3,0 2200 430900 0 110 A 5.0 0,1006 3,0 0,1129 6,0 0.1004 2.0 0.1042 4.0 0,0969 1.0 2200 694500 0 1801 5.0 0,1184 3,0 0.1954 6,0 0,1165 2.0 0.1720 4,0 0,1140 1.0 2912 53377 0 0186 5.0 0,0174 2.0 0.0217 6,0 0,0183 4.0 0,0180 3,0 0,0168 1.0 2912 152000 0 0289 4.0 0,0259 3,0 0,0294 5,0 0.0253 2.0 0,0302 6,0 0,0245 1.0 2912 187700 0 0374 6.0 0,0338 3,0 0.0367 5.0 0,0327 2.0 0.0349 4,0 0,0316 1.0 2912 312657 0 1379 5.0 0,1008 1,0 0.1371 4,0 0,1063 3,0 0.1426 6.0 0,1028 2.0 2912 407900 0 0552 3.5 0.0519 2.0 0.0552 3,5 0,0515 1.0 0,0690 6,0 0,0583 5,0 2912 452180 0 0345 5.0 0.0269 3,0 0.0359 6,0 0,0268 2.0 0.0338 4.0 0,0259 1.0 2912 583700 0 0329 5.0 0,0291 2.0 0.0339 6,0 0.0292 3,0 0.0313 4,0 0,0280 1.0 2912 606500 0 0765 4.0 0.0594 1.5 0.0820 6,0 0,0594 1.5 0,0781 5.0 0,0612 3,0 2912 656500 0 0313 5.0 0.0213 2.0 0.0324 6,0 0,0219 3,0 0.0282 «,0 0.0200 1.0 2912 659205 0 0636 5.0 0.0585 2,0 0.0678 6.0 0,0605 3,0 0.0626 4,0 0.0548 1.0 2912 662800 0 0578 6.0 0.0511 2.0 0.0573 5,0 0,0512 3,0 0.S361 4,0 0.0492 1.0 2912 663600 0 0527 4,0 0.0453 2.0 0.0933 5,0 0.0455 3,0 0.0935 6,0 0,0446 1.0 2912 686300 0 0294 5.0 0.0260 2,0 0,0308 o;o 0,0263 3,0 0.0287 4,0 0,0256 1.0 2912 686700 0 0621 2.5 0,0421 2,5 0.0421 2.5 0.0421 2.5 0.0432 5.9 0,0432 5.5 2912 699600 0 0318 4.0 0,0261 1,0 0,0408 6,0 0,0285 3,0 0.0327 5,0 0,0266 2.0 2912 7Q1220 0 0539 5.0 0,0482 2.0 0.0954 6,0 0.0484 3,0 0.0519 4.0 0.0466 1.0 4xittAMioooirt ooooeooooooeooooon

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HWN00000 OOOOOOOOH H HH HHHHHHH WO«4*<«««lvl«4OOOOOOOO40 44 44 4 4444 auNNNNNNMnnnnnnnnnnnnnnnnnn3O00O>O> »0 OOOOOOOONNNNNNNNNNN THE THE OENOALL COEFFICIENT OF CONCORDANCE IS THE KENDALL 0,44 CHI SOUARE STATISTIC IS 199,39 RANK RANK TOTALS 249,0 133,0 319,0 1*2.9 279,0 122,5 176

t a b l e s PRICE RELATIVE PREDICTION PERFORMANCE WITH RESPECT TO CONSTANT BIAS FOR SIXTY ONE COMPANIES OVER THE TEN TEAR PERIOD 1958 THROUSH 1*67 USING THREE EXPONENTIAL SMOOTHING MODELS WITH BOTH SHITTING SMOOTHING CONSTANTS AND FIXED SMOOTHING CONSTANTS

MODEL NO, 1 MODEL NO, 2 MODEL NO, 3 INDUSTRY COMPANY BEST OVERALL BEST ALPHA BEST OVERALL BEST ALPHA BEST OVERALL BEST ALPHA CODE COOSERROR RANK ERROR RANK ERROR RANK ERROR r a n k ERROR RANK ERROR r a n k 2 0 S 2 4 9 0 3 0 0 0 1 2 8 5 . 0 0 . 0 0 0 1 1 . 0 0 0058 4,0 0,0005 3,0 0 0179 6,0 0.0004 2.0 2 0 02 2 2 9 5 0 0 0 0 0 2 9 1 . 0 0 , 0 0 4 4 3 , 0 0 0044 3,0 0.0044 3,0 0 0050 5.0 0,0073 6,0 20*2 255100 0 0 8 1 8 4 . 0 0 , 0 0 2 0 5 , 5 0 1,0 0,0020 5,5 0 0001 2.0 0.0002 3,0 2 0 * 2 3 2 1 4 1 5 0 0 1 2 9 4 . 0 0 , 0 1 2 1 3 , 0 0 0113 2.0 0,0110 1,0 0 0154 6,0 0,0142 5,0 2 o S 2 3 4 8 8 0 0 0 0 3 1 0 5 . 0 0 , 0 1 8 4 1 . 5 0 0300 4,0 0,0184 1.5 0 0357 6,0 0.0256 3,0 2 0 8 2 5 5 7 1 0 0 0 1 1 4 8 3 . 0 0 , 0 8 8 5 1 . 5 0 1196 4,0 0.0685 1.5 0 1475 6,0 0,1209 5,0 2088 1 6 2 6 0 0 0 0 0 0 3 2 . 0 0 ,.0 0 0 7 4 , 0 0 0002 1.0 0,0005 3,0 0 0019 5,0 0,0050 6,0 2088 1 6 2 9 0 0 0 0 0 5 0 3 . 0 0 . 0 1 0 6 5 . 0 0 0016 1,0 0.0048 2.0 0 0068 4,0 0,0136 8,0 2 0 8 4 2 1 8 9 3 0 0 8 4 4 9 4 . 00 , 0 3 6 3 3 , 0 0 0324 1,0 0,0348 2.0 0 0510 6.0 0.0506 5,0 2088 5 7 6 4 2 5 0 1 . 0 0 . 0 0 1 5 5 , 0 0 0012 4,0 0,0004 3,0 0 oooi 2.0 0,0035 6,0 2086 4 3 1 4 5 0 0 0 0 4 6 2 . 0 0 , 0 0 9 7 5 , 0 0 0031 1,0 0.0070 4,0 0 0052 3.0 0,0126 6,0 2 2 0 0 1 2 2 0 0 0 0 0 3 8 2 . 5 0 . 0 0 4 3 4 , 0 0 0026 1.0 0,0038 2,5 0 0056 6,0 0,0052 5,0 2 2 0 0 7 3 5 0 0 0 0 1 0 5 3 . 0 0 , 0 0 8 7 1 , 0 0 0105 3,0 0,0105 3,0 0 0137 6,0 0.0119 5,0 2 2 0 0 1 4 9 4 6 0 0 0 1 8 8 4 , 0 0 . 1 . 6 0 0202 5,0 0 . 1.5 4 0284 6,0 0,0017 3,0 2 2 0 0 1 7 5 6 0 0 0 0 3 0 1 4 . 0 0 . 0 2 2 5 1 , 5 0 0313 5,00,0225 1.5 0 0320 6,0 0,0242 3,0 2 2 0 0 2 0 3 6 0 0 0 0 0 2 4 2 . 5 0 , 0 0 2 4 2 , 5 0 002* 2.5 0,0024 2.5 0 0065 5,5 0,0065 5,5 2 2 0 0 3 2 2 6 0 0 0 0 0 2 1 2 . 0 0 , 0 1 9 2 4 , 5 0 0008 i;o 0.0192 4,5 0 0035 3,0 0.0240 6,0 2 2 0 0 3 7 4 7 5 0 0 0 0 1 8 1 . 0 0 . 8 3 2 7 5 , 0 0 0042 2.0 0,0487 6,0 0 0048 3,0 0,0191 4,0 2 2 0 0 4 3 8 9 0 0 0 0 0 2 3 6 . 0 0 , 0 0 1 6 3 , 0 0 0020 4.5 0,0014 2,0 0 0006 1,0 0,0020 4,5 2 2 0 0 6 9 4 5 0 0 0 0 0 3 0 4 . 0 0 , 0 0 1 0 2 . 0 0 0065 5.0 0 , 1.0 0 0105 6,0 0.0019 3,0 2 9 1 2 5 3 3 7 7 0 0 0 0 3 4 . 0 0 . 1 , 0 0 0024 4,0 0,0003 4,0 0 0003 4.0 0.0001 2.0 2 9 1 2 1 5 2 8 0 0 0 0 0 2 6 6 . 0 0 , 0 0 0 7 3 . 5 0 0007 3,5 0,0003 1,0 0 0017 5,0 0,0094 2.0 2 9 1 2 1 8 7 7 0 0 0 0 0 7 5 6 . 0 0 . 0 0 1 6 3 , 0 0 0056 5,0 0,0014 2,0 0 0056 4,0 0,0011 1,0 2 9 1 2 3 1 2 4 5 7 0 0 1 3 7 5 . 0 0 . 0 0 4 7 1 , 0 0 0102 4,0 0,0056 2,0 0 0172 6,0 0,0063 3,0 2 9 1 2 4 0 7 9 0 0 0 0 0 6 7 3 , 5 0 , 6 0 2 1 2 , 0 0 0067 3,5 0,0016 1,0 0 0161 6,0 0,0086 5,0 2 9 1 2 4 5 2 1 8 0 0 0 0 0 3 4 . 5 0 , 0 0 0 3 4 , 5 0 0010 6,0 0,0002 3,0 0 0001 2.0 0 , 1,0 2 9 1 1 5 8 3 7 0 0 0 0 0 2 4 4 . 0 0 , 0 0 1 2 3 , 0 0 0017 5,0 0,0005 1,0 0 0014 4,0 0,0007 2.0 2 9 1 2 4 0 4 5 0 0 0 0 0 5 0 2 . 0 0 , 0 0 9 0 4 . 5 0 0039 1,0 0,0090 4,5 0 0060 3,0 0,0198 6,0 2 9 1 2 4 5 6 5 0 0 0 0 0 9 4 4 . 0 0 , 0 0 0 5 2 . 5 0 0078 5,0 0,0003 1,0 0 0063 4.0 0,0005 2,5 2 9 1 2 6 5 9 2 0 5 0 0 0 1 9 6 . 0 0 , 0 0 0 6 4 , 0 0 0002 3,0 0 , 1,0 0 0008 5,0 0,0001 2,0 2912 462100 0 0 0 1 4 4 . 0 0 , 0 0 0 5 3 , 0 0 0003 1,5 0,0007 4,5 0 0003 1,5 0,0007 4,5 2 9 1 2 4 4 3 4 0 0 0 0 0 0 1 1 . 5 0 , 0 0 0 2 3 , 0 0 0006 5,0 o ,o q o i 1,5 0 0008 6,0 0,0004 4.0 2 9 1 2 4 8 6 3 0 0 0 2 . 0 0 , 0 0 0 1 6 , 0 0 2,0 0,0010 6,0 0 2,0 0,0003 5,0 0 , 0 0 0 4 0006 2,5 0,0006 2,5 1 0017 5,5 0.0017 5,5 2 9 1 2 6 8 6 7 0 0 0 0 0 0 4 2 . 5 i >5 0 2912 499400 0 0 0 1 2 1 . 0 0 , 0 0 1 6 3 , 0 0 0028 5,0 0,0037 6,0 0 0013 2.0 0.0024 4,0 2912 701220 0 0 0 0 5 3 . 5 0 , 0 0 1 0 5 , 0 0 0001 1,0 0,8022 4,0 0 0003 2,0 0.0005 3,5 177

TABLE 3>-C0NT|NUED

MODEL MO, 1 MODEL NO, 2 MOOEL NO, 3 i n d u s t r y COMPANY BEST OVERALL BEST ALPHA BEST OVERALL BEST AUPMA BEST OVERALL BEST *UPMA CODE CODE ERROR RANK ERROR RANK ERROR RANK ERROR RANK ERROR RANK ERROR RANK 2912 736700 0069 2.9 0.0069 .9 0.0069 2.9 0.0069 2.9 0.0083 .3 0,0093 9,5 2913 334600 0,0026 4.9 0.0026 4,9 0.0026 4.3 0,0026 4.9 0.0019 1.3 0.0015 1.3 2913 436069 0,0008 4.0 2.0 0. 2.0 0,0011 '6,0 0.0009 3.0 0. 2,0 0,0024 2913 636200 0,0097 6.0 0.0032 3.0 0,0093 9.0 0,0026 2.0 0.0077 4.0 1.0 2913 636900 0,0041 4.0 0.0021 3.0 0,0092 9.0 0,0018 1.0 0.0067 6.0 0.0019 2.0 2913 711282 0,0082 6.0 0.0028 3.0 0,0072 9.0 0.0022 2,0 0.0059 4.0 0,0019 1.0 0,0329 5.3 3000 138700 0.0001 1.9 0.0083 4.0 0,0001 1.3 0,0007 3.0 0.0929 3.5 3000 209080 0.0027 4.0 0.0007 2.9 0,0032 3.0 0.0007 2.9 0,0099 6.0 0. 1.0 0.0084 3000 266300 0.0104 6.0 0.0037 3.0 0.0099 4.0 0.0019 2.0 9.0 0.0018 1.0 3000 317900 0.0001 1.9 0.0016 9.0 0.0029 6.0 0.0001 1.3 0.0003 3.0 0,0008 4.0 3000 318100 0.0177 6.0 0.0098 2.0 0.0172 9.0 0,0068 3.0 0.0146 4.0 0,0097 1.0 3000 449700 0019 2.0 0.0094 4.0 0,0034 3.0 0.0183 6.0 0.0079 9.0 0,0008 1.0 0.0971 3000 487100 0496 3.9 0.0431 1.9 0,0496 3.3 0,0451 1.9 0.0607 6.0 9.0 4.0 3000 738691 1.9 0.0021 6.0 0,0001 3.0 0,0019 9.0 0. 1.9 0,0016 3241 19000 ,0097 3.0 0,0063 4.0 0,0084 5.0 0.0122 6.0 0.0040 1.0 0,0031 2.0 3241 19900 ,0001 1.0 0,0003 2.0 0.0040 4.0 0.0116 3.0 0.0022 3.0 0,0165 6,0 3241 304000 ,0191 6.0 0,0089 3.0 0,0169 3.0 0,0111 4.0 Q .0078 2.0 0.0098 1.0 3241 312800 ,0199 9.9 0.0091 3.0 0,0199 9.3 0,0038 2.0 0,0095 4.0 0,0026 1,0 3241 371200 ,0039 9.0 0.0024 2.9 0,0039 6.0 0,0024 2.3 0,0030 4.0 0.0022 1.0 3241 401126 ,0190 9.0 0.0093 3.0 0,0233 6,0 0,0037 2.9 0.0114 4.0 0,0027 1.0 3241 424300 ,0043 9.0 0.0006 2.0 0,0199 6.0 0,0002 1.0 0.0034 4.0 0,0011 3.0 3241 434800 ,0106 6.0 0,0091 2.9 0,0102 9.0 0,0091 2.9 0,0097 4.0 0.0067 1.0 3241 466600 ,0011 9.9 0.0006 3,3 0,0011 9.9 0,0003 3.9 0.0004 1.0 0. . 1,0 3241 489210 ,0194 6.0 0.0116 2.9 0,0178 9.0 0,0116 2.9 0,0196 6.0 0.0030 1.0 3241 971000 ,0002 6.0 0,0001 4,0 0,0001 4.0 0,0001 4.0 0, 1.5 0. 1,9

RANK TOTALS 239,9 190,0 218 iO 178,9 291,0 200,9

THE FRIEDMAN CHI SQUARE STATISTIC IS 18,76

THERE ARE 5 DEGREES OF FREEDOM

THE UNIT NORMAL APPROXIMATION IS 2,81

THE DENDALL COEFFICIENT OF CONCORDANCE IS 0,06

THE KENOALL CHI SQUARE STATISTIC IS 19,69

THE UNIT NORMAL APPROXIMATION IS 2,96

THE AVERAGE CORRELATION COEFFICIENT IS 0,05 178

TAILS 9 TRICE RELATIVE PREDICTION PERFORMANCE WITH RESPECT TO PROPORTIONAL BIAS FOR SIXTY ONE COMPANIES OVER THE TEN VEa R PERIOD L«SI THROUGH 1«67 US I NO THREE EXPONENTIAL SMOOTHING MODELS WITH BOTH SHIFTING SMOOTHING CONSTANTS AND FIXED SMOOTHING CONSTANTS

INDUSTRY COMPANY BEST OVERALL BEST ALPHA b e s t o v e r a l l BEST ALPHA BEST OVERALL BEST ALPHA CODE CODE ERROR RANK ERROR RANK ERROR RANK ERROR RANK ERROR RANK ERROR RANK aoS2 49030 0 0826 9.0 0.0137 2,0 0 1755 6; o 0.0139 3,0 0,0748 4,0 0. 1.0 2oB2 229900 0 0140 9.0 0. 2,5 0 2.5 0, 2,5 0.0197 6,0 0, 2.5 20*2 299100 0 0174 9.0 0.0012 3,0 0 0476 6,0 0,0010 2,0 0,0139 4,0 0, 1,0 20*2 321419 0 0382 4.0 0,0222 2,0 0 0461 6,0 0,0212 1.0 0.0422 5,0 0.0252 3,0 21)82 348800 0 0093 6.0 0, 2,0 0 0044 4,0 0, 2.0 0.0046 5,0 0. 2,0 2082 997100 0 1309 5.0 0. 2.0 0 1463 6,0 0. 2.0 0.1198 4.0 0, 2.0 2088 162600 0 0496 2.0 0,8567 4.0 0 0604 5,0 0,0749 6.0 0.0422 1.0 0.0559 3,0 208* 162900 0 0494 4.5 0,0337 1,0 0 0518 6,0 0,0358 3,0 0,0494 4,5 0.0347 2,0 2g8A 218930 0 0744 9,0 0,0664 3,0 0 0818 6,0 0,0978 2.0 0,0724 4,0 0.0577 1,0 2088 976429 0 0922 5.0 0,0478 3,0 0 0450 1,0 0,0497 4,0 0,0948 6.0 0.0498 2,0 208* 631490 0 0393 5.0 0,0136 2.0 0 0227 4,0 0,0121 1.0 0.0414 6,0 0,0142 3,0 2200 12200 0 0413 6.0 0,0367 2,0 0 0410 5,0 0,0355 1.0 0,0402 4.0 0,0397 3,0 2200 73500 0 2.8 0,0241 9,5 0 2,5 0. 2.5 0 • 2.5 0.0241 5.5 2200 149460 0 1307 5,0 0. 2,0 0 1334 6,0 0, 2,0 0,1088 4,0 0, 2,0 2200 179600 0 0543 9.0 0. 2.0 0 0591 6,0 0, 2.0 0,0537 4,0 0, 2.0 2200 203600 0 3.5 0. 3,5 0 3,5 0, 3,5 0. 3,5 0, 3,5 2200 322600 0 1156 4,0 0, 2.0 0 1348 6,0 0, 2.0 0.1172 5,0 0. 2,0 2200 374750 0 1727 2.0 0,1831 4,9 0 1808 3,,0 0,2091 6.0 0.1831 4,5 0,1014 1,0 2200 438900 0 0260 9.0 0,0210 3,0 0 0282 6,0 0.0190 2.0 0,0212 4.0 0, 1,0 2200 694500 0 1084 6.0 0,0467 3,0 0 0961 4,0 0,0264 2.0 0.0996 5,0 0,0186 1.0 2912 53377 0 0071 4.0 0,0065 2.0 0 0093 6,0 0.0067 3,0 0.0073 5.0 0,0058 1.0 2912 192800 0 0071 3.0 0,0089 4,0 0 0107 6,0 0.0069 1.0 0.0105 5,0 0,0070 2,0 2912 187700 0 0018 2.0 0,0048 6,0 0 0027 3,5 0,0031 5,0 0.0012 1.0 0.0027 3,5 2912 312657 0 0118 4.0 0,0024 1.0 0 0146 6,0 0.0079 3.0 0.0130 9.0 0,0035 2,0 2912 407900 0 1.5 0,0099 6,0 0 1,5 0,0054 5,0 0.0007 3.0 0,0014 4,0 2912 452190 0 0241 5.0 0,0062 2,0 0 0223 4,0 0,0052 1.0 0.0253 6,0 0,0063 3,0 2912 583700 0 0071 4.0 0,0032 2.0 0 0081 6,0 0,0040 3,0 0,0073 5,0 0,0029 1,0 2912 606500 0 0266 4.0 0. 2.0 0 0337 6.0 0. 2..0 0.0279 5,0 0, 2,0 2912 696900 0 0073 4.5 0,0053 2,0 0 0109 6,0 0,0062 3,0 0,0073 4,5 0,0045 1.0 2912 699209 0 0220 9.0 0,0192 2.0 0 0290 6,0 0.0203 3,0 0.0218 4.0 0,0156 1,0 2912 662800 0 0154 9.0 0,0112 2.0 0 0196 6,0 0.0115 3,0 0,0148 4,0 0,0085 1,0 2912 663600 0 0203 6.0 0,0134 2,0 0 0194 5,0 0,0146 3,0 0,0183 4,0 0,0131 1,0 2912 686300 0 0158 6.0 0,0093 3,0 0 0136 s;o 0. 1.5 0,0118 4,0 0. 1.5 2912 686700 0 3.5 0, 3,5 0 3,9 0, 3,5 0. 3,5 0, 3,5 2912 699600 0 0048 4.0 0,0006 2,0 0 0118 6,0 0,0018 3,0 0,0054 9,0 0.0005 1.0 2912 701220 0 0197 9.0 0,0114 3,0 0 0201 6.0 0,0076 1.0 0.0143 4.0 0.0103 2.0 179

TABLE 9.-CQNT1NUED MODEL NO, 3 COMPANY best o v e r a l l BEST ALPHA b e s t OVERALL BEST ALPHA BEST OVERALL BEST ALPHA INDUSTRY RANK COOE CODE . ERROR RANK ERROR RANK ERROR RANK ERROR RANK RANK err or 3,9 2912 736700 0, 3.9 0, 3,9 0. 3,9 0, 3.9 0. 3.9 0. 3,9 2913 334600 0, 3.9 0, 3,9 0. 3*5 0. 3,9 0. 3.9 0, 2913 486089 0,0091 6.0 0.0090 1.9 0,0060 3,0 0,0083 4.0 0.0087 9.0 0.0090 1.9 2913 686200 0,0004 2.0 0,0061 4.0 0.0002 1.0 0.0091 9.0 0,0006 3.0 0,0090 4.0 2913 686900 0,0238 6.0 0,0087 2,9 0.0178 9.0 0.0087 2.9 0,0136 4.0 0,0080 1.0 2913 711282 0.0003 1.9 0,0023 6,0 o;ooo3 1.9 0,0019 9,0 0,0004 3.0 0,0018 *.0 1.9 3000 188700 0,3379 9.0 0.0487 3,0 0,3694 6,0 0,0792 4,0 0. 1.9 0. 209080 0•0 70S 4.0 0. 2.0 0,0997 9,0 0, 2.0 0,1033 6,0 0, 2.0 3000 0,0089 1.0 3000 266300 0,0136 9.0 0,00*9 3,0 0.0191 6,0 0.0099 2.0 0.0120 4.0 3000 317900 0.0907 9.0 0,0296 3.0 0.0486 4.0 0.02*1 1.0 0 .0*18 6.0 0.02*7 2.0 2.9 3000 318100 0,0092 2.9 0,0127 9,0 0.0128 6,0 0.0124 4,0 0,0078 1.0 0.0002 0.3449 4.0 0,0696 3,0 0,3674 9.0 0, 1.9 0.3808 6.0 0, 1,9 3000 449700 2.0 3000 487100 0,9601 9.9 0, 2,0 0.9401 0, 2,0 0,8483 4.0 0, 3000 738691 0.0943 6.0 0,0078 3,0 0.0*31 9,0 0.0072 2.0 0,0489 4,0 0.0092 1,0 3241 19000 0.0*48 9.0 0,0179 3,0 0.0*17 6,0 0,0149 2,0 0,0488 4,0 0,0142 1,0 3241 19900 0.1334 9.0 0.8764 2.0 O.iJ*! 6,0 0.0830 3,0 0,0980 4,0 0.09*1 1.0 3241 304000 0,0396 4.0 0.0167 3,0 0.0*19 6,0 0.0193 2.0 0.0470 9,0 0,0122 1.0 4,0 0,0069 9.0 0,0076 6.0 0,0012 3,0 3241 312800 0, 1.9 0,0040 0l .. 3241 371200 0.0029 9.0 0. 2.0 0.0097 6,0 0, 2.0 0,0021 4,0 0. 2.0 3241 401126 0.0217 4.0 0,8194 2.0 0,0237 9,0 0.0171 3,0 0.0246 6.0 0,0096 1,0 3241 424300 0,1864 6.0 0,0336 2.0 0.0829 4,0 0,0397 3,0 0.1063i 9,0 0,0329 1,0 3241 434800 0.0009 9.0 0, 2.0 0.0042 6,0 0, 2,0 0.0004 4.0 0. 2.0 3241 466600 0.0164 4.0 0, 2.0 0,01*7 9,0 0. 2.0 0.0207 6,0 0. 2,0 3241 489210 0.0027 4.0 0. 1.9 0,0123 6,0 0, 1.9 0.0006 3.0 0,00*0 9.0 3241 971000 0,0116 6.0 0. 3,0 0. 3,0 0, 3,0 0. 3.0 0. 3,0

RANK TOTALS 266.0 172.0 291,9 166,0 298.9 127,0

THE FRIEDMAN CHI SOUARE STATISTIC IS 104,87 THERE ARE 8 DECREES OP FREEDOH

THE UNIT NORMAL APPROXIMATION IS 11,18

THE DENDALL COEFFICIENT OF CONCORDANCE IS 0,37 THE KENDALL CHI SOUARE STATISTIC IS lliiS7

THE UN|T NORMAL APPROXIMATION IS lli62 THE AVERAOE CORRELATION COEFFICIENT IS 0,36 180

table 10 PRISE RELATIVE PREDICTION PERFORMANCE NITH RESPECT '0 CORRELATION FOR SIXTY ONE COMPANIES OVER THE TEN YEAR PERIOD 1998 THROUGH 1«67 USINS THREE EXPONENTIAL SMOOTHING MOOELS WJTH SOTU SHIFT|NS SM00THIN8 CONSTANTS AND FIXED SMOOTHING CONSTANTS

MODEL NO. 1 MODEL NO 2 MODEL NO, 3 4DUSTRV COMPANY BEST OVERALL BEST ALPHA BEST OVERALL BEST ALRHA BEST OVERALL BEST iL HA CODE CODE ERROR RANK ERROR RANK ERROR RANK ERROR RANK ERROR RANK ERROR R NK 20*2 49030 0 0921 4.0 0.048 1.0 0 094 9,0 0 0490 2,0 0 0919 3,0 0.0997 .0 2082 229900 0 0291 1.9 0.031 4,9 0 031 4,9 0 0316 4,9 0 0291 1.9 0,0316 ,9 20*2 299100 0 0943 2.0 0.094 3,0 0 094 4,9 0 0947 4.9 0 0932 1.0 0,0990 .0 2082 321419 0 0486 3.9 0.047 2.0 0 049 6,0 0 0490 9.0 0 0486 3.9 0.0467 .0 2082 348800 0 1422 1.9 0.142 9,0 0 142 1.9 0 i486 9,0 0 1423 3.0 0.1426 .0 2082 997100 0 1426 2.0 0.198 9,0 0 142 2,0 0 1989 9,0 0 1426 2.0 0.1989 .0 2088 162600 0 1197 9,0 0.100 2.0 0 112 4,0 0 0836 1.0 0 1191 6,0 0,1009 ,0 2088 162900 0 0378 9.0 0.034 2.0 0 036 4,0 0 0391 3,0 0 0379 6,0 0,0333 ,0 2086 218930 0 1731 4.0. 0.169 1.0 0 173 6,0 0 1699 3,0 0 1738 9,0 0.1680 .0 2086 976429 0 0497 9.0 0.028 2,0 0 040 4,0 0 0279 1,0 0 0472 6,0 0,0307 ,0 2086 631490 0 0407 2.0 0.046 9,0 0 049 3,0 0 0469 6,0 0 0404 1.0 0,0498 ,0 2200 12200 0 0600 4.0 0.099 2.0 0 063 6,0 0 0974 3,0 0 0601 9,0 0,0942 .0 2200 73900 0 0421 4.9 0.019 2,0 0 042 4,9 0 0421 4,9 0 0421 4,9 0,0196 ,0 2200 149460 0 2406 1.0 0.287 9,0 0 290 2,0 0 2878 9,0 0 2920 3,0 0.2876 ,0 2200 179600 0 0691 2.0 0.091 9,0 0 069 2,0 0 0913 9.0 0 0691 2,0 0.0913 ,0 2200 203600 0 1124 3.9 0.112 3,9 0 112 3,9 0 1124 3,9 0 1124 3,9 0.1124 ,9 2200 322600 0 0770 2.0 0.096 9,0 0 078 3,0 0 0968 9,0 0 0768 1.0 0,0968 .0 2200 374790 0 2683 3.0 0.174 2,0 0 249 9,9 0 1399 1.0 0 2893 9.9 0.2887 ,0 2200 436900 0 0621 3.0 0,076 1,0 0 062 9,0 0 0800 2,0 0 0624 4,0 0,09*5 ,0 2200 694900 0 0487 2.0 0,070 3,0 0 092 9,0 0 0901 4,0 0 0619 1,0 0,0939 ,0 2912 933-'7 0 0112 9.0 6.010 3,9 0 009 1,0 0 0113 6,0 0 0104 2.0 0,0109 ,9 2912 192800 0 0192 6.0 0.016 1.0 0 016 3,9 0 0181 9,0 0 0180 3,9 0,0171 .0 2912 187700 0 0281 3.9 0.027 1.0 0 028 9,9 0 0282 9,9 0 0281 3,9 0,0278 .0 2912 312697 0 1124 9.9 0.093 3,0 0 112 4,0 0 0926 1.0 0 1124 9,9 0,0930 .0 2912 407900 0 0489 9.9 0.043 1.0 0 049 9,9 0 0443 2.0 0 0482 4,0 0,048i .0 2912 492180 0 0101 2.0 0,020 9,0 0 012 3,0 0 0214 6,0 0 0084 1,0 0,0196 ,0 2912 983700 0 0234 2.0 0.024 9,9 0 023 3,0 0 0247 9,9 0 0226 1,0 0,0244 ,0 2912 606900 0 0449 3.0 0.090 9,0 0 044 2,0 0 0904 9,0 0 0442 1,0 0,0904 ,0 2912 696900 0 0146 2.9 0,019 6,0 0 013 1.0 0 0194 »,0 0 0146 2,9 0,0190 ,0 2912 699209 0 0197 4.0 0.036 2,0 0 038 1,0 0 0402 6,0 0 0400 9.0 0,0391 ■ 0 2912 662100 0 0406 4.0 0.039 2,0 0 6,0 0 0390 1.0 0 0410 9.0 0,0430 ,0 2912 663600 0 0323 4.0 0,031 3,0 0 tit 9,0 0 0306 1,0 0 0344 6,0 0,0311 ,0 2912 686300 0 0136 1.0 0.016 2,0 0 017 4,0 0 0293 9,9 0 0169 3,0 0,0293 ,9 2912 686700 0 0419 3.9 0.041 3,9 0 041 3,9 0 0419 3,9 0 0419 3.9 0.0419 ,9 2912 699600 0 0298 4,0 0.023 1,9 0 026 6,0 0 0230 1.0 0 0260 9.0 0,0237 ,9 2911 701220 0 0937 1<0 0,039 3,9 0 039 2,0 0 0366 6,0 0 0373 9,0 0,0398 ,9 181

t’ABUE 10— CONTINUED MODEL NO, 1 MODEL NO, 2 ' MODEL NO, 3 OUSTRV COMPANY 8EST OVERALL 9EST ALPHA BEST OVERALL BEST alpha BEST OVERALL BEST ALpHA CODE eooE ERROR RANK ERROR RANK ERROR RANK ERROR RANK ERPOR RANK ERROR RANK 2712 736700 0,0406 3.9 0,0406 3,9 0,0406 3.9 0,0406 3,9 0,0406 3.9 0,0406 3,9 2713 334600 0,0220 3.9 0.0220 3,9 0.0220 3,9 0,0220 3,9 0,0220 3.9 0,0220 3,9 2713 4S6Q8S 0,0234 4.9 0.0211 2,9 0,0236 6,0 0,0173 1.0 0.0234 4.9 0.0211 2,9 2713 6S6200 0,0309 9.0 0.0282 1,0 0,0301 4,0 0,0271 3,0 0,0307 6.0 0,02*7 2.0 2713 666900 0,0309 6.0 0,0300 9,0 0,0288 2,0 0,0277 4.0 0.0287 1.0 0,02*3 3,0 2713 711282 0,0273 2.0 0,0276 4,9 0,0268 1,0 0.0278 6,0 0,0279 3.0 0.0276 4,9 3000 186700 0,2178 2.0 0.3941 4,0 0,2169 1,0 0,3780 3,0 0,4344 9,9 0,4344 9,9 3000 209080 0,1311 3.0 0.1848 9,0 0.1166 2,0 0.1846 9,0 0.1102 1.0 0,1*48 9,0 3000 266300 0,0474 9,9 0.0470 1,9 0.0<73 3.9 0,0473 3,9 0.0474 9.9 0.D470 1.9 3000 317700 0,0371 1.0 0.8910 4,0 0,0436 3,0 0,0960 6,0 0.0407 2.0 0,0942 9.0 3000 316100 0,0*97 9.9 0.0996 2.9 0,0994 1,0 0,0999 9,9 0,0997 4,0 0.0996 2,9 3000 447700 0,2420 3.0 0,4092 4,0 0,2386 2.0 0,4627 9,9 0,1926 1.0 0.4629 9.9 3000 487100 0,2771 2.0 0,7942 9,0 0.2791 2,0 0.7742 9,0 0.2791 2.0 0.7942 9,0 3000 738691 0.0397 1.0 0,0929 4,0 0,0364 3,0 0,0936 9,0 0,0398 2.0 0.0942 6.0 3241 19000 0,0*66 2.0 0,0982 6,0 0,0*93 1,0 0,0977 9.0 0.0*86 3,0 0,0979 4.0 3241 17900 0,2069 4,0 0,2090 3,0 0,2099 9,0 0,2000 2,0 0.2103 6.0 0,1973 1.0 3241 304000 0,1222 3.0 0,1271 4,9 0il2l2 2,0 0,1273 6,0 0■1180 1.0 0.1271 4,9 3241 312800 0,0720 9.9 0,0699 2,0 0.0720 9,9 0,0670 1.0 0.0708 3,0 0,0713 4,0 3241 371200 0,0786 2.0 0.0796 9,0 0.0774 1.0 0,0776 9,0 0.0787 3.0 0,0790 9,0 3241 401126 0,0*24 2.0 0.0931 4,0 0.0729 3,0 0,0939 6,0 0.0716 1.0 0.0934 9,0 3241 424300 0,0221 2.0 0,0364 9.0 0,030* 3,0 0,0386 6,0 0.0213 1.0 0,0397 4,0 3241 434600 0,0*12 4.0 0,0312 4.0 0,0288 1.0 0,0312 4,0 0.0312 4,0 0.0312 4,0 3241 466600 0,0731 2.9 0,0830 9,0 0; 0 7 31 2.9 0,0830 9,0 0,0687 1.0 0.0*30 9,0 3241 489210 0,0733 9.0 0,0733 9,0 0.0703 1,0 0,0733 9,0 0.0731 3,0 0,0708 2,0 3241 971000 0,3060 1.0 0.3197 4,0 0.3197 4,0 0,3197 4,0 0.3197 4,0 0.3197 4.0 NK TOTALS 177.9 208,0 203.9 246,9 176.9 227,0

THE FRIEDMAN OH! SQUARE STATISTIC IS 1,02 THERE ARE 9 DESREES OR FREEDOM THE UNIT NORMAL APPROXIMATION [J 0,66 THE DENDALL COEFFICIENT OF CONCORDANCE IS 0,03 THE KENDALL CHI SQUARE STATISTIC IS 0,43 THE UNIT NORMAL APPROXIMATION IS 1,07 THE AVERAOE CORRELATION COEFFICIENT IS 0,02 TABLE IX PRICE 1ELATIVE PREDICTION FERIOWARCE RAKING OF rag* EXPONENTIAL SWOXH2BC MODELS, BODIPOtAXIK SHIFTING AS HELL AS FIXED SMOOTHING CONSTANTS, BT IHDOSTRX FOB THE TEN TEAR PERIOD 1958 THROUGH 1967

MODEL RANKINGS TESTS OF SiqnnCAHCE TOTAL PRICE RELATIVE PREDICTION ERRORS AHD MODEL NO. 1 ADEL MO. 2 MODEL t10. 3 CHI— k e h d a i x KENDALL PREDICTION ERROR BE- SMOOTHING CONSTANT SMOOTHING CONSTANT SMOOTHING CONSTANT SQUARE COEFFICIENT OF CHI-SQOARE COMPOSITION BT IHDCSTET SHIFTING FIXED SHIFTING f i x e d SHIFTING FIXEDSTATISTIC CONCORDANCE STATISTIC

MEAN-SQUARED ERROR: Beverage Producers 4.5 2.0 4.5 1.0 6.0 3.0 41.75* 0.77 42.22* Textile Producers 4.0 2.0 6.0 1.0 5.0 3.0 25.70* 0.62 27-72* Petroleua Producers 5.0 2.0 6.0 3.0 4.0 1.0 61.92* 0.S9 64.69* Tire and Rubber Goods 5.0 2.0 6.0 3.0 4.0 1.0 33.77* 0.86 34.38*. Ceaent Producers 5.0 2.0 6.0 3.0 4.0 1.0 47.62* 0.90 49.55* All companies 5.0 2.0 6.0 3.0 4.0 1.0 187.80 0.64 195.39 CONSTANT BIAS: Beverage Producers 3.0 4.0 1.0 2.0 5.0 6.0 16.23* 0-30 16.53* Textile Producers 3.5 2.0 3.5 1.0 6.0 5;0 8.92 0.21 9.49 Petroleua Producers 6.0 3.0 4.0 1.0 5.0 2.0 9.77 0.10 10.47 Tire and Rubber Goods 3.0 4.0 5.0 2.0 6.0 1.0 4.27* 0.11 4.38* Ceaent Producers 5.0 2.0 6.0 4.0 3.0 1.0 26.57 0.50 27.72 All companies 5.0 2.0 4.0 1.0 6.0 3.0 18.76 0.06 19.69 PROPORTIONAL BIAS: Beverage Producers 5.0 2.0 6.0 3.0 4.0 1.0 25.61J* 0.49 26.94^* Textile Producers 5.0 3.0 6.0 2.0 4.0 1.0 12.38* 0.31 13.83* Petroleua Producers * 4.0 2.5 6.0 2.5 5.0 1.0 24.20* 0.23 25.56* Tire and Rubber Goods 5.0 3.0 6.0 2.0 4.0 1.0 22.43* 0.60 23.88* Ceaent Producers 4.0 2.0 6.0 3.0 5.0 1.0 26.64* 0.52 28.41* All Companies 5.0 3.0 6.0 2.0 4.0 1.0 104.57 0.37 111.57

^Chi-square values above 20.52 are significant at the 99.9 percent level of confidence. *Chi-square values above 15.09 are significant at the 99 percent level of confidence. Chi-square values above 11.07 are significant at the 95 percent level of confidence. 182 TABLE 11 (continued)

uwn Mwrnay; TESTS OF SIGNIFICANCE TOTAL PUCE RELATIVE PREDICTION EUDBS AID MODEL » - 1 M H g L SO. 2 MODEL HO. 3 CHI- KENDALL KENDALL PREDICTION ERROR DE- SMDOtHIBG COKSMT SJBOTHIBC CONSTANT SMOOTHING CONSTANT SQDARE COEFFICIENT OF CHI-SQOARE catosmai sr imuksiiT SHIFTING H if.il SHIFTING FTTFn SHIFTING FIXED STATISTIC CONCORDANCE STATISTIC m i m i T T f ST1E 8GTH: Beverage Producers 2.0 1.0 6.0 3.0 4.0 5.0 2.27 0.05 2.48 Textile Producers 1.0 2.0 5.5 4.0 3.0 5.5 3.41 0.09 3.89 Petroleun Producers 5.0 1.0 3.0 6.0 4.0 2.0 2-41** °-02 2-57** Tire and lubber Goods 2.5 4.0 1.0 6.0 2.5 5.0 12.09 0.33 13.12^ Ceaent Producers 3.0 5.0 1.0 6.0 2.0 4.0 10.39 0.21 11.59 All Cneqianles 1.0 4.0 3.0 6.0 2.0 5.0 8.82 0.03 9.63

.Cli-sqoare values above 20.52 ace significant at the 99.9 percent level of confidence. ^Chi-square values above 15.09 axe significant at the 99 percent level of confidence. Chi-square values above 11.07 axe tlpiificait at the 95 percent level of confidence.

\, TABLE 12

COMPARATIVE PRICE RELATIVE PREDICTION RESULTS USING FIXED SMOOTHING CONSTANTS VERSUS SHIFTING SMOOTHING CONSTANTS FOR SIXTY-ONE COMPANIES FROM 1958 THROUGH 1967

Average Mean-! Squared Error Smoothing Model Smoothin ? Constants Difference. Significance Tests Type Shifting Fixed Absolute Percentage z-valuec t-valued

Constant - No. 1 0.142 0.119 0.023 16.21 -6.57a 6.22b

Linear Trend - No. 2 0.148 0.120 0.028 18.9% -6.39® ' 5.84b

Constant Exponent - No. 3 0.140 0.120 0.020 14.3% -6.45® 6.39b

a. Values of z ^ -5.50 are significant at the 99.99999 percent confidence level. b. Values of t > 3.232 are significant at the 99.9 percent confidence level. c. The Wilcoxon matched-pairs, signed ranks test was used to compare the mean-squared error foreach respective model where each paired observation consisted of the mean-squared prediction error for that model when the best fixed smoothing constant was used and the mean-squared error that resulted from using the smoothing constant which had performed best in the previous ten years. For sample sizes larger than 25, the sum of the like-signed ranks (T) is approximately normally distributed, so the standardized normal variate is

z = (T - M)/S, where TABLE 12— Continued

H = N(N + l)/4

- J m . + 1 H 2 N + 1) 24

N = the number of paired observations, 61.

d. Hie t-test can also be applied to the paired observations by considering the differences, where

D± = (MSEis - MSEif),

M = zi=l Di /N

S2 = E? D?/N - M2 x=l i N = the number of paired observations, 61.

MSE^s = the mean-squared prediction error for the company 1 using shifting smoothing constants.

MSEj^ = the mean-squared prediction error for the company 1 using a fixed smoothing constant for each company.

Null hypothesis: There is no difference between the average mean-squared prediction error using shifting smoothing constants for each company and the average mean-squared prediction error using the optimal fixed smoothing constant for each company.

Research hypothesis: The average mean-squared prediction error using shifting smoothing constants for each company is greater than the average mean-squared prediction error using a fixed smoothing constant for each company. 185

The original observations of the mean-squared prediction error for each company are in tables one through six. TABLE 13

COMPARATIVE ABSOLUTE PREDICTION ERRORS USING FIXED SMOOTHING CONSTANTS VERSUS SHIFTING SMOOTHING CONSTANTS FOR SIXTY-ONE COMPANIES FROM 1958 THROUGH 1967

Mean Average Absolute Error Q Smoothing Model Smoothing Constants Difference Significance Tests Type Shifting Fixed Absolute Percentage z-value t-value

Constant - No. 1 0.271 0.248 0.023 8.52% -6.46a 7.57b

Linear Trend - No. 2 0.275 0.250 0.025 9.09% -6.133 5.93b

Constant Exponent - No. 3 0.268 0.247 0.021 7.84% -6.23a 6.76b

a. Values of z < -5.50 are significant at the 99.99999 percent confidence level. b. Values of t > 3.460 are significant at the 99.95 percent confidence level for a one-tailed test. c. See the notes to Table 12 for a more nearly complete discussion of the Wilcoxen test and the t-test for differences in means.

Null hypothesis: There is no difference between the mean average absolute prediction error using shifting smoothing constants for each company and the mean average absolute pre­ diction error using the optimal fixed smoothing constant for each company. 186 TABLE 13— Continued

Research hypothesis: The mean average absolute prediction error using shifting smoothing constants for each company is greater than the mean average absolute prediction error using the optimal fixed smoothing constant for each company.

The original observations of the average absolute prediction error for each company are in tables one through six. 188

TABLE 14

KOLMOGOROV-SMIRNOV ONE SAMPLE TEST FOR THE NORMALITY OF THE CONSTANT EXPONENT SMOOTHING MODEL’S PREDICTIONS FOR SIXTY-ONE COMPANIES FROM 1958 THROUGH 1967 WHERE SHIFTING SMOOTHING CONSTANTS ARE USED

Observed Cumulative Theoretical Interval Proportions Proportion • Proportion Difference ofe z a °i Ci Fi Di Less than -1.15 0.0541 0.0541 0.1250 0.0709 -0.68 0.0934 0.1475 0.2500 0.1025 -0.32 0.1131 0.2606 0.3750 0.1144 0.00 0.1934 0.4540 0.5000 0.0460 0.32 0.2738 0.7278 0.6250 0.1028 0.68 0.0951 0.8229 0.7500 0.0729 1.15 0.1246 0.9475 0.8750 0.0725 + 09 0.0525 1.0000 1.0000 0.0000

aThe smoothing model price relative predictions were standardized by means of the transformation

Zj = (Xj - M)/S, where

Xj is a particular price relative prediction (1 £ j <_ 610)

M is the mean of all such predictions

S is the standard deviation of the predictions

C^ is the cumulative proportion of the standardized price relative

predictions that are less than the upper bound of interval

i (i - 1, 2, ..., 8)

F^ is the theoretical proportion of the standardized price relative

predictions that are less than the upper bound of interval i (i «

1, 2, ..., 8) assuming the price relative predictions are normally

distributed. 189

Is the absolute value of the difference between the theoretical

cumulative proportion and the observed cumulative proportion.

Null hypothesis: The observed cumulative proportion of price relative

predictions is not unequal to the theoretical cumulative

proportion for a normally distributed random variable.

Research hypothesis: The smoothing model price relative predictions are

not normally distributed.

The maximum value of D^, denoted MD, is equal to 0.1144from the above table. At the 99% confidence level, we have

Pr(MD > 1.63/V610)■ 0*01, or

Pr(MD >. 0.021) - 0.01

The null hypothesis must be rejected at the 99% confidence level, and the research hypothesis is accepted. 190

TABLE 15

KOLMOGOROV-SMIRNOV ONE SAMPLE TEST FOR THE NORMALITY OF THE CONSTANT EXPONENT SMOOTHING MODEL'S PREDICTION ERRORS FOR SIXTY-ONE COMPANIES FROM 1958 THROUGH 1967 WHERE SHIFTING SMOOTHING CONSTANTS ARE USED

Observed Cumulative Theoretical Interval Proportions Proportion - Proportion Difference of za F. °i Ci l Di Less than . -1.15 0.0770 0.0770 0.1250 0.0480 -0.68 0.1361 0.2131 0.2500 0.0369 -0.32 0.1820 0.3951 0.3750 0.0201 -0.00 0.1557 0.5508 0.5000 0.0508 0.32 0.1361 0.6869 0.6250 0.0619 0.68 0.1279 0.8148 0.7500 0.0648 1.15 0.0852 0.9000 0.8750 0.0250 + » 0.1000 1.0000 1.0000 0.0000

aThe smoothing model price relative prediction errors are standardized by the transformation

= (Xj - M)/S, where

Xj is a particular prediction error (1 <_ j £ 610)

M is the mean of all such prediction errors.

S is the standard deviation of the prediction errors

C^ is the cumulative proportion of the standardized price relative

prediction errors that are less than the upper bound of interval

i(i - 1, 2, 8)

F^ is the theoretical proportion of the standardized price relative

prediction errors that are less than the upper bound of interval

i(i « 1 , 2, ..., 8) assuming the errors are normally distributed.

D^ is the absolute value of the difference between the theoretical 191

cumulative proportion and the observed cumulative proportion.

Null hypothesis: The observed cumulative proportion of price relative

prediction errors is not unequal to the theoretical

cumulative proportion for a normally distributed

random variable.

Research hypothesis: The smoothing model's price relative prediction

errors are not normally distributed.

The maximum value of D^, denoted MD, is equal to 0.0648 from the above

table.. At the 99% confidence level, we have

Pr(MD _> 1.63Aj/610) = 0.01, or

Pr(MD > 0.021) - 0.01

•The null hypothesis must be rejected at the 99% confidence level, and the research hypothesis accepted. 192

TABLE 16

KOLMOGOROV-SMIRNOV ONE SAMPLE TEST FOR THE NORMALITY OF THE CONSTANT EXPONENT SMOOTHING MODEL’S PREDICTIONS FOR SIXTY-ONE COMPANIES FROM 1958 THROUGH 1967 WHERE FIXED SMOOTHING CONSTANTS ARE USED

Observed Cumulative Theoretical Interval Proportions Proportions Porportion Difference of za C. °i i Fi Di Less than -1.15 0.0705 0.0705 0.1250 0.0545 -0.68 0.0934 0.1639 0.2500 0.0861 -0.32 0.0902 0.2541 0.3750 0.1209 , 0.00 0.2180 0.4721 0.5000 0.0279 0.32 0.2262 0.6983 0.6250 0.0733 0.68 0.1246 0.8229 0.7500 0.0729 1.15 0.1082 0.9311 0.8750 0.0561 + “ 0.0689 1.0000 1.0000 0.0000

The smoothing model price relative predictions were standardized by means of the transformation

Zj = (X^ - M)/S, where

Xj is a particular price relative prediction (1 <, j <. 610)

M is the mean of all such predictions

S is the standard deviation of thd predictions

C^ is the cumulative proportion of the standardized price relative

predictions that are less than the upper bound of Interval

i (1 = 1, 2...... 8)

F^ is the theoretical proportion of the standardized price relative

predictions that are less than the upper bound of interval i (i =

1, 2, 8) assuming the price relative predictions are normally

distributed. 193

Is the absolute value of the difference between the theoretical

cumulative proportion and the observed cumulative proportion.

Null hypothesis: The observed cumulative proportion of price relative

predictions is not unequal to the theoretical cumulative

proportion for a normally distributed random variable.

Research hypothesis: The smoothing model price relative predictions are

not normally distributed.

The maximum value of D^, denoted MD, is equal to 0.1209 from the table above. At the 99% confidence level, we have

Pr(MD > 1.63/-/610) = 0.01, or

Pr(MD 21 0.021) = 0.01

The null hypothesis must be rejected at the 99% confidence level, and the research hypothesis is accepted. 194

TABLE 17

KOLMOGOROV-SMIRNOV ONE SAMPLE TEST FOR THE NORMALITY OF THE CONSTANT EXPONENT SMOOTHING MODEL'S PREDICTION ERRORS FOR SIXTY-ONE COMPANIES FROM 1958 THROUGH 1967 WHERE FIXED SMOOTHING CONSTANTS ARE USED

Observed Cumulative Theoretical Interval Proportions Proportion - Proportion Difference - a of z °i Ci Fi Di Less than -1.15 0.0852 0.0852 0.1250 0.0398 -0.68 0.1311 0.2163 0.2500 0.0337 -0.32 0.1803 0.3966 0.3750 0.0216 0.00 0.1426 0.5392 0.5000 0.0392 0.32 0.1361 0.6753 0.6250 0.0503 0.68 0.1492 0.8245 0.7500 0.0745 1.15 0.0853 0.9098 0.8750 0.0348 + “ 0.0902 1.0000 1.0000 0.0000 aThe smoothing model price relative prediction errors are standardized by the transformation

Zj » Xj - M)/S, where

Xj Is a particular prediction error (1 _< j _< 610)

M Is the mean of all such prediction errors.

S is the standard deviation of the prediction errors

C^ is the cumulative proportion of the standardized price relative

prediction errors that are less than the upper bound of interval

1(1 - 1 , 2, ..., 8)

F^ is the theoretical proportion of the standardized price relative

prediction errors that are less than the upper bound of interval

i(i - 1, 2, 8) assuming the errors are normally distributed.

D^ is the absolute value of the difference between the theoretical 195

cumulative proportion and the observed cumulative proportion.

Null hypothesis: The observed cumulative proportion of price relative

prediction errors is not unequal to the theoretical

cumulative proportion for a normally distributed

random variable.

Research hypothesis: The smoothing model's price relative prediction

errors are not normally distributed.

The maximum value of , denoted MD, is equal to 0.0745 from the above table. At the 99% confidence level, we have

Pr(MD j> 1.63/V^F) = 0.01, or

Pr(MD >_ 0.021) = 0.01

The null hypothesis must be rejected at the 99% confidence level, and the research hypothesis accepted. 196

TABLE 18

CONSTANT EXPONENT SMOOTHING MODEL'S MEAN-SQUARED ERROR VERSUS PRICE RELATIVE VARIANCES AS MEASURES OF UNCERTAINTY

Wilcoxon matched-pairs signed-ranks test

T = 749, where T is the smaller sum of like signed ranks.

When the number of observations is greater than 25, T is approximately normally distributed with

Mean M = N(N+1), for N cases, and

Standard deviation S = . /N(N+1)(2N+1) ■V* 24

The standardized variate becomes,

Z = ? = T - (N2 + N)/4 s N(N+l)(2N+1) 24

Null hypothesis: There is no difference between the mean mean-squared prediction

error and the mean price relative variance for sixty-one

companies.

Research hypothesis: The mean mean-squared prediction error is greater

than the mean price relative variance for sixty-one

companies.

For N = 61, and T = 749, Z = -1.411

Pr[Z < -1.411] - 0.0793

The null hypothesis cannot be rejected at the 95% confidence level, so the null hypothesis is accepted. t - test for paired observations

For paired observations, the differences are tested where Di = (MSEi - VAR*)

M -

s2 - ei . 1d12/n “2 -

N ■ the number of pairs, 61.

MSE^ « the mean-squared prediction error for the company i

VAR^ - the variance (estimate) of price- relatives about the mean

price relative for company 1

SM ■ the standard error of the mean, which is equal to S/

M - E(M) ^ - 1 “ -SM---

t, - 1.412 60 From tables of the t distribution:

Prftgo 1.303] » 0.10

and

Pr[t6Q >, 1.684] - 0.05

The null hypothesis, as stated above, cannot be rejected at the

95% confidence level, so the null hypothesis is accepted. 198

TABLE 19

ANALYSIS OF COVARIANCE TABLE FOR THE PRICE RELATIVE VARIANCE AND THE CONSTANT EXPONENT SMOOTHING MODEL’S MEAN-SQUARED PREDICTION ERROR FOR SIXTY-ONE COMPANIES USING SHIFTING SMOOTHING CONSTANTS

Source of CM Mean

* '2 Variation df Exy Iy2 df . zy Square

Among means 4 0.3877 0.4080 0.4490 4 0.0196 0.0049

Within groups 56 0.8751 0.9136 1.0138 55 0.0601 0.0011

Total 60 1.2628 1.3216 1.4628 59 0.0797

Model tested: Y . . - y + t . + g (X.. - X) + E.., where ij 1 ij ij 1 ■ 1, 2, ..., t (treatment groups) and

J ■ I, 2, (cases for group i)

Regression coefficients: Means 1.0525

Within 1.0439

Total 1.0466

F Ratios:

Mean-squared error, mean effects 6.2014 (4, 56 degrees of freedom)

Price relative variance, mean effects 6.2008 (4, 56)

Mean-squared error, mean effects

after adjusting for regression 4.4843 (4, 55)

Within regression coefficient 872.8520 (1, 55)

Pr (F^ 55 > 3.68) - 0.01

Pr (F^ 55 > 2.54) - 0.05

Pr (Fx 55 > 7.12) - 0.01 199

TABLE 20

ANALYSIS OF COVARIANCE TABLE FOR THE PRICE RELATIVE VARIANCE AND THE CONSTANT EXPONENT SMOOTHING MODEL'S MEAN-SQUARED PREDICTION ERROR FOR SIXTY-ONE COMPANIES USING FIXED SMOOTHING CONSTANTS

Source of Mean 2 0 '2 Variation df Ex2 Exy zy df Ey Square

Among means 4 0.3877 0.3742 0.3675 4 0.0063 0.0016

Within groups 56 0.8751 0.8600 0.8706 55 0.0255 0.0005

Total 60 1.2628 1.2342 1.2381 59 0.0318 i

Model tested: Y = y + + g (X^ - X) + E ^ , where

i = 1, 2, t (treatment groups) and

j = 1, 2, ...» n^ (cases for each group 1)

Regression coefficients: Means 0.9654

Within 0.9827

Total 0.9774

F Ratios:

Mean-squared error, mean effects 6.2014 (4, 56 degrees of freedom) Price relative variance, mean effects 5.9104 (4, 56)

Mean-squared prediction error,,, mean

effects after adjusting for regression 3.4062 (4, 55)

Within regression coefficient 1823.7726 (1, 55)

Pr (F^ 55 > 3.68) - 0.01

Pr (F4 55 > 2.54) = 0.05

Pr (Fj 55 > 7.12) = 0.01 200

TABLE 21

ANALYSIS OF COVARIANCE TABLE FOR PRICE RELATIVES AND THE CONSTANT EXPONENT SMOOTHING MODEL'S PRICE RELATIVE PREDICTIONS FOR SIXTY-ONE COMPANIES FROM 1958 THROUGH 1967 ' USING SHIFTING SMOOTHING CONSTANTS

Source of '2 Mean Variation df Ex2 Exy . Zy2 df zy Square

Among means 4 0.3583 0.2385 2.5377 4 2.7166 0.6791

Among groups 605 8.7140 -2.8304 67.9500 604 67.0306 0.1110

Total 609 9.0723 -2.5918 70.4877 608 69.7472

Model tested: = y + + 8 ( X ^ - X) + E ^ , where

i = 1, 2, t (treatment groups) and

j * 1, 2, . ^ (cases for group i)

Regression coefficients: Means 0.6657 Within -0.3248 Total -0.2857

F Ratios: Price relatives, mean effects 6.2191 (4, 605 degrees of freedom) Predicted price relatives, mean effects 5.6486 (4, 605) Price relatives, mean effects after adjustment for regression 6.1196 (4, 604) Within regression coefficient 8.2839 (1, 604)

Pr (F4 120 > 4.95) = 0.001 Pr (F4*00 > 4.62) = 0.001

Pr (Fj^* 12q > 6*85> = °*01 Pr CF1 0 0 > 6 *64> = °*01 Pr > U *38) a °-001 201

TABLE 22

ANALYSIS OF COVARIANCE TABLE FOR PRICE RELATIVES AND THE CONSTANT EXPONENT SMOOTHING MODEL'S PRICE RELATIVE PREDICTIONS FOR SIXTY-ONE COMPANIES FROM 1958 THROUGH 1967 USING FIXED SMOOTHING CONSTANTS

Source of Mean Variation df Ex2 Exy V df v '2 Square Among means 4 0.5534 0.5464 2.5377 4 2.3147 0.5787

Among groups 605 4.5079 0.8385 67.9500 604 67.7940 0.1122

Total 609 5.0613 1.3849 70.4877 608 70.1087

Model tested: Y. . = p + t . + 6 (X.. - X) + E.., where ij i ij iJ 1=1, 2, t (treatment groups) and

j = 1, 2 (cases for group i)

Regression coefficients: Means 0.9872

Within 0.1860

Total 0.2736

F Ratios:

Price relatives, mean effects 18.5695 (4, 605 degrees of freedom) Predicted price relatives, mean effects 5.6486 (4, 605)

Price relatives, mean effects after

adjustment for regression 5.1557 (4, 604)

Within regression coefficients 1.3896 (1, 604) 202

TABLE 23

KOLMOGOROV-SMIRNOV ONE SAMPLE TEST FOR THE NORMALITY OF GROWTH MODEL PREDICTION ERRORS FOR SIXTY-ONE COMPANIES FROM 1958 THROUGH 1967

Observed Cumulative Theoretical Interval Proportion Proportion Proportion Difference of za 0. F. i Ci X Di Less than -1.15 0.0934 0.0934 0.1250 0.0316 -0.68 0.1066 0.2000 0.2500 0.0500 -0.32 0.1262 0.3262 0.3750 0.4880 0.00 0.1410 0.4672 0.5000 0.0328 0.32 0.1525 0.6197 0.6250 0.0053 0.68 0.1475 0.7672 0.7500 0.0172 1.15 0.1344 0.9016 0.8750 0.0266 + 00 0.0984 1.0000 1.0000 0.0000 aThe price relative prediction errors were standardized by means of the trans formation

Zj = (Xj - M)/S, where

Xj is a particular prediction error (1 '<_ j _< 610)

M is the mean value of the prediction errors,

S is the standard deviation of the prediction errors,

C^ is the cumulative proportion of the standardized price

relative prediction errors that is less than the

upper bound of interval i (i - 1, 2, ..., 8)

F^ is the cumulative proportion of a normally distributed

random variable that is less than the upper bound of

interval i (i ■ 1, 2 ...... 8)

is the difference between the observed cumulative pro­

portion and the cumulative proportion of a normally distributed random variable.

Null hypothesis: The observed cumulative proportion of price relative

prediction errors is equal the cumulative proportion

of a normally distributed random variable.

Research hypothesis: The prediction errors are not normally distributed.

The maximum value of D^, denoted MD, is equal to 0.0500 from the above

table. At the 99% confidence level, we have

Pr(MD >. I.63/4/ 6IO) - 0.01, or

Pr(MD J> 0.021) = 0.01

The null hypothesis must be rejected at the 99% confidence level, and the research hypothesis accepted. 204

TABLE 24

KOLMOGOROV-SMIRNOV ONE SAMPLE TEST FOR THE NORMALITY OF GROWTH MODEL PRICE RELATIVE PREDICTIONS FOR SIXTY-ONE COMPANIES FROM 1958 THROUGH 1967

Observed Cumulative Theoretical Interval Proportion Proportion Proportion Difference of za °i Ci Fi Di Less than -1.15 0.1738 0.1738 0.1250 0.0488 -0.68 0.1213 0.2951 0.2500 0.0451 -0.v32 0.1180 0.4131 0.3750 0.0381 0.00 0.0721 0.4852 0.5000 0.0148 0.32 0.1049 0.5901 0.6250 0.0349 0.68 0.1066 0.6967 0.7500 0.0533 1.15 0.1311 0.8278 0.8750 0.0472 + <» 0.1722 1.0000 1.0000 0.0000

aThe predicted price relatives were standardized by means of the trans­ formation

Zj - (Xj - M)/S, where

is a particular price relative prediction (1 <. j <. 610)

M is the mean predicted price relative

S is the standard deviation of the predicted price relatives.

C^ is the cumulative proportion of predicted price relatives-

that is less than the upper bound of interval 1(1 ■ 1, 2, ..., 8)

F^ is the cumulative proportion of a normally distributed

random variable that is less than the upper bound of

of interval i (i - 1, 2...... 8)

Null hypothesis: The cumulative proportion of the predicted price relatives

is equal to the cumulative proportion of a normally dis­ 205

tributed random variable.

Research hypothesis: The predicted price relatives are not normally

distributed.

The maximum value of D^, denoted MD, is equal to 0.0533 from the table above. At the 99% confidence level, we have

Pr(MD >_ 1.63/V610) = 0.01

Pr(MD j> 0.021) = 0.01

The null hypothesis must be rejected at the 99% confidence level, so the research hypothesis is accepted. 206

TABLE 25

KOLMOGOROV-SMIRNOV ONE SAMPLE TEST FOR THE NORMALITY OF PRICE RELATIVES FOR SIXTY-ONE COMPANIES FROM 1958 THROUGH 1967

Observed Cumulative Theoretical Interval Proportion Proportion Proportion Difference of za °i Ci Fi Di

Less than -1.15 0.0951 0.0951 0.1250 0.0299 -0.68 0.0672 0.1623 0.2500 0.0877 -0.32 0.1377 0.3000 0.3750 0.0750 0.00 0.1344 0.4344 0.5000 0.0656 0.32 0.1656 0.6000 0.6250 0.0250 0.68 0.1787 0.7787 0.7500 0.0287 1.15 0.1475 0.9262 0.8750 0.0512 + 00 0.0738 1.0000 1.0000 0.0000

The price relatives were standardized by means of the transformation

Zj * (Xj - M)/S, where

Xj is a particular price relative (1 <. j <. 610)

M is the mean price relative

S is the standard deviation of the price relatives.

C^ is the cumulative proportion of the standardized price

relatives that is less than the upper bound of interval

i (i - 1, 2 ..... 8)

F^ is the cumulative proportion of a normally distributed

random variable that is less than the upper bound of

interval i (i = 1, 2, ..., 8)

Null hypothesis: The cumulative proportion of price relatives is equal

the cumulative proportion of a normally distributed 207

random variable on an interval by interval basis,

except for chance variations.

Research hypothesis: The price relatives are not normally distributed.

The maximum value of D^, denoted MD, is equal to 0.0877 from the table above. At the 99% confidence level, we have

Pr(MD ._> 1.63/^/610) = 0.01

Pr(MD >_ 0.021) - 0.01

The null hypothesis must be rejected at the 99% confidence level, so the research hypothesis is accepted. 208

TABLE 26

ANALYSIS OF COVARIANCE TABLE FOR PRICE RELATIVE VARIANCES AND THE GROWTH MODEL'S MEAN- SQUARED ERRORS FOR SIXTY-ONE COMPANIES

Source of Mean . . . 2 . . . . . 2 .. . . . ’. 2 . ■ • Variation df EX ■'Exy ' 2y df sy . Squared

Among means 4 0.3875 0.2380 0-. 2445 4 0.1416 0.0354

Among groups 56 0.8751 0.1864 1.0674 55 1.0277 0.0187

Total 60 1.2626 0.4243 1.3118 59 1.1692

Model tested: Y ^ ■ p + ^ + 3 0 ^ - X) + e^, where

1-1, 2, t (treatment groups), and

J - 1, 2, ..., n^ (cases for group 1)

Regression coefficients: Means 0.6141.

Within 0.2130

Total 0.3361

F Ratios

Mean-squared error, mean effects 6.1996 (4, 56 degrees of freedom)

Price relative variance, mean effects 3.2068 (4, 56)

Mean-squared error, mean effects after

adjustment for regression 1.8940 (4, 55)

Within regression 2.1244 (1, 55)

Pr(P4 55 >, 3.68) - 0.01

Pr(F > 2.54) - 0.05 4,55 “

Pr(Fl,55 - 7,12) " 0,01 209 TABLE 27

ANALYSIS OF COVARIANCE TABLE FOR PRICE RELATIVES AND THE PRICE RELATIVE PREDICTIONS PRODUCED BY THE GROWTH MODEL FOR SIXTY-ONE COMPANIES FOR 1958 THROUGH 1967

Source of '2 Mean Variation df Ix2 Zxy zy2 df zy Square

Among means 4 0.9083 1.4341 2.5377 4 2.6739 0.6685

Among groups 605 31.8039 -2.2210 67.9500 604 67.7949 0.1122

Total 609 32.7122 -0.7869 70.4877 608 70.4687

Model tested: Y^ - p + + P(X^ - X) + , where .

1-1, 2, t (treatment groups), and

J - 1, 2, ..., n^ (cases for group 1)

Regression coefficients: Means 1.5790

Within -0.0698

Total -0.0241

F Ratios

Price relatives, mean effects 4.3195 (4,605 degrees of freedom)

Predicted price relatives, mean effects 5.6486 (4,605)

Price relatives, mean effects after

adjustment for regression 5.9555 (4,604)

Within regression coefficient 1.3819 (1,604)

Pr (F^, 120 >. 4.95) - 0.01 210

TABLE 27— continued

Pr(F4 ^ > 4.62) = 0.001

Pr(Fl,120 - 6,85) = 0,01

Pr(F. > 6.64) - 0.01 i,°° -

Pr(Fl 120 - 11,38) " °*001 TABLE 28

ANALYSIS OF VARIANCE TABLE FOR PRICE RELATIVES REGRESSED ON PRICE RELATIVE PREDICTIONS FROM THE GROWTH MODEL FOR SIXTY-ONE COMPANIES FROM 1958 THROUGH 1967

Source of Sum of Mean Variance df Squares Square F Ratio

Regression 1 814.937 814.937 4738.256

Residual 609 104.743 0.172

Equation: R^fc = 0.96149 (Rit + a), where

R^t is the price relative for company i (i = 1, 2, ...

and for period t (t = 1958, 1959, ..., 1967)

a R^t is a prediction of R^fc

, a is the constant 0.03002 which is R - R

Coefficient standard error - 0.03197

Multiple correlation, squared - 0.8861 212

TABLE 29

THE OPTIMAL LINEAR CORRECTION OF THE PRICE RELATIVE PREDICTIONS MADE BY THE GROWTH MODEL FOR SIXTY-ONE COMPANIES FROM ' 1958 THROUGH 1967

Systematic prediction bias can be eliminated from a set of predictions by means of the linear correction factor

A Rit = a + b Rifc, where

R^t is the price relative for company i (i = 1, 2, 61) and

year t (t = 1958..... 1967),

a, b are the linear correction factors, and

A R^fc is the prediction of R ^ .

To minimize the mean-squared error with respect to a and b, we have

b a . s ( £ -i).

A A A I(R - R)

A a = R - bR

For this particular set of predictions,

a - 1.2367799

b = -0.0494832

Henry Theil, Applied Economic Forecasting (Chicago: Rand McNally & Company, 1966), p. 34. 213

YABLE SO PRICE RELATIVE PREDICTION ERROR DECOMPOSITION FOR THE POSTULATED DROUTH MODEL AND FOR AN EXPONENTIAL SMOOTHING MOOEL TESTED ON SIXTY ONE COMPANIES OVER THE TEN YEAR PERIOD 1*5* THROUGH 1*67

MEAN.SGUARED ERROR CONSTANT BIAS PROPORTIONAL BIAS CORRELATION COMPANY g r o w t h SMOOTHING GROWTH SMOOTHING g r o w t h SMOOTHING GROWTH SMOOTHING NUMIER MODEL MODEL MODEL MODEL MODEL MODEL MOOEL m o d e l 4*030 0,141* 0,0(01 0 0010 0,0004 0,0832 0, 0.0577 0 0397 22*300 0,0777 0,03(* 0 0054 Si0JO73 0,0444 0, 0,027* 0 031* 233100 0|0((( 0,0532 0 0002 0.0002 0,0133 0, 0.0331 0 0330 321413 0,144* O.OSd 0 0003 0.0142 0,0*71 0,0252 0,04*3 0 0467 341(00 0,1*34 0,1(82 0 0134 0.025* 0,03*5 0, 0.1423 0 142* 3ST10O 0,3772 0,31** 0 034* 0,120* 0,14(1 0, 0.1*43 0 1*8* 1(2600 0,35(( 0,1*14 0 0002 0.0030 0,2532 0,055* 0,1032 0 1005 142*00 0,0*25 0,0(1* 0 003* 0.013* O', 0431 0,0347 0.0418 0 0333 2i««30 0,3(17 0,17(3 0 0325 0.050* 0,138* 0,0377 0,170* 0 1*80 37(423 0,1072 0,0100 0 0001 0.0035 0,047* 0,045* 0.03** 0 0307 (31450 0,131* 0,072* 0 0273 0.0126 0,0704 0,0142 0,0342 0 0458 12200 0,1233 0,0**1 0 0154 0.0032 0,0241 0,0397 0.08*1 0 0342 73300 0,07*3 0,033* 0 00*7 0.011* 0,02*8 0,0241 0,03*8 0 01** 14*4*0 0,3707 0,tt«5 0 0022 0.0017 0,0*32 0, 0,2833 0 287* 173(00 0,21S( 0,1153 0 01(7 0.0242 0,1135 0, 0,0837 0 0*13 203(00 0,223* 0,11S« 0 0034 0.00(3 0,1147 0, 0,1038 0 1124 322(00 0,13(1 0,120* 0 0001 0.0240 0,0(44 0, 0,0*33 0 0**8 374730 0,3312 0,40*2 0 014* 0,01*1 0,027* 0,1014 0.28(7 0 28*7 431*00 0,1522 0,09*5 0 0415 0.0020 0,020* 0, 0,08*7 0 0*43 (*4300 0,17*7 0,1140 0 0034 0.001* 0,06*3 0,01*4 0,1070 0 0*33 33377 0,0217 0,014* 0 0005 0.0001 0,010* 0,0038 0,010* 0 010* 132(00 0,0103 0,0243 0 00(3 0.0004 0,0340 0,0070 0,017* 0 0171 1(7700 0,10*7 0,031* 0 0014 0.0011 0,0(3* 0,0027 0,0247 0 0278 312(37 0,2234 0,102* 0 0003 0.00(3 0,1315 0,0033 0,0*34 0 0*30 407*00 0,0*21 0,03*3 0 0043 0.00*8 0,0410 0,0014 0,0473 0 04*1 431110 0,1040 0,023* 0 00(4 O' 0,0*10 0,00(3 0,01*3 0 01»* 5(3700 0,031* 0,02(0 S 000* 0.0007 0,0100 0,0029 0,0207 0 0244 *0*300 0,0(43 0,0*12 0 0078 0.010* 0,0117 0, 0,0430 0 0304 (3*300 Q,0I7( 0,0200 0 0033 0.0003 0,0705 0,0045 0,0138 0 0130 (39205 0,0*42 0,034* 0 0034 0.0001 0,0384 0,013* 0.0424 0 03*1 ((2(00 0,0*74 0,04*2 0 0044 0,0007 0,038* 0,0083 0,0443 0 0400 214

TABLE 30— CONTINUED

M£AN*SQUa RED ERROR CONSTANT BIAS RRORORTICNa L BIAS CORRELATION c s m r a n y GROWTH SMOOTHING g r o w t h s m o o t h i n g g r o w t h s m o o t h i n g g r o w t h s m o o t h i n g NUMBER MODEL MOOEL m o d e l m o d e l MODEL MODEL m o d e l MOOEL 663600 01022 0,044 0,0098 0,0004 o.oon • 0131 o 0171 .031 686300 01068 0,029 0,0013 - 0,0003 0,041 0, 0291 ,029 686700 0i09l 0,043 0,0294 0.0017 ,026 0, 0396 ,041 696600 0 1073 0,026 O',0014 0,0024 ,091 0 ,0009 0226 ,023 701220 0,090 0,046 0,0019 0,0009 0 ,046 0 10103 0427 0..039 736700 0,079 0,048 0,0407 0,0063 0 ,009 0, 0299 0,040 334600 0,046 0,023 0,0041 0,0019 0 ,044 0 0163 0,022 486088 0,066 0,026 0,0013 0, 0 ,049 0 0090 0196 0,021 666200 0,038 0,036 0,0044 0.00'24 0,008 0,--0090 - 0241 0 ,028 666300 0,066 0,039 0,0076 0.0019 0__,027 0,0060 0319 0 .029 711212 0,092 0,031 0,0101 0,0019 0,019 0,0018 0279 0,027 188700 0.999 0,487 0 0406 0.0929 0,063 0, 4336 0.,434 - 208060 0.310 0,184 0,0041 0, 0,129 0 , 1768 0,184 266300 0,142 0,037 0,0040 0.0018 0,093 0 .,0089 0491 0.047 317900 0,097 0,081 0,0029 0.0008 0,031 0 ,,0267 0626 0,094 3U100 0.172 0,070 0,0067 0.0097 0,126 0,,0092 0394 0.099 449700 0,794 0,463 010140 0.0008 0,499 0 . 2816 0.462 487100 1,022 0,991 0;0999 0,0971 0,179 0 . 7471 0,794 738681 0,199 0,061 0,0001 0.0016 0,099 0,0092 0960 0.034 18000 0,129 0,114 0,0043 0,0091 0,030 0,0142 0947 0,097 19900 0,299 0,269 0,0011 0.0169 0,074 0,0941 2198 0,199 304000 0,166 0,149 0,0164 0,0096 0,041 0,0122 1270 0,127 312600 0.149 0,079 0,0029 0.0024 0,096 0,0012 0662 0,071 371200 0,098 0,061 0,0000 0,0022 0,004 0 , 0948 0,079 401126 0,161 0,109 0,0109 0,0027 0,078 0,0096 0924 0.093 424300 0,110 0,069 0; 0027 0,0011 0,066 0,0329 0404 0,039 434800 0,090 0,039 0,0016 0,0067 0,020 0, 0279 0,031 464600 0,197 0,063 0,0009 0 , 0,074 0 , ..0829 0,063 488210 0,109 0,078 0,0023 0.0036 0,036 0,0040 0,0667 0.070 971000 0,399 0,319 0,0000 0 , 0,049 0 , 0,3137 0.319 AVCRAOII 0,17479 0,12009 0i01030 0,00933 0,04912 0,01143 0,09934 0,09927 i T t pH CN

H O n o H o o n **aini>) nNa)o«»in9 O viN noooP' Khuiq (\>v4n<^o>vr^r<-«-io«r«^hhno ouiaN«nKin«4n(MNoNNN4 «oonN«o«n« NAviNninN (ZZUJHOOOOONOHOOOOOOvlOOOOOvlwt^O^CavlOO ac u. - Ui i*. o oooooooooooeoooooooooooooooooo• Ui Q _J ►- Ui * s t r 3 0 x »«u) _iz *4in(7 0 c3 NP>KoP)NP-r> Hno>HCN o _i m Q«iv«i'pii>H(> k p> c □ ft »o « ru no«(iOHrooviooNnKNO|a b. UI N a)HOHHHNM *NNnNNIM ni« NIVN«IMNH*4««Nn*inn«oNW N«N««it«KN« a: 4 0 0 >h Jh iM««an\innoftCNNino«niMNK«o>fl«t n«noN » o u k 4 2 Ui H N \noo«ono4 om n»«i^ 4 o u. V) i oop>nniP)r)4irNtiMMP>(MM«PiP}n«ion«4(y(u« i u o i h h n i v m or 3uii> a o ...... UI V) z 0X 0000000000000000000000000000000 a a; o o ui r> (A Z >>•» UI O l> H U m j x ^ m KnoN4N»4ooh«o c HONmNcuio □ □r)K o>(vnKoo«(VF) M O M OZUIOOOOOOHOOOOOOO*4«iOOOOOOO«iOOOOOOO a i ft a or u. - ...... u> o ui a i k . f t ft K D X _ i f t O (9 UIOUI4 UIZ > k Diu at —•_» *4onwiv0 4'ono«oi)ir\ino>MviriooinoiDn oonoon •"■»€!> 4 XUIOOOOBO «4H«OM»OOi 000€>00000 —> ui 13 ui ...... UlhZI- • T OOOOODOOQOOOOOOOOOOOOOODCOOOOOO ft Z (A J 3 IU *4 Ui D P- X US «J •- Cl _ XI •— CO O *-_!<► r » « 0 >4(V«OP> OiO m m r- W O rlNNNKWN4 O Q a i n < I N « ft o x 'jiuiv4P'(M'iisHV\(> oooo *rru4 .i«« p. 4 r>. O ft VI 0 0 4 KOVftKtnftO OIONKP' H N O nO K nonN O «.»*•» OIDOO ffo*too«< wn nop)T(«i«ioniv(VHO ttwoowno >«»

x a eo o o if\o o o o o tfto o o o o o o o o c > * * o o * i»oc*«.'>c.->t.'*ino 4Ui n ooH ooootoN iixoo'Oaoooo Oh oo«\ooocoooi f t O o m » i 4 0Hft44(yin4««»Ov«(OI>>(4MOO«4(MIO»ffil9N404inwNKr» (VIOO« O N 0 = 3 4 Ntntv4inc>H\ir>« O 2 NNnni<>pi«4NOO «4*iNnn40 HHn44||\«<0«<0 TABLE 31— eONTlNUED

MEAN-S3UARE0 ERROR AVERAGE ABSOLUTE ERROR COMPANY GROWTH SMOOTHING NET g r o w t h SMOOTHING NET n u m b e r MOOEL MODEL DIFFERENCES MOOEL MODEL DlFFEREN 463400 0 0239 0,0446 •0,0217 0,1296 0,1713 •0.0477 466300 0 0663 0.0296 0,0427 0,1973 0,1339 0,0638 666700 0 0911 0.0432 0,0479 0,2303 0,1849 0,0494 699600 0 0799 0.0246 0,0493 0.2313 0,1438 0,0679 701220 0 0906 0.0469 0,0441 0,2460 0,1997 0,0903 736700 0 0794 0,0419 0,0269 0.2071 0,1730 0,0341 334600 0 0664 0.0234 0.0430 0.2161 0,1416 0,0743 466069 0 0667 0.0260 0,040'7 0,2266 0,1269 0,1021 666200 0 0366 0.0361 0,0029 0,1667 0,1613 0,0094 666900 0 0668 0.0392 0,0276 0,2094 0,1724 0,0330 711282 0 0927 0.0314 0,0213 0,1670 0,1614 0,0096 188700 0 9992 0,4873 0,0719 0,6630 0,9841 0.0799 209060 0 3106 0,1846 0,1298 0,4977 0,3478 0,1099 266300 0 1426 0.0977 0,0649 0,3490 0,1977 0,1913 317900 0 0973 0.0816 0.0199 0,2491 0,2174 0,0317 318100 0 1723 0,0709 0,1016 0,3790 0,2209 0.1969 449700 0 7948 0,4636 0,3312 0,6939 0,9213 0,1722 467100 1 0228 0,8913 0,1719 0,6619 0,6029 0.0794 736691 0 1993 0,0610 0,0943 0,3361 0,1898 0,1463 19000 0 1299 0,1169 0,0126 0,2749 0,2860 •0,0111 19900 0 2993 0,2699 0,0294 0,3737 0,4199 -0,0462 304000 0 1669 0,1492 0',0413 0.3896 0,3941 0,0317 312800 0 1696 0,0791 0,0907 0,3123 0,2124 0,0999 371200 0 0988 0,0619 •0,0231 0,2009 0,2200 •0,0191 401126 0 1619 0,1097 0,0798 0,3473 0,2844 0,0629 424300 0 1100 0,0693 0,0407 0,2669 0,2144 0,0921 434600 0 0901 0,0399 0,0102 0,1694 0,1632 0,0262 466600 0 1979 0,0830 0,0749 0,3494 .0,2162 0,1292 489210 0 1094 0,0786 0,0306 0,2706 0,2336 0,0372 971000 0 3994 0,3197 0,0437 0,4931 0,4036 0,0499

AVERAGES 0 .1746 0,1201 0,0947 0,3111 . 0,2467 0,0644 217

T A B L E 32 PRICE RELATIVE PREDICTION ERROR DECOMPOSITION FOR THE POSTULATED GROWTH MODEL AND FOR AN exponential s m o o t h i n o m o d e l u s i n g s h i f t i n g SMOOTHING CONSTANTS FOR SIXTY.ONE COMPANIES OVER THE TEN YEAR «ER!OD 1958 THROUGH 1967

MEAN.SQUARED ERROR CONSTANT BIAS PROPOR AL BIAS CORRELATION COMPANY GROWTH SMOOTHING GROHTH SMOOTHING GROWTH OQTHING GROWTH SMOOTHING NUMBER MODEL MODEL MODEL m od el MODEL MODEL MODEL MODEL 49030 0.1419 0,1446 0,0010 0.0179 0,0832 0,0748 0.0577 0.0519 229500 0,0777 0,0458 0 i 0054 0.0050 0.0444 0,0157 0.0278 0,0251 259100 0,0686 0,0672 0,0002 0.0001 0,0153 0,0139 0.0531 0,0532 321415 0,1469 0,1062 0,0005 0,0154 0,0971 0,0422 0.0493 0,0486 348800 0.1954 0,1828 0,0134 0.0357 0,0395 0,0048 0.1425 0.1423 557100 0,3772 0,4099 0 i 0346 0,1475 0,1481 0,1198 0,1945 0,1426 182600 0,3566 0,1631 0.0002 0.0018 0,2532 0,0422 0.1032 0,1191 182900 0,0925 0,0901 0 i 0056 0,0068 0,0451 0,0454 0.0418 0,0379 218930 0,3617 0,2972 0i0525 0.0510 0,1386 0,0724 0.1706 0,1738 5’6425 0,1072 0,1021 0,0001 0,0001 0,0476 0,0548 0,0596 0.0472 j J1450 0,1319 0,0870 0,0273 0.0052 0,0704 0,0414 0,0342 0,0404 12200 0,1255 0,1059 0,0154 0,0056 0,0241 0,0402 0.0861 0,0601 73500 • 0,0793 0,0558 0 i 0097 0,0137 0,0298 0, 0,0398 0,0421 149460 0,3707 0,3892 0.0022 0.0284 0,0852 0,1088 0,2833 0.2520 175600 0.2158 0,1508 0,0167 Oi 0320 0,1155 0,0537 0.0837 0.0651 203600 0,2259 0,1189 0|0054 0,0065 0,1147 0, 0.1058 0,1124 322500 0,l58i . 0,1975 0.0001 0,0035 0,0644 0,1172 0,0935 0,0768 374750 0,3312 0,4772 0.0149 0,0048 0,0276 0,1831 0,2887 0,2893 438900 0,1522 0,1042 0,0415 0.0006 0,0209 0,0212 0.0897 0.0824 694500 0,1797 0,1720 0.0034 0.0105 0,0693 0,0996 0.1070 0.0619 53377 0,0217 0,0180 0.0005 0.0003 0,0108 0,0073 0.0104 0,0104 152800 0,0803 0,0302 0;0083 0,0017 0,0540 0,0105 0.0179 0,0180 187700 0,1097 0,0349 O',0014 0,0056 0,0836 0,0012 0.0247 0.0281 312657 0,2254 0,1426 0.0005 0.0172 0,1315 0,0130 0,0934 0,1124 407900 0,0928 0,0650 0,0043 0.0161 0,0410 0,0007 0,0475 0,0482 452180 0,1040 0.0338 0,0066 0,0001 0,0810 0,0253 0,0165 0,0084 983700 0,0316 0,0313 0|0008 0.0014 0,0100 0,0073 0,0207 0,0226 606500 0,0.645 0,0781 0,0078 0.0060 0,0117 0.0279 0,0450 0,0442 656500 0,0276 0.0282 0 j 0033 0,0063 0,0705 0,0073 0,0138 0,0146 659205 0,0842 0,0626 0,0034 0.0008 0,0384 0,0218 0.0424 0,0400 662800 0,0874 0,0561 0.0044 0.0003 0,0386 0,0148 0,0445 0,0410 218

TABLE 32--C0NTIMJED MEAN-SQUARED ERROR CONSTANT BIAS PROPCRT B t AS CORRELATION COMPANY GROWTH SMOOTHING GROWTH s m o o t h i n g GROWTH 00THIN3 GROWTH SMOOTHING NUMSER MOOEL MODEL MOOEL m o d e l MODEL MOOEL MODEL MODEL 663600 0,0229 0,0535 Q ;0058 0,0008 0,0000 0,0183 0,0171 0.0344 686300 0,0683 0,0287 0,0013 0. 0,0419 0,0118 0.0251 3,0169 686700 0,0911 0,0432 0,0256 0.0017 0,0260 0 , 0,0396 0.0*19 689600 0,0759 0,0327 0.0014 0.0013 0,0518 0,0054 0,0226 0,0260 701220 0,0906 0,0519 OiOOiS 0.0003 0,0463 0,0143 0.0*27 0,0373 736700 0,0754 0,0489 0.0407 0,0083 0,0051 0 , 0.0295 0,0*06 334600 0,0664 0,0235 0.0041 0.0015 0,0441 0 , 0.0183 0,0220 486089 0,0647 0.0330 0.0013 0,0.009 0,0498 0.0087 0.0156 0.023* 686200 0,0386 0,0390 0.0044 0,0077 0,0081 0,0006 0.0261 0.0307 686S00 0,0668 0,0490 0 i 0078 0.0067 0,0271 0,0136 0,0319 0.0297 711282 0,0527 0,0338 o;oioi 0.0059 0,0151 0,000* 0.0279 0,0275 188700 0,5592 0.4873 0.0606 0.0529 0,0650 0 . 0,4336 0.4344 209080 0,3106 0,2194 0.0041 0.0059 0,1297 0,1033 0.1768 0,1102 266300 0,1426 0,0478 0.0040 0.0084 0,0935 0,0120 0.0*51 0,0*7* 317900 0,0973 0,0928 0.0029 0.0003 0,0318 0,0518 0.0626 0.0*07 3i8l00 0,1723 0,0780 0,0067 0.0148 0,1263 0,0075 0,0394 0.0557 449700 0,7948 0,5413 0.0140 0,0079 0,499* 0,3808 0,2816 0,1526 487100 1,0228 0,9881 0,0958 0.0607 0,1799 0.5483 0.7471 0,2791 738651 0,1553 0,0847 0,0001 0 . 0,0993 0,0489 0,0560 0,0358 15000 0,1295 0,1414 o;oo43 0,0040 0,0305 0,0488 0,0947 0,0886 19900 0,2953 0,3075 0,0011 0,0022 0,0743 0,0990 0,2199 0,2103 304000 0 ,1869 0,1728 0,0184 0,0078 0,0411 0,0470 0,1270 0,1180 312800 0,1658 0,0879 0,0029 0.0095 0,0967 0,0076 0,0662 0.0708 371200 0,0588 0,0838 O',0000 0,0030 0,0040 0,0021 0.0548 0.0787 401126 0,1619 0,1276 0.0109 0.0114 0,0785 0,0246 0.092* 0.0916 424300 0,1100 0,1310 0.0027 0.0034 0,0668 0,1063 0.0*06 0,0213 434800 0,0501 0,0413 0,0016 0.0097 0,0207 0,0004 0,0279 0,0312 466600 0,1579 3,8900 0,0809 8.0084 0,3744 0,0207 0,0925 0,0689 485210 0,1094 0,8893 0,0023 0.0154 0,0383 0.0006 0,0697 0.0731 971000 0,3594 0,3157 0,0800 0. 0,0457 0 , 0.3137 0.3157

AVERAOEf 0,17475 0,13989 0;01030 0.01149 0,06912 0.04700 0.09934 0.081*0 T A B L E 33 M I C E RELATIVE PREDICTION PERFORMANCE OF THE POSTULATED 3ROHTH MODEL VERSUS AN EXPONENTIAL SMCOTHINS MOOEL WITH SHITTING SMOOTHING CONSTANTS POR SIXTY-ONE COMPANIES OVER THE TEN VEAR PERIOD 1958 THROUGH l’»7

HEAN-SOUARED ERROR AVERAGE ABSOLUTE ERROR c o m p a n y GROHTH s m o o t h i n g NET growth smoothing NET NUMBER MOOEL m o d e l DIFFERENCES MODEL MODEL DIFFERENCES 49030 0.1419 0,1446 •0,0027 0,3175 0,3372 •0,0197 229500 0.0777 0,0458 0,0319 0,2528 0,1834 0,0694 255100 0(0686 0,0673 0,0013 0,2366 0,2349 0,0017 321415 0,1469 0,1061 0,0408 0,3062 0,2563 0.0499 3483:3 0,1954 0,1828 0,0126 0.3085 0,2875 0,0210 557100 0.3772 0,4099 -0,0327 0,4656 0,4908 -0,0252 162600 0,3566 0,1631 0,1935 0.5059 0,2544 0,2515 162900 0,0925 0,0901 0,0024 0.2531 0,2475 0,0056 218930 0,3617 0,2972 0,0645 0,5083 0,4091 0,0992 576425 0,1072 0,1021 0,0051 0,2493 0,2774 -0,0281 631450 .0,1319 0,0871 0,0448 0,3081 0,2603 0,0478 12200 0,1255 0,1060 0,0195 0,2520 0,2307 0,0213 73500 0,0793 0,0557 0,0236 0,2323 0,2085 0,0238 149460 0,3707 0,3892 -0,0185 0.4951 0,4905 0,0046 175600 0.2158 0,1508 0,0650 0.3606 0,3213 0,0393 203600 0,2259 0,1189 0,1070 0,3647 0,2632 0.1015 322600 0,1581 0.1975 -0,0394 0,3030 0,3637 -0.0607 374750 0,3312 0,4772 -0,1460 0.4529 0,5234 •0,0705 438900 0,1522 0,1042 0,0480 0,3020 0,2929 0.0091 694500 0,1797 0,1719 0,0078 0.3477 0,3512 -0,0035 53377 0,0217 0,0181 0,0036 0,1161 0,1142 0,0019 152803 0,0803 0,0303 0,0500 0,2692 0,1077 0,1615 187700 0,1097 0,0349 0,0748 0,2986 0,1601 0,1385 312657 0,2254 0,1426 0,0828 0.4287 0,3264 0,1023 407900 0,0928 0,0649 0,0279 0,2346 0,2186 0,0160 452183 0,1040 0.0338 0,0702 0,2831 0,1396 0,1435 583703 0,0316 0,0314 0,0002 0,1464 . 0,1401 0,0063 636500 3,0645 0,0782 -0,0137 0.1938 0,2044 -0.0106 656500 0,0876 0,0281 0,0595 0.2787 0,1468 0,1319 659205 0,0842 0.0626 0,0216 0,2472 0,2003 0,0469 662800 0,0874 0,0562 0,0312 0,2096 0,1925 0,0171 TABLE 33— CONTINUED

m e a n -s o u a r e d ERROR a v e r a s e a b sd UTE ERROR COMPANY SROW U SM09TMINB NET SROUTm Sm OOTm I S NET NUMBER MODE MODEL DIFFERENCES MODEL MODEL DIFFEREN 643600 0,022 0.0934 • 0,0309 0.1296 0,197 -0,0714 686300 0,066 0,0287 0,0396 0,1973 0,140 0,0966 466700 0,091 0,0432 0,0479 0,2303 0,184 0,0494 699600 0,079 0,0327 0,0432 0,2313 0,161 0,0694 701220 0,090 0,0919 0,0367 0,2460 0,202 0,0434 734700 0,079 0,0469 0,0269 0,2071 0,173 0,0341 334600 0,066 0,0234 0,0430. 0.2161 0,141 0,0743 466069 0,046 0,0329 0,0336 0,2286 0,0707 664200 0,036 0,0319 •0,0003 0,1667 0,197 0,0088 666900 0,066 0,0490 0,0176 0,2094 0,180 0,0246 711262 0,092 0,0337 0,0190 0,1670 0,167 •0,0003 166700 0,999 0,4673 0,0719 0,6630 0,984 0,0769 209080 0,310 0,2194 0,0912 0,4977 0,371 0,0869 266300 0,142 0,0676 0,0746 0,3490 0,199 0,1493 317900 0,097 0,0927 0,0046 0.2491 0,241 0,0078 318100 0,172 0,0760 0,0943 0,3790 0,227 0,1911 449700 0,794 0,9413 0,2939 0,6939 0,944 0,1466 467100 1,022 0,6861 0,1347 0,6819 0,410 0,0719 736691 0,199 0,0846 0,0709 0,3361 0,223 0,1126 19000 0,129 ' 0,1414 •0,0119 0.2749 0,312 •0,0371 19900 0,299 0,3079 •0,0122 0,3737 0,406 •0.0326 304000 0,166 0,1728 0,0137 0,3696 0,389 •0,0033 312800 0,169 0,0879 0,0779 0,3123 0,248 0,0636 371200 0,099 0,0837 •0,0249 0,2009 0,227 •0,0261 401126 0,161 0,1276 0,0939 0,3473 0,317 0.0296 424300 0,110 0,1309 •0,0209 0,2669 0,294 •0,0279 434800 0,090 0,0412 0,0069 0,1894 0,169 0,0241 466600 0,197 0,3899 0,0660 0,3494 0,228 0,1169 469210 0,109 0,0893 0,0201 0,2708 0,246 0,0226 971000 0,399 0,3197 0,0437 0.4931 0,403 0,0499

AVERAQES 0,1741 0,1369 0,0349 0,3111 0,2479 0,0432 221

TABLE 34

PREDICTION PERFORMANCE OF PREDICTION MODEL NO. 1 VERSUS PREDICTION MODEL NO. 3 USING SHIFTING SMOOTHING CONSTANTS

Null hypothesis: There is no difference between the mean-squared pre­

diction errors resulting from the use of Model No. 1 and those

resulting from the use of Model No. 3.’

Research hypothesis: The mean-squared prediction errors resulting from

prediction Model No. 3 are less than those of Model No. 1.

Wilcoxen matched-pairs signed ranks test

I is the smaller ,sum of like-signed ranks

T - 797.5

For T ■ 797.5 and N ■ 61, z ■ -1.063

Pr(Z < -1.063) - 0.1439

t - test for paired observations

*60 “ lA12 From tables of the t distribution:

Pr(t60 > 1.296) - .10

Pr(t6Q > 1.671) - .05

For a more complete presentation of these tests, see Table 18.

These results are somewhat inconclusive, because at the 90% confi­ dence level, the null hypothesis is rejected by the t - test. The

Wilcoxen test results indicate that the null hypothesis cannot be re­ jected at the 90% confidence level. These results indicate that the null hypothesis cannot be rejected with a high level of confidence. 222

TABLE 35

PERFORMANCE OF PREDICTION MODEL NO. 1 VERSUS PREDICTION MODEL NO. 3 USING FIXED SMOOTHING CONSTANTS

Null hypothesis: There is no difference between the mean-squared pre­

diction errors resulting from the use of Model No. 1 and those

resulting from the use of Model No. 3.

Research hypothesis: The mean-squared prediction errors resulting from

prediction Model No. 1 are greater than those resulting from No. 3.

Wilcoxen matched-pairs signed ranks test

T is the smaller of like-signed ranks

T = 926.5

For T = 926.5 and N = 61, Z = -0.14

Pr(Z < -0.14) = .4443 t-test for paired observations

t60 = -0.954

From tables of the t distribution:

Pr(t6Q > 1.296) = 0.10

Pr(t6Q > 1.671) = 0.05

For a more complete description of these tests, see Table 18.

The results of these two tests are consistent. The null hypothesis cannot be rejected at the 90% confidence level by the Wilcoxen test or by the t-test. TABLE 36

PRICE RELATIVE VARIANCE VERSUS THE MEAN- SQUARED PREDICTION ERROR OF MODEL NO. 3 USING SHIFTING SMOOTHING CONSTANTS

Null hypothesis: The price variance relative for company i is equal to

the mean-squared prediction error for company i (i = 1, 2, ..

Research hypothesis: The mean-squared prediction error for company i

is greater than the price relative variance forcompany i (i

2, ..., 61).

Wilcoxen matched-pairs signed ranks test

T is the smaller sum of like-signed ranks

T - 99

For T ■ 99 and N “ 61, Z ■ -6.08

Pr(Z <_ - 6.08) < 0.0001 t-test for paired observations

t60 - 2.996

From tables of the t distribution:

Pr(tgg _> 2.660) ■ 0.005

**(t60 > 3.232) - 0.001

For a more complete presentation of these two tests, see Table 18.

The results of these two tests are consistent in that the null hypothesis is rejected at the 99.9% confidence level. TABLE 37

MEAN-SQUARED ERROR VERSUS VARIANCE FOR SIXTY-ONE COMPANIES

Prediction Average Mean- Average Price Difference Significance Tests Model Type Squared Error Relative Variance Absolute Percentage z-value t-value

Growth Model 0.1748 0.1158 0.0590 33.8% -6.60 8.868

Model No. 3 - Shifting Smoothing Constants 0.1399 0.1158 0.0241 17.2% -6.08° 5.062c

Model No. 3 - Fixed Smoothing Constants 0.1201 0.1158 0.0043 3.6% -1.413b 1.412b

Mean of Past Ten Price Relatives 0.1331 0.1158 0.0173 13.0% -6.324a -6.129a

a Significant at the 99.5% confidence level. b Significant at the 90% confidence level, but not significant at the 95% level.

N3S3 225

TABLE 38

PREDICTION PERFORMANCE OF PREDICTION MODEL NO. 3 USING SHIFTING SMOOTHING CONSTANTS VERSUS THE POSTULATED GROWTH MODEL

Null hypothesis: The mean-squared prediction errors resulting from the

constant exponent model using shifting smoothing constants are

equal to the mean-squared prediction errors resulting from the

use of the postulated growth model.

Research hypothesis: The mean-squared prediction errors resulting from

the use of the postulated growth model are less than those re­

sulting from use of the constant exponent model.

Wilcoxen matched-pairs. signed ranks test

T is the smaller sum of like-signed ranks

T = 255

For T - 255 and N - 61, z = -4.9597

Pr (Z < -4.9597) < .0001

t-test for paired observations

t,, ® 4.8816 61 From tables of the t distribution:

Pr(t6Q > 3.232) - 0.001

For a detailed presentation of the mean-squared errors, see Tables 30 through 33. For a more detailed presentation of these tests see Table 18.

These results are conclusive, the null hypothesis can be rejected at the 99.9% confidence level. However, the research hypothesis, as stated above, cannot be accepted. It must be concluded that the mean-squared prediction errors resulting from use of the growth model are larger, not 226 smaller, than those errors resulting from use of the constant exponent model using shifting smoothing constants. 227

TABLE 39

MEAN-SQUARED PREDICTION ERRORS USING THE MEAN OF PAST PRICE RELATIVES VERSUS THE PRICE RELATIVE VARIANCE AS A MEASURE OF UNCERTAINTY

Null hypothesis: The mean-squared prediction errors resulting from the use

of the arithmetic mean of past price relatives are equal to the

variances of the price relatives for the companies tested.

Research hypothesis: The mean-squared prediction errors resulting from the

use of the arithmetic mean of past price relatives are greater

than the variances of the price relatives for the companies

tested.

Wilcoxen matched-pairs, signed ranks test

T is the smaller of like-signed ranks

T = 65

For T = 65 and N = 61, Z = -6.3244

Pr (Z < -6.3244) < .0001 t-test for paired observations

tgl = -6.129

From tables of the t distribution:

Pr(t6Q < - 3.232) = 0.001

For a more detailed presentation of these two tests of signficance, see

Table 18.

The results are consistent and relatively conclusive — the null hypoth­ esis is rejected at the 99.9% confidence level and the research hypothesis accepted. 228

. TABLE 40

PREDICTION PERFORMANCE OF THE ARITHMETIC MEAN OF PAST PRICE RELATIVES VERSUS MODEL NO. 3 USING FIXED SMOOTHING CONSTANTS

Null hypothesis: The mean-squared prediction errors resulting from the use

of an 'arithmetic mean of past price relatives are equal to the

mean-squared prediction errors resulting from the use of the con­

stant exponent model using fixed smoothing constants.

Research hypothesis: The mean-squared prediction errors resulting from the

use of an arithmetic mean of past price relatives are greater

than ti: mean-squared prediction errors resulting from the use of

the constant exponent model using fixed smoothing constants.

Wilcoxen matched-pairs, signed-ranks test

T is the smaller sum of like-signed ranks

T - 267

For T - 267 and N - 61, Z - -4.8735

Pr (Z < -4.8735) < 0.00001 t-test for paired observations

tgl - -4.2955

From tables of the t distribution:

' Pr (t6Q < -3.232) = 0.001

The results are conclusive. The null hypothesis is rejected at the

99.9% confidence level and the research hypothesis accepted. 229

TABLE 41

PREDICTION PERFORMANCE OF THE ARITHMETIC MEAN OF PAST PRICE RELATIVES VERSUS MODEL NO. 3 USING SHIFTING SMOOTHING CONSTANTS

Null hypothesis: The mean-squared prediction errors resulting from the use

of an arithmetic mean of past price relatives are equal to the

mean-squared prediction errors resulting from the use of the con­

stant exponent model using shifting smoothing constants.

Research hypothesis: The mean-squared prediction errors resulting from the

use of an arithmetic mean of past price relatives not equal to

the mean-squared prediction errors resulting from the use of the

constant exponent model using shifting smoothing constants.

Wilcoxen matched-pairs, signed-ranks test

T is the smaller sum of like-signed ranks

T = 677

For T = 677, and N = 61, Z = -1.9285

Pr (Z < -1.93 or Z < 1.93) = 0.0536 t-test for paired observations

t6i = 1.734

From tables of the t distribution for a two-tailed test:

Pr(-1.671 < t6Q < 1.671) = 0.1

Pr(-2.00 < t6Q < 2.00) = 0.05

The null hypothesis cannot be rejected at the 95% confidence level.

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