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Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 RENNIE, Henry George, 1940- AGGREGATION THEORY, INVESTMENT BEHAVIOR AND RATIONAL LAG FUNCTIONS.

The Ohio State University, Ph.D., 1973 Economics, theory ti

University Microfilms,A XEROX Company, Ann Arbor, Michigan

© Copyright by

Henry George Rennie

1973

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. AGGREGATION THEORY, INVESTMENT BEHAVIOR

AND RATIONAL LAG FUNCTIONS

DISSERTATION

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

By Henry George Rennie, B.Sc., M.A.

The Ohio State University 1973

Approved by ACKNOWLEDGMENTS

I t ts my pleasure to acknowledge the assistance of various persons who contributed to this study. Special thanks are due my advisers, Professors E. Bal tens per ger, W. L. L'Esperance, and

P. M attila, who gave me continual feedback throughout this study.

I should, in particular, lik e to thank Professor L'Esperance for his encouragement from the inception to the completion of this study.

An essential part of a study of this type is computer programming and I would lik e to thank my very competent assistants,

Molly Garrett, Paul Hart, Andrew Hochstein, and John Irmen. To my typists, Mary Jane Donaldson, Mari Kanavel, and Carol Kaufman, thank you.

The sacrifice involved in reaching the final product was borne by my family. To Heather, who went from speaking no English a t a l l , to saying, "daddy is making book," to, "is your dissertation finished, father?" and to Matthew who was apparently oblivious to it a l l , T.K.& H. And to my w ife, Phoebe, for her encouragement, thank you.

11 VITA

8 October 1940 Born - Greenock, Scotland, Gt. Britain

1963 ...... B.Sc., The Ohio State University, Columbus, Ohio

1964 ...... Research Assistant, Bureau of Business Research, The Ohio State University, Columbus, Ohio

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

1965-1969 . . . Teaching Associate, Department of Economics, The Ohio State University, Columbus, Ohio

1969-Present. Assistant Professor, Department of Economics, The University of Toledo, Toledo, Ohio

PUBLICATIONS

"Bayesian probability and the General pascal Distribution," American Statistical Association, Proceedings of the Business and Economic Statistics Section, 1972, 431-434.

FIELDS OF STUDY

Monetary Theory and Policy

Econometrics

International Trade and Development

11 f TABLE OF CONTENTS

Page ACKNOWLEDGMENTS ...... i i

VITA ...... , ...... 1ii

LIST OF TABLES...... vi

LIST OF ILLUSTRATIONS...... 1x

INTRODUCTION ...... 1

Chapter

I . AGGREGATION BIAS IN RATIONAL LAG FUNCTIONS ...... 7

1. Aggregation and the aggregation problem 2. The meaning of "consistent aggregation" 3. Necessary and sufficient conditions for consistent aggregation 4. Approaches to the aggregation problem 5. The analogy approach to aggregation 6 . The relations between micro and macro parameters 7. A measure of the aggregation bias in macro-parameters 8 . The concept of a distributed lag 9. The rational lag function 10. The general Pascal distributed lag function 11. Parameter bias in rational lag forms 12. The relation between the estimated and the implied macro-parameters in rational lag functions 13. The effect of aggregation on the lag.structure and moments of a general Pascal lag function 14. Aggregation, minimum residual variance and model choice

II. INVESTMENT...... 48

1. Introduction 2. Generalized accelerator mechanism 3. Theories of capital demand 4. Replacement investment 5. Models of Investment behavior

1 v Chapter Page

I I I . ESTIMATION OF A RATIONAL LAG FUNCTION...... 6 8

IV. EMPIRICAL: INVESTMENT THEORY PERFORMANCE...... 75

1. Introduction 2. The sample 3. Measurement 4. Micro performance 5. Macro performance

V. EMPIRICAL: AGGREGATION BIAS...... 103

1. Bias is the unconstrained macro coefficients 2. Bias in the lag structure and average lag 3. Bias in the aggregate residual variance 4. Bias and theory selection 5. The empirical literature: A comparison of findings

VI. SUMMARY AND CONCLUSIONS...... 169

1. Summary 2. Conclusions

APPENDIX

A. . 177

B...... 211

C...... 224

D...... 231

E...... 238

F...... 261

G...... 265

H...... 277

1...... 284

J...... 288

BIBLIOGRAPHY...... 295

V LIST OF TABLES

Table Page

1. Components of bias in the estimated aggregate parameters of a rational distributed lag function ...... 37

2. Sample firms, their ranking by sales, assets, and net income, and th eir OBE-SEC industry group ...... 77

3. Size distribution of firms within groups by average investment and capital stock ...... 78

4. Calculated replacement coefficients for twenty-seven firms and six industry groups ...... 82

5. Goodness of f it statistics-firms ...... 90

6 . Number of times, out of twenty-seven, a theory had a lower standard error than competing theories .... 96

7. Number of desired capital stock coefficients and number at least twice th eir standard errors for twenty-seven companies ...... 98

8 . Goodness of f it statistics-aggregates ...... 100

9. Number of times, out of six, a theory had a lower standard error than competing theories ...... 101

10. Number of desired capital stock coefficients and number at least twice th eir standard errors for six aggregates...... 1 0 2

11. Test results of parameter vector homogeneity by theory and group...... 108

12. Average size of |(E-T)/E| over six industries by theory and coefficient ...... 114

13. Analysis of variance of |(E-T)/E| over six industries by theory and coefficient ...... 117

14. Analysis of variance of T/E by theory and coefficient. . 118

vi Table Page

15. Analysts of variance of average [E-T| over five theories by bias component and coefficient .... 1 2 0

16. Analysis of variance of E-T by corresponding-noncorresp- onding components and coefficient ...... 121

17. Autocorrelation statistics-firms ...... 124

18. Autocorrelation statistics-aggregates ...... 129

19. A summary of results using the Durbin-Watson statistic by theory and level of aggregation ...... 130

20. A summary of results using Durbin's h statistic by theory and level of aggregation...... 134

21. Analysis of variance of T/E by estimator (OLS and D-TSLS) and c o e ff ic ie n t ...... 137

22. Analysis of variance of T/E by bias component and constrained coefficients ...... 141

23. Estimated and true lag structures and th eir bias by industry group. Neoclassical I and I I theories . . 143

24. Average lags and proportion of investment completed after two years for various values of B4 and consistent with a general Pascal lag ...... 149

25. Sign of the differences between the estimated and true aggregate residual variances by theory and industry . 153

26. Ratio of the estimated to the true aggregate residual variance, ^ /{U 'U /T ), by theory and industry . . . 154

27. Analysis of variance of ^/(L I'U /T ) by theory and industry ...... 155

28. Ratio of the bias components to the estimated aggregate residual variance ...... 156

29. Average of the absolute values of the ratios of the components to S 2 ...... 158

30. Number of times, out of six, a theory had a lower standard error than competing theories by true and estimated residual variance...... 159

v ii Table Page

31. Investment theories ranked by true and estimated residual variance...... 160

32. Kendall's t correlation of the true and estimated rankings of five investment theories ...... 161

33. Ratio of the components of the aggregate estimated coefficients to E. Boot-DeWIt ...... 163

34. Ratio of the components of the aggregate estimated coefficients to E. Gupta ...... 166

35. Kuh's equation f o r m s...... 167

36. Ratio of the true to the estimated aggregate coefficients. K uh ...... 168

v iii LIST OF ILLUSTRATIONS

Figure Page

1 . 95% confidence ellipse from constrained estimates of B4 and Bg. Neoclassical I ...... 146

ix INTRODUCTION

1. Micro theory and macro data: The aggregation problem

I t frequently happens in economics that several competing the­

ories exist to explain an economic variable. Fixed capital invest­

ment behavior is a case in point where there are at least four compet­

ing theories ... accelerator, expected profits, liquidity, and neo­

classical J Each theory and its variations has been proposed and def­

ended as explaining movements in fixed capital investment; however, a

gap exists between these theories and their empirical testing. While

the theories are developed at the micro or individual firm level they

are quite often tested at a more aggregative or industry level. 2

Usually this is done without any discussion of or ju stificatio n for

the rationale of applying a theory of individual firm behavior seriatim

at the aggregative level.^

^These theories are discussed in Chapter I I .

^See Robert Eisner, "A Distributed Lag Investment Function," Econometrica. XXVIII (January, I960), 1-29; W. H. L. Anderson, Corpor­ ate Finance and Fixed Investment (Boston: Division of Research, Grad­ uate School of Business Administration, Harvard University, 1964); Dale W. Jorgenson and James A. Stephenson, "Investment Behavior in U. S. Manufacturing, 1947-1960," Econometrica, XXXV (April, 1967), 67- 89; J. R. Meyer and R. R. Glauber, Investment- Decisions, Economic Fore­ casting, and Public Policy (Boston: Division of Research, Graduate, School of Business Administration, Harvard University, 1964).

^For example, C. E. Ferguson in his excellent book The Neo­ classical Theory of Production and Distribution (Cambridge: Cambridge University Press, 1969) says at the outset, "But the macroeconomic theory discussed is the macroeconomic theory constructed by analogy with the corresponding microeconomic theory," 5. 1 Economists are aware that traversing the ground between a well developed micro theory and a postulated analogous macro theory is an act of fa ith . The assurance which would come from proving that a specific set of micro equations implies an analogous macro equation is absent in the theory of investment behavior.^ R. M. Solow takes a very expedient attitude when he says:

"Before going on* le t me be explicit that I would not try to ju s tify what follows by calling on fancy theorems on aggregation and index numbers. Either this kind of aggregate economics appeals or i t doesn't. Personally I belong to both schools. If i t does, I think one can draw some crude but useful conclusions from the results."5

Economic theory can explain or predict only when confronted with empirical data. The paucity of the data at the individual (firm, consumer, e tc .) level necessitates the use of grouped data. The method of aggregation then becomes important. We shall discuss some aggregation methods in Chapter I; however, a fundamental problem common to most methods is that the micro and macro empirical results

^ It is not absent in the theory of consumer behavior. I t can be shown "that i f the prices of a group of goods change in the same proportion, that group of goods behaves as i f i t were a single commo­ dity." See J. R. Hicks, Value and Capital (2d ed.; Oxford: Oxford University Press, 1946), 312-3T3. Also, Paul A. Samuel son, Foundations of Economic Analysis (Cambridqe: Harvard University Press, 1$47), l4T- m — ------—

5r. M. Solow, "Technical Change and the Aggregate Production Function," Review of Economics and Statistics, XXXIX (Auqust, 1957), 312. are inconsistent . 6 That is, knowledge of the individual relations and

their arguments w ill not usually imply the same value of the aggregate

dependent variable as w ill knowledge of the aggregate relation and its

arguments. The consistency conditions (discussed in 1.4.) are so

severe that they are seldom satisfied in practice.^ The practical question then becomes one of determining, empirically, the magnitude of the errors created by aggregation.

2. Purpose and scope of this study

It is common practice among economists to select one theory over competing theories on the basis of minimum residual variance of

a regression (or maximum R^).® Indeed, i t has been shown that the

estimated residual variance of an incorrect maintained hypothesis w ill

be larger than that of a correct alternative hypothesis; 9 however, 1 t

appears not to have been recognized that i f the hypotheses refer to

aggregated data, the test is innappropriate because the estimated

6The phrase "consistent aggregation" appears to have been fir s t used by M. McManus in "General Consistent Aggregation in Leontief Models," Yorkshire Bulletin of Economic Research, V III (June, 1956), 28-48. The concept of consistent aggregation is explored in Chapter I where a rigorous definition is presented. In addition, various def­ initions of consistency are quoted and shown to be essentially the same, albeit a loose paraphrasing, of the rigorous definition. A distinction can be made between "totally consistent" and the less restrictive "partially consistent" concepts in Yuji Ijir i, "Funda­ mental Queries in Aggregation Theory," Journal of the American Stat­ istical Association, LXVI (December, 197l), 766-782.

^For the necessary and sufficient conditions for consistent aggregation and th e ir derivation, see Andr£ Nataf, "Sur la Possibilite de Construction de Certaines Macromodeles," EcOndmetrica, XVI (July, 1948), 232-244. These conditions are presented in Chapter I. aggregate residual variance is biasedThus, there will be errors

in model selection pertaining to micro data i f these models are

selected using the minimum residual variance of aggregate data. One

purpose of this study w ill be to determine the size of the error in

the aggregate residual variance.

Nataf's paper^ has shown that micro and macro parameter

estimates w ill be inconsistent unless a ll parameters for a given

variable are identical. This study, after specifying a "true" aggreg­ ate parameter, w ill measure the size of this aggregation bias. Micro

and macro structures will be chosen from the rational distributed lag

class of functions . ^ 2

8As an example, see Dale W. Jorgenson and Calvin D. Siebert, "A Comparison of Alternative Theories of Corporate Investment Behavior," American Economic Review, L V III (September, 1968), 681-712.

^Henri Theil, Economic Forecasts and Policy (2d ed.f rev.; Amsterdam: North-Holi and, 1961 ), 326-334.

10I t w ill be shown in Chapter I that the sample variance of the estimated macro disturbance equals a "true" sample variance plus the sum of the variances and co-variances of aggregation bias com­ ponents. Thus, a model selected over competing models using the minimum residual variance criterion at the macro level w ill not necessarily be the same model, selected on the same basis, i f the micro data were available.

^N ataf, Econometrica, XVI.

12'The rational lag function is discussed in Chapter I. Est­ imation problems and techniques are discussed in Chapter I I I . For the development of the rational lag function, see Dale W. Jorgenson, "Rational Distributed Laq Functions," Econometrica, XXXIV (January, 1966), 135-149. I f the micro and macro parameters are constrained to conform to those of a general Pascal distributed lag function^ we shall be interested in the errors among the estimated aggregate, "true" aggre­ gate, and individual lag structure and average lag.

The scope of this study is limited to a specific set of com­ peting theories . . . those of fixed capital investment behavior. A spirited debate is taking place in the current economic literature as to which of four investment theories is correct . . . the accelerator, the expected profits, the liquidity, or the neoclassical th e o r y .^

We shall review these theories in Chapter I I and c ritic a lly assess th eir empirical performance in Chapter IV. 1 5 These theories are used solely as a vehicle for application of the concepts developed in

Chapter I which concepts have wider applicability than to the field of investment behavior.

The study is further confined to a specific set of micro and macro functions. Specifically, to the class of rational lag funct-

13 Jorgenson, Econometrica, XXXIV, 135-149.

^Jorgenson and Siebert, American Economic Review, L V III, 681-712.

^The reader is referred to Dale W. Jorgenson, "Econometric Studies of Investment Behavior: A Survey," Journal of Economic Literature, IX {December, 1971), 1111-1147 fo r a review of empirical studies in fixed capital investment. This a rtic le is reviewed in Chapter I I . ionsJ6 This is not a severe constraint for two reasons: (a) The class of rational lag functions subsumes those of the finite, arith­ metic, geometric, and Pascal as special cases; (b) the theory of aggregation bias developed applies to non-distributed lag functions as well as to distributed lag functions. Estimation of the rational lag function is discussed in Chapter I I I . ^

Answers to these problems are relevant to three, not necess­ a rily mutually exclusive, groups of persons: The policy maker; the theorist; and the empirical economist.

The policy maker as forecaster and controller is interested in the magnitude, speed, and s ta b ility of policy variable changes on investment. I f the results of the model d iffe r greatly because of aggregation bias or because of incorrect model choice then his results are misleading.

The economic theorist is also interested in answers to the .• above three questions. The wide debate in the literature concerning the merits of various competing investment theories w ill be largely settled by empirical performance. Although aggregation affects this performance, very l i t t l e discussion of the direction or extent of this lft effect has yet taken place in the literatu re of investment behavior. 10

^Jorgenson, Econometrica, XXXIV, 135-149.

l^See Phoebus J. Dhrymes, Distributed Lags: Problems of Estimation and Formulation (San Francisco: Holden-Day, 1971), Chapter 9 for a discussion of the problems of estimating the parameters of a rational lag function.

18See 0. C. 6 . Boot and G. M. DeWit, "Investment Demand: An Empirical Contribution to the Aggregation Problem," International Economic Review, I (January, 1960), 3-30 who measure the dlhdttlbn ana extent of aggregation bias for ten heterogeneous firms, one investment theory (expected p ro fits ), and non-distributed lags. This article w ill be reviewed in Chapter V, CHAPTER I

AGGREGATION BIAS IN RATIONAL LAG FUNCTIONS

1. Aggregation and the aggregation problem

The term "aggregation" refers to the process of reducing a set of quantities and/or relations to a smaller set of quantities and/or relations. 1 For example, we may wish to aggregate demand functions for each commodity and each consumer in the U. S. into a macro consump­ tion function for the entire economy; or, we may wish to aggregate investment functions for individual firms into an industry investment function.

Having defined the ter’m "aggregation" we now explain what is meant by the phrase "the aggregation problem." This phrase is used by

^ h is is the essence of definitions of the term "aggregation" given by: Lawrence R. Klein, "Macroeconomics and the Theory of Rational Behavior," Econometrica, XIV (A pril, 1946), 93; E. Malinvaud, Statistical Methods of Econometrics (Chicago: Rand McNally, 1966), 118; Kenneth May, "The Aggregation Problem for a One-Industry Model," Econometrica, XIV (October, 1946), 285; Kenneth May, "Technological Change and Aggregation," Econometrica, XV (January, 1947), 51.

2See R. G. D. Allen, Mathematical Economics (2d ed.; London: Macmillan, 1966), Chapter 20 which is entitled "The Aggregation Problem" and in particular section 20.1 which is entitled "The Problem"; T. M. Brown, Specification and Uses of Econometric Models (London: Macmillan, 1970)^ Section 8.2 is entitled "The Nature of the Mathematical Problem of Aggregation"; H. Theil, Linear Aggregation of Economic Relations (Amsterdam: North-Holi and, 1954), section 1 ."I is entitled "Aspects of the Problem." I t is not implied here that the authors were solely or even generally concerned with the consistency of aggregation but a ll start by recognizing i t as "the problem." As I j i r i says, "total consistency is the most basic problem in aggregat­ ion theory." See Yuji I j i r i , "Fundamental Queries 1n Aggregation Theory," Journal of the American S tatistical Association, LXVI (December, 19/1), 756. 7 many authors, for example Allen, Brown, and Theil, 2 and refers to the fact that the results of aggregation are generally inconsistent,

While a rigorous definition of consistency w ill be presented in section 1.3. i t w ill be helpful to present an in tu itiv e discussion of the aggregation problem at this point.

Aggregation involves relationships between three sets of functions: (a) A set of functions for individuals (firms, consumers, etc .) relating an endogenous variable to exogenous variables. For example, the set of individual demand functions or the set of firm investment functions in the above examples. I f we assume J indiv­ idual functions in K variables we may write the jth individual equat­ ion as: f1*1*) yj = V xl j xKj) [j[ " ]■ ***’ jjj

(b) A set of functions relating the aggregate variables to the underlying individual variables:

( 1 . 2 .) x^ - X(^ (x|^*|, . . . iX|^j) (j — 1 , . . . ,J) (k = 1 , . . . ,K)

(1.3.) y " y(yj, ... »yj) (j ~ 1» ••• »J) (k — 1 , . . . , K)

(c) An aggregate function relating the aggregate variables:

(1.4.) y ■ Ffx'j, ... »X|^)

The above four equations define ah aggregation structure which consists of a microsystem ( I . I . ) , a macrosystem (1 .4 .), and aggregation functions (I.2-1.3.). To understand the nature of the aggregation problem we refer to the following schematic diagram of an aggregation structure:.4

(d = 1, ... ,J) (k = 1, ... ,K)

{xk} -

where: {*kj ) s micro exogenous variables.

{y j> = micro endogenous variables.

{xk> = macro exogenous variables.

{y} = macro endogenous variables,

f j = microrelation.

F = macrorelation.

*k = aggregation function fo r the exogenous variables, y = aggregation function fo r the endogenous variables.

From the aggregation structure schematic, we can see that given the micro-variables {xkj, yj> and microrelation fj as well as the aggregating functions xk, y we have defined the macro-variables

yd. There is thus an implied relation between {x^} and {y}. I f , in addition, we explicitely relate {x^} and {y} independently of F it

^ I j i r i , Journal of the American Statistical Association, LXVI. 766.

4 I j i r i , Journal of the American Statistical Association, LXVI, 768. 10 would be unlikely that the two functions (Implicit and explicit) would

be identical. The implicit and explicit functions would be inconsist­

ent. The same problem of inconsistency would arise i f we allowed the

macro endogenous variable (y> to be implied by the micro variables

{xkj» y j) and relation f j and aggregating function y while indepen­

dently specifying a macrorelation F and exogenous aggregating function

y -

2. The meaning of "consistent aggregation"

The adjective "consistent" is not used by all authors. Some

choose to use words such as exact, intrinsic, perfect, or totally C consistent. Be that as it may, we shall now present definitions of

consistency by various authors.

1. Green says, "Aggregation w ill be said to be consistent when the use of information more detailed than that contained in the aggregates would make no difference to the results of the problem a t hand. " 6

2. Fisher says, "... the term 'exact simplification' is used here to describe the situation where the detailed data X and the simplified data K coincide and there is no loss."7

3. I j i r i says, " . . . i f an aggregation structure is to ta lly consistent, the behavior of the microsystem can be completely identified from the behavior of the macro­ system . " 8

5Walter D. Fisher, Clustering and Aggregation in Economics (Baltimore: Johns Hopkins Press, 1969), uses the term "exact." TFTeil, Linear Aggregation, uses"perfect." I j i r i , Journal of the American Statistical Association, LXVI, uses "totally consistent."

6 H. A. John Green, Aggregation in Economic Analysis (Prince­ ton: Princeton University Press, 1964), 3.

^Fisher, Clustering and Aggregation in Economics, 18.

8 I j i r i , J. of the American S tatistical Association, LXVI, 766. 11

Whether the definition of consistency 1s expressed in the language of information theory via loss functions or in the language of deductive logic they are a ll saying the same thing. An aggregation structure consisting of a microsystem, a macrosystem, and an aggreg­ ation function is consistent if it contains no formula such that both the formula and its negation can be derived from the structure. 9

3. Necessary and sufficient conditions fo r consistent aggregation

Are the four functions (I.1 .-I.4 .) consistent? Is the analogy approach to aggregation consistent? In the process of answering these questions we shall rigorously define what is meant by "consistent."

Nataf^ 0 derived the necessary and sufficient conditions fo r the aggregation of the individual functions ( 1 . 1 .) to the aggregate function (1 .4 .). According to Nataf's Theorem^ necessary and suff­ icient conditions for the aggregation of the functions ( 1 . 1 .) to the function (1.4.) are that there exist functions G, H, g^, hj, G^, Hj,

^kj* suc^ that:

9This is the Post criterion for consistency and is one of four properties of formal deductive systems. See Irving M. Copi, Symbolic Logic (2d ed.; New York: Macmillan, 1965), 180.

l^Andre Nataf, "Sur la Possibilite de Construction de Certalnes Macromodeles," Econometrica, XV (July, 1948), 232-244.

^See Green, Aggregation in Economic Analysis, 36-38 for a statement and proof of this theorem. (1.5.) y a H[h1 (y 1) + . . . + hj(yj)3 = G[gj(xj) + ... + gk(xk)] where: = HjChi j ( xl j ) + ••• + hKj^xK j^ and xk = G[gkj(xkj) + ... + 9kj(xkj)3

for (j = , *... ,J) and (k= 1 , . . . ,K)

What do Nataf's conditions mean for linear aggregation?

According to Nataf's Theorem there must exist functions G, H, gk» hj,

Gk, Hj, gkj, hkj satisfying the equation (1.5.). Thus, for linear aggregates we can re-w rite equation (1 .3 .) as:

(1 .6 .) y = y(y-), ... ,yj) = zy. = H[zhj(yj)]. J J

Similarly, we can re-write equation (1.2.) as:

(I<7*) xk = xk(xk l xkj) = *;xkj " Gk^ 9 kj^xk j^ * J u

I f equation (1 .1 .) Is linear we can deduce the functions Hj and hkj =

( 1 . 8 .) yj “ fjUij. ... “ aj + Jbkjxkj K

Hj^ hkj^xkjJ3.

Finally, if equation (1.1.) is linear we can deduce the func­ tions G and gk from equations (1.1.) and (1.3.):

(1 .9 .) y = Eyj = saj + ZSbkxkj . J J JK 13

We conclude, therefore, that the necessary and sufficient

conditions for consistent aggregation of a set of Individual firm

investment functions into an industry investment function are that

the individual functions must be linear with identical slopes.

4. Approaches to the aggregation problem

Various approaches to the aggregation problem have been pro­

posed. These approaches d iffe r in their handling of the four equat­

ions ( I .1 .- I . 4 . ) discussed above. The consistency approach to aggre­

gation places primary importance on the consistency of these four equations. 12 For example, Klein1 3 takes a micro-theory and certain

properties which the macro-theory must f u lf i ll as given. The prob­

lem then is to construct consistent aggregates of the micro-variables.

May 1^ starts with a micro-theory and aggregates, then poses the question of what macro-theory is consistent with them.

^There is a difference between consistent aggregation and the consistency approach to aggregation. Consistent aggregation, referring to the final result of an aggregation structure determines if both a formula and its negation can be deduced from an aggregation structure. On this, see section 1.9. The consistency approach to aggregation is concerned with the derivation of variables and/or relations which are consistent with other parts of an aggregation structure. On th is, see the difference between May and Klein (presented below) where i t is shown that two different structures are developed following the con­ sistency approach but are both (internally) consistent.

1 3 Klein, Econometrica, XIV, 93-108.

1 4 May, Econometrica, XIV, 285. Theil1 5 begins his theory of aggregation by stating that while

the Klein and May approaches lead to consistent aggregation they

happen not to be the commonly used methods of aggregation. He

therefor presents the analogy approach to aggregation. From a given micro-theory and the aggregates which are simple or weighted sums of

the micro-variables he constructs a macro-theory by analogy. For example, i f an individual firm 's investment demand is a function of changes in desired capital stock then we would postulate that the

industry investment demand is a function of the sum of the firms change

in desired capital stock. As mentioned in the Introduction to this study, this appears to be the most common approach to aggregation although i t is seldom exp licitely stated. I t is the analogy approach which we wish to explore in this study.

5. The analogy approach to aggregation

Theil^ 6 begins his study of the analogy approach by aggregat­ ing over one set of individuals. We shall follow his analysis in this section but recast the relations in the convenient matrix notation of KloekJ7

Let us assume that micro economic theory determines a function­ al relationship between a set of variables. We shall not be concerned

l 5 Theil, Linear Aggregation of Economic Relations.

1^Thei1, Linear Aggregation of Economic Relations.

1 7 T. Kloek, "Note on Convenient Matrix Notations in M ulti­ variate Statistical Analysis and in the Theory of Linear Aggregation," International Economic Review, I I (September, 1961), 351-360. 15 at this point with the economic content of the theory but merely state that we have a set of J linear relations for J individuals (firms, consumers, e tc .):

( 1 . 1 0 .) Yj * xjBj + Uj = 1 ...... where: Yj a a (Txl) vector of dependent variables for the jth relation.

Xj = a (TxK) matrix of exogenous variables for the jth relation.

Bj - a (Kxl) vector of unknown parameters for the jth relation.

Uj = a (Txl) vector of random elements with zero expectation for the jth relation.

We introduce the analogous aggregate relationship:

(I.11.) Y = XB + U with Y and X defined as the simple sum aggregates:

(1.12.) Y = 2Y- = a (Txl) vector of the aggregated dependent J J variable over the J relations.

(1.13.) X = SX,* = a (TxK) matrix of aggregated independent J variables over the J relations. and B = a (Kxl) vector of unknown parameters.

U = a (Txl) vector of random elements with zero expectation.

The above four equations describe, as Allen succinctly states,

"the translation of linear micro-relations into a linear macro-relation lfl by means of linear aggregates."

^BAllen, Mathematical Economics, 696. 16

6 . The relations between micro and macro-parameters

The above consistency conditions are very re stric tiv e. What i f the individual parameters are not Identical? What aggregation error does the analogy approach imply? In short, how are the parame­ ters of the aggregate relationship ( I . 11.) related to the parameters of the underlying micro relations (I.10.)? Our purpose in this section is to relate the aggregate parameters, B, to the individual parameters,

To relate B to Bj we introduce a set of auxiliary equations in which each variable in the Xj matrix is a linear function of all the aggregate independent variables:

(1.14.) Xj = XDj f Vj (j = 1, ... ,J) where: Xj = a (TxK) matrix of exogenous variables in the jth relation.

X = ZXj = a (TxK) matrix of aggregated independent variables J over the J relations.

Dj = a (KxK) matrix of auxiliary coefficients relating each variable in Xj to all variables in X.

Vj = a (TxK) matrix of residuals J 9

I f we solve for Dj by the method of least-squares we get:

-1 (1.15.) Dj = (X'X) X'Xj

We note that the least-squares estimate of B is:

-1 (1.16.) ti = (X'X) X'Y

^9The residuals, Vj, are non-stochastic due to our assumption that X j, and hence X, is non-stochastic. 17 and that:

"1 (1.17.) B = E(B) = E[(X'X) X'Y] by the assumption that E(U) = 0.

Substituting equations (1 .1 2 .), ( I . 10.). and (1.15.) in (1.17.) we get:

-I (1.18.) B = E[(X'X) X'E (XjBj + Uj)] = EDjBj. J J What meaning is to be attached to equation (1.18.)? First, the macro-parameters, B, are those implied by the macro-relation

( I . 11.) and the specified aggregates ( I . 12.- I . 13.). Second, the macro­ parameters, B, are functions of the underlying micro-parameters, Bj.

But what is the nature of these functions?

To answer the above question, we expand equation (1.1 8.):

V ^ llj D12j ... Dlkj ... D1Kj b2 °21 j B2j * = SD.B-- = Z • * * J 3 3 J • * * • • ... DKkj ... DKKj< bk. ?Klj °K2j ?Kj.

The matrix Dj are the least-squares parameters of the regression of

Xj on X. That is , for a given firm ( j ) we regress each variable

(column) of Xj on all the variables in X. Thus, Dgkj represents the coefficient of the regression of the kth micro-variable of the jth firm on the gth macro variable. I t follows from this that the fir s t row in Dj is the set of constants from each regression. 18

We can now write the implied intercept term as:

(1.20.) B1 “ £Dlk jBkj ft • • • f K J 0

The implied aggregate slope parameter is:

Bk = j j DgkjBkj ,K .K • J

These results must be disappointing to a macro-economist.

First, the implied macro-parameters are specific to the period of time covered by the data. Even i f the micro-parameters, B-, remain J unchanged outside the sample period, the implied macro-parameters may change. This is so because the macro-parameters, B, depend on Dj which in turn depends on the micro-variables, Xj. Therefor, if the relation between X- and X changes outside the sample period but B-: J J remains unchanged we can have changes in the macro-parameters.

A second problem with the implied macro-parameters is that they are functions not only of the corresponding micro-parameters but of the non-corresponding micro-parameters as well. That is , the Bkth macro-parameter is a function not only of the corresponding micro­ parameters, Bkj (j = 1, . . . ,J) but of the non-corresponding micro­ parameters Bgj (g f k, g - 1, ... „K), (j = 1, ... „J). A welcome exception is that the micro constants do not affect the implied macro slope parameters. 19 7, A measure of the aggregation bias 1n macro-parameters

Equation (1.18.) relates the aggregate parameters to the under­ lying micro-parameters. For consistent linear aggregation, the corres­ ponding micro slope parameters must be identical. 2 0 Thus, a measure of aggregation bias in macro-parameters could be derived from the definition of consistency and the actual relationship between the macro and micro-parameters. The implied macro-intercept term in equation (1.20.) can be written as the sum of the micro-intercepts plus certain covariance terms:2^

(1.22.) B] = EB-| j + JjCov(BkjDlkj ) (k = 2...... K) J K ( j = 1 * • •« »J)

The implied macro slope parameters in equation (1.2 1.) can be written as the average of the corresponding micro slope parameters plus certain covariance terms : 2 2

i (1.23.) Br, = J EBl.., + J 2Cov(BifiDaki) (k =2 ...... K) J K 3 9 3 (g - 2 K ( j - 1...... J)

Theil has shown that it is always possible, by a suitable type of fixed weights aggregation, to obtain consistency between the micro

20Using Nataf's definition of consistency. See section 1.3. Also, for the same result see; Allen, Mathematical Economics. 709; Green, Aggregation in Economic Analysis, 40; Theil, Linear Aggregation of Economic Relations, 142.

2 ^Theil, Linear Aggregation, 16; Allen, Mathematical Economics, 711.

22Thetl, Linear Aggregation, 16. 20 and macro equations. That is , " if aggregation is performed such that a ll microvalues . . . are weighted proportionally to th eir microparame­ ters . . . both the intercept . . . and the rates of change . . . of the macro-equation depend on corresponding microparameters only."23

Our purpose in this study is not to obtain consistent aggre­ gation but, rather, to measure the errors when linear aggregates are used. Thus, if we define the "true"2^ macro-intercept as:

(1.24.) 2 B-| j (j = 1...... J) J then the aggregation bias of the macro intercept is:

(1.25.) JSCov(BkjDlkj-) (k = 2...... K) K ( j = 1, . . . »J)

I f we define the "true" macro slope parameter as:

(1.26.) UBkj* (k = 2, . . . ,K) JJ (j — 1 , . . . |J) then the aggregation bias of the macro slope parameter is :2®

(1.27.) JZCov(BkjDgkj) fk = 2, . . . ,K) K (g - 2 ...... K) (j = 1 J)

We can separate the aggregation bias into a bias attributable to corresponding and non-corresponding micro-parameters. F irs t, we re-w rite the covariance terms of both the intercept and slope parame-

^3 Theil, Linear Aggregation, 18.

2^The word "true" was used by J. C. G. Boot and G. M. DeWit, "Investment Demand: An Empirical Contribution to the Aggregation Problem," International Economic Review, I (January, 1960), 3-30. 25Note that absence of aggregation bias does not imply consis­ tency using Nataf's definition of consistency as explained in section 3. 21 ters equations (1.25. and 1.26.):

(1.28.) 0Kc°v (DkgjBgj > - ® kgjBgJ - 0DkgBg ■ 1. ... ,K] = 2, ... ,Kj [j - 1 , . . . ,JJ where DjJg = 1ZDkgj and B* = lEBgj JJ Now, since^®

(1.29.) J1 SDka1B; 0 TJ kgJ 9

LB*KJ

Then, combining equations (1.28.) and (1 .2 9 .), we get the aggregation bias of the macro-intercept:

(1.30.) ( g = 2 ...... K) (4 = 1...... 0) and the aggregation bias of the macro slope parameters:

(1.31.) Jk.... =.2 ...... K] iDkgjBgj " Bk [g = 2 , . . . ,K [j = 1, ... .J]

26We make use of the fact that z D.,- 0 0 ., 0 J Kg 1 0 .. 0 0 1 0 0

0 ... 1.

See Allen, Mathematical Economics. 710. 22 The aggregation bias of the macro slope parameters attributable

to the corresponding micro-parameters 1 s:

(1.32.) £°kkjBkj " Bk (j * 1* *" *jj while the aggregation bias of the macro slope parameters attributable to the non-corresponding micro-parameters is:

(1.33.) £BkcHBai (k s 2» . . . ,K) J K9J 9J (g * 2.. . . . |K) jk t g) (j - 1 , . . . »J)

I t w ill be noted from equation (1.30.) that the entire aggreg­ ation bias of the macro-intercept is attributable to the non-corres­

ponding micro-parameters.

8 . The concept of a distributed lag

The concept of a lag in economics is required whenever a cause

in one time period has its effect in another (future) time period.

If the total effect of one variable on another does not take place in

the same (future) time period we refer to this as a distributed lag effect.

We may write a distributed lag function as:

( 1 . 3 4 .) Yt =sJjwsXt _s where and Xt are observed values of the dependent and independent

variables at time t and the ws are the unknown parameters.

For estimation we impose the requirement that a fin ite change 23 in the independent variable results in a fin ite change in the depen­ dent variable:

(1.35.) sҤws = ^ +0I>

A more restrictive constraint is:

(1.36.) s=ows = ®

9. The rational lag function

Several authors of distributed lag schemes have been motivated by their desire to approximate the in fin ite lag model (1.34.) with a more manageable model.2? They have generally sought...

(a) "To approximate the true lag coefficients closely.

(b) To do so with as few parameters as possible, while keeping the estimation scheme relatively simple."28

2?For example, the geometric lag model is an approximation to (1.34.) by replacing the infinite set of coefficients, ws, by two par­ ameters, a, b, where ws = abs, s = 0 ,1 ,2 , . . . , 0 Jan Tinbergen, Econometrics (New York: McGraw-Hill, 1951). The Pascal distributed lag model is obtained from (1.34.) by replacing the coefficients with ws = [(r+u- 1 ) ! /( u ! ( r - 1 ) ! ] ( l- c ) rcU where 0

2 8 Phrymes, Distributed Lags, 44. Also see Dale W. Jorgenson, "Rational Distributed Lag Functions," EcOndmetrica, XXXIV (January, 1966), 135. 24 2q Jorgenson 3 has recently.Introduced a class of distributed

lag function purporting 3 0 to satisfy the two properties cited above.

We shall not follow his development but, rather, show that the infin­

ite distributed lag model (1.34.) may be written as the ratio of two

finite polynomials:3^

(1.37.) Yt = W(L)Xt * A|L|Xt

where: Yt = a dependent variable in period t .

X^ = an independent variable in period t.

L « the lag operator such that LX^ s ^t-l» *■' *t = *t-2 » etc*

W(L) = w0 + w-jL + w2 L2 + . . . = EwsLs s= 0 m A(L) = ag + a“]L + a2 L2 +»••■*■ amL = E ®i^-

n B(L) = 1 + bnL + b2 L2 + . . . + bnLn = E b

To interpret equation (1,37.) we must assign meaning to the

rational expression, A(L)/B(L). By virtue of the isomorphism between

the algebra of polynomial operators, R(L), and the algebra of poly­

nomial functions,R(t), we shall first assign meaning to A (t)/B (t),

where R(L) = rg +rjL + r2 L2 + ... , R(t) = r 0 + r-jt + r 2 t 2 + . . . ,

A (t) = a0 + a^t + a2 t 2 + . . . + amtm, B(t) = 1 + b-jt + b£t2 + . . . +

bntn„ L is the lag operator, t is a real number, and {r.,*}, {a^} , {bj>

2 9 Jorgenson, Econometrica, XXXIV, 135-149.

30We say "purporting" because the claims have recently been criticized . See p. 26 of this study for these criticisms.

3lThe equations in the following development w ill be written in deterministic form. A stochastic term and the concomitant estimation problems w ill be discussed in Chapter I I I . 25 are sequences of real numbers. After assigning meaning to A (t)/B (t) we then translate the results into the polynomial operator, A(L)/B (L).3^

Assume the equation B(t) = 0 has n distinct roots t j , t 2, ...» t n. Then, 3 3

n (1.38.) B (t) = n ( t - t . ) 1=1 1 which equals

(1.39.) B(t) - [ S (-t^K S (1-t/tj)] i=l 1=1 and34

n (1.40.) n (-t,) = (-1)" n t. = i. i= l 1 1 = 1 1

Therefor,

n (1.41.) B(t) = n (l-t/t.) 1=1 1 and

(1.42.) 1/B(t) = 1/C n d -t/t^ ] = n [l/(l-t/t|)]. 1 = 1 i=l

Now, if = 1/t-j»|A-( |<1» then, for |t]

32See Dhrymes, Distributed Lags, 23-24. ■io ^See Dhrymes, Distributed Lags, 26-27. The derivation which follows is adapted from Dhrymes, but instead of transforming B(t) into a monic polynomial we begin with the f ir s t coefficient of B (t), bfl, normalized a t unity (see equation 1.37.). In addition, we assumeuthat A(t) and B(t) have no characteristic roots in common. 26 we have

p 00 00 (1.43.) 1/B(t) = H [ 2 (Aft)k] = 2T4tJ. 1=1 k= 0 j = 0 J Therefor,

m . 00 , » (1.44.) A(t)/B(t) = ( 2 a^t )( 2 r^J) = 2 w.ts i = 0 j = 0 J s= 0 s s where: ws= 2 a.jrs-.j , s = 0 , 1 , 2 , . . . . ,m

tn = 2 a,.r_ . , s = m+1 , ntf-2 , . . . 1 = 0 1 s * '1

By virtue of the isomorphism between the polynomial algebras R(L) and

R(t) we may write equation (1.44.) as:3®

(1.45.) A(L)/B(L) = 2 w_Ls. s= 0

We conclude that

(1.46.) Yt = ^ j Xt = J QwsLSxt

We have shown above that an in fin ite lag model (1 .3 4 .) can be deduced from a rational lag model (1 .3 7 .). Jorgenson set out to show that an in fin ite lag model can be "approximated to any desired degree

0 7 of accuracy"0' by a rational lag model. Sims has criticized

3 5 Dhrymes, Distributed Lags, 28.

3 ®Thus, "L" performs two duties, i . e . , as an operator and as an algebraic value.

3 7 Jorgenson, Econometrica, XXXIV, 135. Also, see his Approx imation Theorem 1, p. 140. 27

Jorgenson's approach to the rational lag function because Jorgenson has shown that any real-valued function on the non-negative Integers can be uniformly approximated arbitrarily well by rational lag functions . 3 8 The problem, as succinctly stated by Dhrymes, is that

"we do not obtain coincidence between any subset of lag coefficients in the 'tru e' and approximate structure, except by accident . 3 0

Dhrymes offers an alternative rationale for using the rational lag function besides that of approximating an in fin ite lag function ; 4 0 however, i t happens not to be the rationale used by Jorgenson in the models we shall be using (see Chapter IV) and therefor w ill not be used here.

10. The general Pascal distributed lag function

We shall be concerned in Chapter V with estimates of the aggregation bias of the time-form and average lag of the rational lag function. Recall that when we fir s t introduced the distributed lag function Y+ = ewsLsX. we left open the option of constraining s= 0

38christopher Sims, "The Role of Approximate Prior Restrictions in Distributed Lag Estimation." Manuscript dated June 1970, p. 3. '

3 0 Dhrymes, Distributed Lags, p. 51.

4 0 0hrymes„ Distributed Lags, p. 237. 28

the sequence {ws} to correspond to the probability distribution of

a non-negative, integer-valued, random variable, so that:

(1.47.) ws>0 (s-0,1,2,...) and E ws = 1. s= 0

If this probability distribution has a rational generating function

i t is called, by Jorgenson, the general Pascal d is trib u tio n .^

I f the coefficients ws conform with (1.47.) then we can deduce the time-form or probability distribution of the rational lag function as follows:^

From (1.44.) and multiplying by B(t) we get

(1.48.) A(t) = B(t) E wsts = Eb^ EWstS s= 0 j= 0 J s = 0 which equals

n . x . - ratn(q.n) , (1.49.) A(t) = E EWfib ;tS+J = E (EW- ^ b j* , j = 0 s= 0 J q= 0 j=Oq‘ J J

where q=s+j.

Since the coefficients of t s+J and t q must be equal in equation

(1.49.) then,

^Jorgenson, Econometrica, XXXIV, p. 139.

42yhis is the normalized lag structure as adapted from Dhrymes, Distributed Lags, pp. 28-29. Also, see Jorgenson, Econometrica, XXXIV. pp. 146-148. Therefor, from the recursive system (1.50.) and given a.j,bj we can compute as many ws coefficients as needed. These ws values along with the integer values, s = 0 , 1 , 2 ,... form a probabil­ ity distribution or time-form of the effects on the dependent variable, Yt , of changes in the independent variable in time t-s ,

Xt-s*

11. Parameter bias in rational lag forms

An assumption was made in section I I . 5. which does not apply to relations chosen from the class of rational lag functions.

Hitherto we have assumed that X = SX.. This is true only when, J J coincidentally the polynomials Aj(L) and Bj( L) are of the same order for each j (firm ). Consequently, there is an additional bias intro­ duced into the aggregate relation. I t 1s the purpose of this section to develop the nature of this bias.

Theil's analogy approach requires all the micro equations and the macro equation to have the same set of included variables. In contrast, a micro equation chosen from the class of rational lag functions by the minimum residual variance criterion ^ 3 neither has the same number nor the same set of included variables. For example,

43jhe minimum residual variance criterion selects that regression from the rational lag class which has the smallest "unexplained" variance. 30 assume the number of firms in an industry is J = 2. Assume that we choose a specific structure from the rational lag class for each firm by minimizing the residual variance. Thus, for j = 1 and from the class ni=2 , n= 2

( I 51 1 Y - Al (L) xl t + ui t ( 1 u " s^uT where = an observation, in period t , on the dependent variable of the fir s t firm.

X-jt = an observation, in period t , on the independent variable of the first firm.

A-j ( L) = aQ-j + a^-jL +

B^L) = 1 + bn L + b2 1 L2

u-jt = a disturbance tern, in period t, for the first firm. (Specification of this term is discussed in Chapter I I I . )

44 assume the structure with the smallest residual variance is

(1.52.) Ylt = = Boi; »nL3 x1t

B](L) [ 1 + b^L + b2 ]L^]

Likewise, for j=2 and from the class m=2, n=2,

( 1 . 5 3 .) Y2t = A2 (LlX2 t + u2 t

B2 (L) assume the structure with the smallest residual variance is

^4See Chapter I I I for a discussion of estimation of the rational lag function. (1.54.) Y2t = A2 (L)X2t = [a0 2 + S2 2 L2] X2t.

B2 (L) [ 1 + b ,2 L]

Equations (1.52.) and ( I.54.) In final form^® are, respectively:

« A A (r.55.) Ylt * a01Xit + al l xl,t-l "bll Yl,t-l ~b21Yl,t-2

A A A (1.56.) Y2t = a02X2(t + +a22x2,t-2 -b12Y2.t-l

Equation (1.55.) excludes X-| ^-t - 2 while equation (1.56.) includes x2,t-2 but excludes X2

(1.57.) Xt - X1#t + X2ft

xt-l = x1,t-l + 0 xt-2 = 0 + x2,t-2

Yt-1 = Yl,t-1 + Y2,t-1 Yt-2 “ Y^t-2 + 0

The result of the above discussion is that instead of testing the aggregate model:

(1.58.) Y = XB + U where X is the matrix of simple sum aggregates used by Theil, (1 .1 4 .), and discussed in section 1,5. we in fact test:

^Jorgenson, Econometrica, XXXIV, p. 138 32

(r . 5 9 .) y = xir + tt

where X in the previous example, would be

. xl , l + X2 ,l xl , 0 + X2 , 0 xl (- l + x 2 ,-l

X1 , 2 + X2 , 2 Xl, l + X2 ,l Xl , 0 + x 2 , 0

• • •

• * #

« • •

Xl, t + x2 , t X1 , t - l + X2 , t - 1 Xl, t - 2 + X2 , t - 2

We shall, in section 1.12,, develop measures of aggregation bias

for the parameters of the rational lag function just as we did in section

1 . 6 , for the parameters of non-distributed lag functions; however, one necessary preliminary is to develop the relation between the vector of estimated coefficients in the misspecified model (1.59.) B, and the population parameters, B, of the true model ( I . 5 8 ,).46

The least squares estimator of B is:

(1.60.) B = ” 1 X1 Y

Substituting (1.58.) in (1.60.) we get:

(1.61.) I = (X1^)"1! 1 [XB + U]

46For a discussion of specification error analysis see Phoebus J. Dhrymes, Econometrics: S tatistical Foundations and Applications. (New Yoriel Harper and Row, 1970)• 222-229. 33

I f the misspecified variables Y are independent of the error term

U, then:

A (1 .62.) B = E(B) = O ^ X r^ X B or:

(1.63.) B = PB where:

(1.64.) P = ( X ^ P X

P is the (KxK) matrix of least-squares regression coefficients of the explanatory variables in the true model, X, on the explanatory variables in the misspecified model, "X:

(1.65.) X = SP + W

W is a (TxK) vector of non-stochastic residuals. Thus, we conclude that, in general, B is a biased estimator of B and that the specification bias of B is given by:

(I.66.) B - B = (P-I)B where I is an identity matrix of order KxK. K is the maximum of the number of explanatory variables in the true or misspecified models.

The specification bias, as the aggregation bias, can be split into two parts attributable to corresponding and non-corresponding parameters respectively. Recalling the definition of the 34 specification bias as (P-I)B we can write the specification bias due to corresponding parameters for each coefficient as:

(1 .67.) i f k = g = diagonal element. “9 k (k = l ...... K) ■ (g=l L). (K>L).

The specification bias due to non-corresponding parameters for each coefficient is:

(1.63.) i f k ^ g = non-diagonal element. Kg (k=l,...,K), {g=l,...,l), (K=L).

12. The relation between the estimated and the implied macro- paraineters in rational lag functions

We are now ready to combine the theory of aggregation bias developed in 1.5-6. with the theory of rational distributed lag functions developed in 1.9-11. In this section we shall separate the estimated macroparameters in a rational lag function into four components . . . a true value, an aggregation bias, a specification bias, and a sampling error.

Assume the J individual equations in the rational lag function

Wj(L ) d - 69-> h j = £ * :]...... U"' •••• |JJ

I f we add a constant and a residual we have, in matrix terms:

(1.70.) Yj = XjBj + Uj ( j = l , . . . , J )

Assuming equal weighted linear aggregation of the variables we have: 35

(1.71.) Y « (j=l...... J) J

X = zXj J

The analagous macro model is:

(1 .7 2 .) Y = XB + U

Instead of applying the model in (1.72.) we use:

(1 .7 3 .) Y = ‘XB + 0

We have seen in equation (1.63.) that 15 is related to B by:

(1.74.) B = PB

From equation (1.19.) B » sDjBi * therefor, we have the implied j 3 macroparameters:

(1.75.) B = PZO.B. J J J

From equation (1.61.) we have the estimated values of B:

A (r.76.) B = (X1^)”1)!1 [XB + u]

Combining (1.75.) and (1.76.) we get:

(1 .7 7 .) B = PB + CX1X)“ 1X1U

Recalling the definition of specification bias (1.66.), (P-I)B, we have: 36

(r.7 8 .) B = (P-I)B + B + (715?)“ 1X1U

where the implied sampling error 1 s:

(1.79.) (X 1 X)"1X1U

Now from (1 .1 9 .), B can be written as sDjB. which is composed of a J true value and an aggregation bias. Therefore,

A (1.80.) B = (P-I)ED.Bj + nDjBj + ( ^ x r ^ U . J 3 J

We can summarize our theory of aggregation bias in rational lag functions by the following table in which we lis t the component parts of the estimated and implied macroparameters, their formulas, and th eir equation numbers: (See page 37.)

13. The effect of aggregation on the lag structure andlnonients of a general Pascal distributed lag function

In section (1.12.) we introduced the rational lag function for the j th firm , Wj(L). I f Wj(L) is the generating function of an integer-valued random variable we can, as shown in section ( I . 10.) determine the probabilities, ws, from the following recursive relations:^

^We are now assuming a rational lag function constrained to conform to a general Pascal lag function. The constraints are listed and discussed in Jorgenson, Econometrica. XXXIV, pp. 146-148. This is also the normalized lag structure as presented in Dhrymes, Distributed Lags, pp. 28-29. 37

TABLE 1

Components of bias in the estimated aggregate parameters of a rational distributed lag function

(k -Z,... •K) COMPONENTS CONSTANT (g = 2 ,... .K) SLOPE (g=2 »... ,K) (equation #) (j= l.....J) (equation #) ( j - l t •*. ,J)

TRUE VALUE f i j (1.24.) (1.26.) J T j kj

AGGREGATION BIAS zDlg jBgj (1.30.) sDkgjBgj“B(t (1.31.) J J

DUE TO CORRES­ Zero £DkkjBkj"Bk (1.32.) PONDING J

DUE TO NON­ (1.30.) EDkgjBgj (1.33.) CORRESPONDING J J k/g

SPECIFICATION b',- b, ( 1 . 6 6 .) C l.6 6 .) 5 k-Bk BIAS

DUE TO CORRES­ tP ,g-DB, (1.67.) (1.67.) PONDING C kg -1)Bk k=g

DUE TO NON­ ( 1 . 6 8 .) pkgBk ( 1 . 6 8 .) CORRESPONDING V i k?*g

IMPLIED SAMPLING 1 st element of: k th element of: ERROR (3t1 X>“ 1 X1U (1.79.) ( x V x ^ u (1.79.)

A ESTIMATED (1.80.) (1.80.) COEFFICIENT 8"l K

IMPLIED gl (1.75.) B'k ( r .75.) COEFFICIENT Assume we aggregate J firms into an industry as described in section ( 1 . 1 2 .) having constrained the coefficients in both the micro and macro models to conform to a general Pascal lag function as described in section ( I . 10.). What is the relationship between the aggregate and individual time forms and th eir expected values and variances? Answers to these questions w ill be the purpose of this section.

Since the value-set of the general Pascal probability dis­ tribution is the same (integers from 0 to + ®) in both the micro and macro models, the question centers on the individual probabili­ ties :

(1.82.) w0 ,j» wl,j* w2 , j »• • • and the aggregate probabilities:

( I •83.) Wq , W|i »• • •

From the recursive equations we can see that the estimated probabilities of the aggregate equation are the coefficients of the rational expression, W(L). These, in turn, are functions of the coefficients of A(L) and B(L); however, these coefficients are biased. The nature of this bias was developed in section (1.12.) where we summarized the component parts o f the estimated aggregate

A parameters, B, in Table 1. We shall find i t convenient to introduce

the following short-hand notation for the component parts of

(1.84.) B = T + AB + SB + SE

where:

B = a (Kxl) vector of estimated aggregate coefficients in the misspecified model (1.73.).

T = a (Kxl) vector of true aggre­ gate coefficients described by equations (1.2 4., 1.26.).

AB = a (Kxl) vector of aggregation biases described by equations (1.31-32.).

SB = a (Kxl) vector of specification biases described by equation (1.66.).

SE = a (Kxl) vector of sampling errors described by equation (1 .7 9 .).

Since B is the coefficient vector fo r the rational d is tri­

buted lag function (1.37.) in final form (1.55-56.) we shall name

the coefficients of the numerator:

(1.85.) B

and those of the denominator

4®Note that we are now using coefficients of the rational lag function constrained to conform to a general Pascal lag function as described in section 1,10. Also see footnote 47. 40

(r.8 6 .)

With the above results we can relate the estimated aggregate probabilities, ws, and the true aggregate probabilities, ws, as follows:

(1.87.) w0 = aQ = Tj + (AB1 +SB|+SE1>

w, = a1 -w0 6 1 - T2 + (AB2 +SB2 +SE2 )-w0 [Tm+,

+(™m+l+SBm+l+SEm+1)]

w2 = a2 -Wob2 -Wi6 i = etc.

Thus, when we present our empirical results in Chapter V we shall present a table lis tin g the true and estimated time-forms of the general Pascal distributed lag function and a measure of the bias component (AB + SB + SE).

The expected value, E, of the general Pascal distribution:

oo (1.88 .) E = EWgS p=0

49 can be computed from knowledge of the generating function W(L):

(1.89.) E = W '(l)

4 9 Feller, I, 266. 41 where: W'(l) = the first derivative of the generating function with respect to L and evaluated at L=l.

Therefore, since W(L) = A(L)/B(L) then:

(1.90.) E ° W'(l) ■ B(1)A'(1) - A( 1 )B‘( 1 )

[B (l ) ] 2

Now the constraint A(1)=B(1) is in effect and therefore (1.90.) reduces to:

(1.91.) E = B( 1)

The bias between the estimated aggregate expected value, E, and the true aggregate expected value, E, is:

(1.92.) E-E

2 The variance, V , of the distribution:

(1.93.) V2 * ew [s-E] 2 s= 0 can be computed from knowledge of the generating function W(L)^®

(1.94.) V2 = W"(l) + W'(l) - CW'(l) ] 2

where: W'( 1 ) and W "(l) are the fir s t and second derivatives of the generating function with respect to L evaluated at L = 1.

5°Feller, I, 266 42

The bias between the estimated aggregate variance, V , and

2 the true aggregate variance, V , is:

( I . 95.) V2 -V2

14. Aggregation, minimum residual variance, and model choice

Economists frequently select a regression equation from a set

of possible choices on the basis of maximum R2. This selection

procedure is based on the intu itive feeling that an equation is

better the more variation in a dependent variable correlates with

movements in the independent variables.

A more rigorous explanation of basing a selection procedure ? 51 on maximum R has been offered by Theil. He does this by setting

up two hypotheses . . . an incorrect maintained hypothesis H and a

correct hypothesis H. Theil then shows that i f H is correct but R

is tested, the estimated residual variance of H will be larger than

that of H. Let:

(1.96.) H: Y = XB + U

(1.97.) H: Y = XB + 0

5^Henri T h eil, Economic Forecasts and Policy. (2nd ed. rev: Amsterdam: North-Holland Publishing Co., 19dl). 43

where H - correct hypothesis.

H = incorrect maintained hypothesis.

Y = (Txl) vector of values of the dependent variable.

X = (TxK) matrix of values of independent variables included in the correct model (1 .9 6 .).

X = (TxK) matrix of values of independent variables included in the incorrect maintained model (1 .9 7 .).

B = (Kxl) vector of parameters to be estimated in (1.96.).

B = (Kxl) vector of parameters to be estimated 1n (1.97.).

U = (Txl) vector of residuals in (1.96.).

U = (Txl) vector of residuals in (1.97.).

The estimated sum of squared residuals of the incorrect maintained hypothesis is:

(1.98.) O1!) = ( Y-X5) ^ ( Y-XB)

We can relate this to the residuals of the correct hypothesis by substituting XB+U for Y

(1.99.) U]0 = (XB) 1 M(XB) + U^MU where:

(r.io o .) M = I - vt X(X'X)"'X v W l v l

Now, lettin g a be the variance of U we have:

( I . 101.) Ed^MU) = a2trM = a2 (T-K) 44

Therefor

(T.102.) E ^ U ) > o2 (T-K) or

(1.103.)

The conclusion is straightforward. If a specification is incorrect, the residual variance is upward biased. On the average the criterion of minimum residual variance leads to the correct choice of the specification.

Does the criterion of minimum residual variance lead to the correct choice of specification when the structures are chosen from the class of aggregate rational lag functions? I t w ill be shown below that the estimated residual variance of an aggregate equation .is generally biased. The result is that i f one of a set of competing theories is chosen on the basis of minimum residual variance we can have misspecification error. We shall now derive the components of this error.

I f we assume that the micro structures are correct and that the Uj's are the true residuals we can reformulate the problem as follows: What is the relationship between the estimated macro variance and the true micro variances?

From equation (1 .7 0 .), Yj = XjBj+Uj, and (1 .7 1 .),

Y = eYj, we get: J Substituting equation (r.1 5 .J , Xj = XDj+Vj, and ( t . lB .) ,

Dj * (X^rVXj*. in (1.104.), we get:

(r.lO S .J Y = XeDjB . + zVjBj + eUj J J J

But from equation (1 .6 5 .), X = XP+W, and (1 .7 3 .), Y = XB+U, we get:

(1.106.) Y = XPEDjBj + WEDjBj + EV^K + Elf. J J J J

Recalling equation (1 .1 9 .), B = eDjB ^ and (1 .7 4 .), B^PB, J 3 J we get:

(1.107.) Y = XB + WB + eVjBj + EU, J 3 J J J

Equating equation (1 .7 3 .), Y=XB+L) and (1 .1 0 7 .), we get:

(1.108.) U = WB + EV,B. + eIL J 3 3 J 3 where: WB = the specification bias

EV.-B. = the aggregation bias J J 3

ElL = the true aggregate residual. J 3 46

Now, from equation (1.77.), B = BH^X)"1)^ , and (1.97.).

U - Y-XB, we get:

C l.109.) U = Y-75

= Y-X [l - ( X1 X) “ 1X10 ]

= Y-XB + X(X1X )'1X10

= 6 + xCx1^)-1^1^)

That is:

( 1 . 110.) U = [I-X (X 1X)“ 1X1]U

A We shall designate the implied sampling error U-U as E:

(1.111.) e = U-U = -X(X1X )"1X1U

Summarizing, we can relate the estimated macro-disturbance A U to the underlying micro residuals and parameters by:

A (1.112.) 0 = SU, + EV jB j + WB + E J J J 3 3 or for convenience by:

A (1.113.) U = U + V + Z + E where:

(1.114.) U = EU. J J

(1 .1 15.) V = EV.B. j J J (1.116.) Z = WB

I f we define the sample variance of the estimated macro -2 disturbance S as:

A A

(1.117.) S2 = 1 iPu T

then we can write this sample variance as the sum of the variances and co-variances of its bias components:

A A (1.118.) S 2 = 1 u'O

s ^ [U]U + v \ + Z*Z + E]E]

[u'v + l^Z + l^E + + V*E + Z ^ ] CHAPTER I I

INVESTMENT

1. Introduction The purpose of this study is to measure errors caused by

aggregation in estimated coefficients, time-form and average lag, and

residual variance. Towards this goal we have shown, in Chapter I , that both the estimated aggregate coefficients and residual variance

of the rational lag function are composed of a true value, an aggre­

gation bias, a specification bias, and an implied sampling error.

The next step, before empirically measuring these errors, is to

select a set of economic theories to which our aggregation theory can be applied.

As a vehicle for obtaining empirical measures of the above errors

I have selected four investment theories...an accelerator theory, an expected profits theory, a liquidity theory, and two versions of a neoclassical theory. The area of investment behavior is particularly relevant to the purposes of this study since the investment variable has been one of the most d iffic u lt to predict, Jorgenson has said,

"there is no greater gap between economic theory and econometric practice than that which characterizes the literature on business

48 49 investment in fixed capital."^ One possible cause of the poor performance of investment models is aggregation bias. For example,

Jorgenson has said of his neoclassical model, "we conclude that aggregate industry groupings such as total manufacturing, total durables, and to tal nondurables cannot be employed without aggnega- 2 tion error." He does not, however, measure the effect of this error on the coefficients, time-form and average lag, and residual variance of his model.

We have also chosen a set of investment theories rather than one particular theory fo r study because of the growing debate in the literature as to which of several theories is correct. One criterion used for ranking theories is empirical performance as measured by the residual variance. I f the aggregate estimated residual variance differs from the true aggregate residual variance because of aggrega­ tion bias then this performance criterion can give perverse results.

We shall obtain empirical estimates of this bias in Chapter VI based on the aggregation theory in Chapter I .

^Dale W. Jorgenson, "Capital Theory and Investment Behavior," American Economic Review, L III (May, 1963), p. 247.

2Dale W. Jorgenson and James A. Stephenson, "Investment Behavior in U.S. Manufacturing. 1947-1960." Econometrica, XXXV (April, 1967), p. 215. 50

2. Generalized accelerator mechanism

In Its most uncompromising form, the acceleration principle 4 may be written:

(II.1.) I? s h - Kfl = a W t - <>t-l> where:

N I t = net investment in time period t .

Kt - capital stock in time period t.

Q. = output of finished products in time period t .

a = the accelerator. An assumed constant ratio between AKt and AQt .

That is, the rigid accelerator mechanism ... AKt = aAQt ... implies an instantaneous response of net investment to changes in finished product output.

The rigid accelerator has not fared too well statistically.

Tinbergen has pointed out two possible reasons for th is . Net

3The name “acceleration" appears to have been f ir s t used by J. M. Clark, “Business Acceleration and the Law of Demand: A Technical Factor in Economic Cycles," Journal of P b litical Economy, XXV (March, 1917), p. 217. ^A.D. Knox, "The Acceleration Principle and the Theory of Investment: A Survey," Economica, New Series, XIX (August, 1952), 269-297. 1

5See, for example, S. Kuznets, "Relation Between Capital Goods and Finished in the Business Cycle," in Economic Essays in Honor of Weslev Clair M itchell. (New York: Columbia university press, iy,s:>), 248-267.

^Jan Tinbergen, "Statistical Evidence on the Acceleration Principle," Economica. V (May, 193B), 164-176. 51

investment has both an upper bound (capacity of the Investment goods

Industry) and a lower bound (maximum disinvestment is usually

limited to replacement investment).

Another reason fo r the poor s ta tis tic a l record of the rig id

accelerator is that while it explains the quantity of investment,

i t does not explain the speed of adjustment of old to new capacity. 7 ft Chenery and Koyck developed a fle x ib le accelerator model in which

changes in desired capital are transmitted into investment. Invest­ ment then becomes distributed over time dependent on the speed of

adjustment of actual to desired capital:

(n.2.) Kt - Kt-i - (l-X)DcJ - Kt„-,] where K* is desired capital in the present time period t, is t actual capital, and (1 - a) is the proportion of the discrepancy between present desired capital and last period's actual capital transmitted into investment in period t .

We can write (II.2.) as:

(II.3.) Kt = (l-x)LK* - Kt-1] + Kt-1 which reduces to:

7Hollis B. Chenery, “Overcapacity and the Acceleration Principle," Econometrica, XX (January, 1952), 1-28. O L. M. Koyck, Distributed Lags and Investment Analysis, (Amsterdam: North-Holland Publishing Co., 1954). 52

( I I . 4.) Kt = (l-A)K* + XK^.,

We can also w rite:

( r r .5 .)

Kt-2 = (1_x)Kt-2 + xKt-3

Substituting ( II.5.) into (II.4.) we get:

(H .6 .) Kt = (1"X) ^ A^Kt_ -j i=0 where i is the number of time periods before the present time period,

t . I f we lag equation ( I I . 6.) one time period and substract the

result from ( II.6.) we get:

( I I . 7.)

where I t is gross investment in period t (defined as Kt - K t.]) and

AK*t- iIs the change in desired capital stock from time period

t-i-1 to t - i .

Equation ( I I . 7 .) is the flexib le accelerator in which gross

investment in period t is a distributed lag function of changes in desired capital. I t w ill be noticed that the function is geometric and is implied by assuming that the actual change in capital stock 53

is a fixed proportion of the difference between desired capital in

period t and actual capital in period t-1 f (equation I I . 2 .).

While the flexible accelerator model ( II.7.) is an improvement

over the rig id accelerator ( I I . l . ) ,

" ... the resulting empirical characterization of the time structure of investment behavior is im plausible."9

For example, Kuh^® estimates the average lag for model ( I I . 7.J to be between fiv e to ten years. Mayer^ using survey resu lts, estimates

two years.

Jorgenson and Stephenson^2 have also estimated the average lag

to be approximately two years. They did this by generalizing the

flexible accelerator ( II.7.) That is, instead of imposing a geometric

9Dale W. Jorgenson and Calvin D. Siebert, "A Comparison of Alter­ native Theories of Corporate Investment Behavior," American Economic Review. L V III (September, 1968), 687.

^°Edwin Kuh, Capital Stock Growth: A Micro-Econometric Approach (Amsterdam: North-Holi and Publishing Co., 1963), 293. For the average firm, Kuh estimated a reaction coefficient (i.e ., 1-x) of .15 as an upper lim it and .08 as a typical value. The expected value of the geometric generating function is ( /dL - a/ 1—X. Therefore, .85 = 5.7 d n r years and .92 = 11,7 years. .08

llMayer, "Plant and Equipment Lead Times," Journal of Business, XXXIII (April, 1960), 127-132.

l^Dale W. Jorgenson and James A. Stephenson, "The Time Structure of Investment Behavior in United States Manufacturing, 1947-1960," Review of Economics and S tatistics. XLIX (February, 1967), 21-22. 54

distributed lag function between Investment and changes in desired

capital they wrote:

(H.8.) It = + 6Kt_1

where is an, as yet, unspecified distributed lag function. Equation I I . 8. is the generalized accelerator mechanism. It

has the advantage of clearly stating a common framework for the four

investment theories to be developed in this chapter; that is , i t

summarizes the three essential parts of an investment theory: it desired capital stock, K ; replacement investment, 6K; a series of

distributed lag weights, u^. When K*,

w ill have a theory of investment behavior. We shall see that the

four theories to be developed in this chapter differ only in their

specification of K* and that the specifications of 5K and u^ are

common to all our theories.

l ^ 3. Theories of capital demand °

To develop specific investment theories from the generalized ic accelerator mechanism, I I . 8,, we must specify desired capital stock, K .

^3This entire section rests heavily on the works of Jorgenson and Siebert. See Dale W. Jorgenson and Calvin D. Siebert, "A Comparison of Alternative Theories of Corporate Investment Behavior," Working Paper No. 115, (September, 1967), Center for Research in Management, Institute of Business and Economic Research, University of California, Berkeley. Dale W. Jorgenson and Calvin D. Siebert, "A Comparison of Alternative Theories of Corporate Investment Behavior," American Economic Review, LVIII (September, 1968), 681-712. Dale W. Jorgenson and Calvin D. Siebert, "Optimal Capital Accumulation and Corporate Investment Behavior," Journal of P olitical Economy. LXXVI (November/December, 1968), 1123-1151. 55

Four specifications of K* are of particular importance since they discriminate among the four competing investment theories being debated in the economic lite ra tu re .. .accelerator, expected pro fits,

liquidity, and neoclassical. We shall discuss the specification of

K* for each of these theories in turn. a) Accelerator theory - We have introduced the accelerator theory in section IE.2. first as the rigid accelerator of J. M. Clark then as the more flexible versions of Chenery and Koyck. More recently,

Eisner^ has tested the accelerator theory. In a ll of these versions the accelerator theory hypothesizes that desired capital stock is a fixed proportion of output. We may write:

(II.9.) K* = bQ

★ where: K = the desired stock of capital, b = the capital coefficient. Q = real output. b) Liquidity theory - The liquidity theory states that financial variables constrain the firm from attaining the desired level of capital stock.^ I t has also been suggested that the p ro fit level

^R. Eisner, "Capital Expenditures, Profits and the Accelera­ tion Principle," in Models of Income Determination. Studies in Income and Wealth, Vol. 28, (Princeton: Princeton University Press, 1964).

^Kuh, Capital Stock Growth: A Micro-Econometric Approach, 208. ------56 relative to long-term debt restricts outside sources of finance.^®

Therefor, we shall write desired capital stock as a function of a measure of liq u id ity , L (discussed in section IV .3 .):

(II.10.) K* = qL c) Expected profits theory - The next hypothesis of the determinants of capital stock is the expected profits theory. Grunfeld^ has presented evidence that the market value of the firm is a good proxy for expected profits. I t is assumed in the expected profits model that desired capital stock is a linear function of expected profits.

This hypothesis may be written:

(11.11.) K* = e + fVt where: K* = desired capital stock in time period t .

Vt = market value of the firm in time period t . A proxy variable for expected p rofits. d) Neoclassical theory - The neoclassical theory of optimal capital accumulation assumes that the firm maximizes the u tility of a

^Michael Kalecki, "A New Approach to the Problem of Business Cycles," Review of Economic Studies. XVI (1949-1950), 57-64.

^Yehuda Grunfeld, "The Determinants of Corporate Investment," in A. C. Harberger, The Demand for Durable Goods. (Chicago: The Univeristy of Chicago Press, l^edj, 211-266. 57

consumption stream subject to a production function. Among the

proponents of this theory, D. W. Jorgenson has made the neoclassical

theory operational. The two versions of the neoclassical theory

presented here are based on his works.

The following theory answers the question: "What value of desired capital stock in time period t,l<£, maximizes net worth of the corporation subject to the constraints of a production function and the definition relating the rate of change of capital stock, gross investment, and replacement investment?"

Assume a production function relating, at time period t , the flows of output, Q(t), labor services, L(t), and capital services,

K (t):

(11.12.) F [QCt), L(t), K(t)] = 0

We further assume that F is twice - differentiable, convex, has positive marginal productivities, and positive marginal rates of substitution between inputs.

The time rate of change of capital in time period t , K (t), is net investment, defined as the difference between total invest­ ment in period t , I ( t ) , and replacement investment in period t ,

6K (t). That is ,

^Dale w. Jorgenson, "Capital Theory and Investment Behavior," American Economic Review. L III (May, 1963), 247-259. Dale W. Jorgenson, “the Theory of Investment Behavior," in Universities - NBER Committee for Economic Research, Determinants of Investment Behavior (New York: National Bureau of Economic Research, 1967), TEPSST 58

Cir.i3,) K(t) = I(t) - 6K(t)

Met revenue before taxes in period t, R(t), is the difference

between the present value of output, p(t)Q (t) and the cost of labor

input, w(tjL(t), and investment, q (t)I(t). That is:

(11.14.) R(t) = p(t)Q(t) - w(t)L(t) - q(t)I(t)*

Tncome used for computing income taxes differs from net

revenue, R {t). We recognize four adjustments to total revenue.

F irst, the labor cost is deducted before computing income tax.

Second, a proportion, v, of depreciation may be deducted from income

before computing taxes. I f replacement investment is a constant

fraction of capital s to c k , ^ 5K, then the current value of replace­

ment is

vsqK. Third, a proportion w of the cost of capital is deductible

from income for tax purposes. Let r (t) be the opportunity cost of

invested cap ital, then the to tal cost of capital in period t is

r(t)q(t)K (t). The amount of the cost of capital deductible for

tax purposes is wr(t)q(t)K(t). Finally, a portion, X, of capital

gains on assets, q(t)K (t), where q(t) is the time rate of change

in the price of capital goods in period t is deductible from income

for tax purposes, X q (t)K (t).

^The distribution of replacement investment approaches a constant fraction of capital stock for any in itia l age distribution of capital stock and for (almost) any distribution of replacement investment over time. See section I I . 4. 59

Let the rate of tax be u(t) then taxes on income are:

(11.15.) T (t) = ;)6KCt)

The present value of the firm is the integral of discounted

net revenue after taxes:

(ir.1 6 .) V = /V 6 t|r(s>ds [R(t) - T(t)]dt 0

We assume the firm's goal is to maximize present value.

This is subject to two constraints: (a) The rate of change of

capital services, £ (t), is proportional to the flow of net invest­

ment, I( t ) - SK(t). That is ,

(11.17.) R(t) = z(t)[I(t) - 5K(t)].

The proportionality factor, z(t), can be interpreted as the time

rate of utilization of capital stock. Equation 11.17. makes it

clear that i t is not the amount of net investment in a given period

of time which is important for changes in production (output) but

the services of this addition to capital stock. It also points out

that a given net investment can provide differing amounts of services

depending on its utilization rate. Thus, the flow of capital

services is a (proportional) function of net investment with the

"constant" of proportionality, z ( t) , varying over time. Data on

z(t) are unavailable at the firm level and so we make the 60 simplifying assumption that z (t) is unity.**® (b) the second constraint is that outputs and inputs are related by a production function:

CII.18.} F[Q(t),L(t),K(t)] = 0

To maximize present value subject to the constraints

C ir.12.) and (11.13.) we introduce the Lagrangian expression:

(II.19.) i - Ae-^r(s)ds[R(t)-T{t)]+Vt)K(t).I(t)+aC(t)] 0 +X2(t)F [q (t).L(t),K(t)]fdt

00 (11.20.) = ; f(t)dt 0

The Euler necessary conditions for (11,20.) to be a maximum subject to the constraints (11.12.) and (11.13.) are:

r^r(s)ds ( I I .21.) 3f = e“/o [l-u(.t)]p(t) + Ag(t ) 3F * 0 W Tt7 W TtT

(11.22.) 8f = -e^or(s)ds[l-u(t)]w(t) +A2aF = 0 atTET atTHT

(11.23.) af -e^o^^qft) - X-, (t) = 0 W t T

The assumption that z (t) is unity is im plicit in Jorgenson's neoclassical model. For example, see equation 2 in Jorgenson and Stephenson, Econometrica. XXXV (A p ril, 1967), p. 174. For an heroic attempt at measuring capital services in relation to capital stock see Zvi Griliches and Dale W. Jorgenson, "Sources of Measured Pro­ ductivity Change: Capital Input," American Economic Review. LI (May, 1966), p. 60, We obtain 3f/3K(t) by application of Euler's equation:^

(11.24.) 3f - d af = e"^or^s^ S u(t)q(t)rvd+w(t)r(t)-X(t)Q(t) 3K(tT dtakftT q(t)

+ A2(t)3F + Ai(t)6-dx (t)] * o c WCtT 3 T 1

(rr.2 5 .) 3f = £(t) - I(t) + 6K(t) = 0 3K TH T

(11.26.) af = F[Q(t),L(t),K(t)] * 0 3A2(t)

By a theorem of implicit differentiation:

(11.27.) 3Q = af/3l_(t) = s(t) a ifa r " s r /w w p ro

Noting from equation (1.23.) that:

(11.20.) X-|(t) = -e-/Jr(S,dSq(t) then:

(11.29.) dXi(t) = -e^or(s)dSq*(t) + r(t)e‘/ or^S^ Sq(t) dt------

(11.30.) = e-^^fottM t) - q(t)]

21 A necessary condition fo r thexfunction f ( X) to yield a rela tiv e maximum for the definite integral J = /xlF[X,f(x) ,fHx)]dx is that f(x) satisfy the differential equation: jjF. - d (aF-.) = 0 . See Edouard 3y 3x 3y Goursat, A Course in Mathematical Analysis, Translated by E. R. Hedrick (New York! Dover Publications, inc., 1964), III, 231-236, 62

Substituting (11.28.) and (11.29.) in (11.24.) we get:

a i -31° * 11‘HCfcjd, ♦ C ^ r(t) -

(11.32.) ■ c(t) m where c (t) is the numerator of (11.31.) and may be interpreted as

the implicit (shadow) price of capital services in period t.

I f we assume that the production function is Cobb-Douglas:

(11.33.) Q » L(t)bK*(t)a

then the e la s tic ity of output with respect to desired capital

services in period t,K (t), is:

(11.34.) e[Q(t)»K*(t)] = a

Noting that the marginal productivity condition (11.32.) holds in equilibrium we shall say that equation (11.32.) determines

the optimum level of capital stock, K (t). Actual capital stock,

K (t), is determined by equation (11.13.). We may then write:

(H .35.) aQ(t) = c(t) K *(t) P W

This may be solved for desired capital stock:

(11.36.) K*(t) = ap(t)Q(t) C(t) 63

In the above neoclassical formulation of the determinants of

desired capital we e x p lic itly included the rate of capital gains and

losses, ${t)/q(t), in the price of capital services. We shall term this model Neoclassical I . I f we exclude capital gains and losses

as a determining factor in the demand for capital services we have

the Neoclassical II model. In this model c(t) is:

(11.37.) c(t) * g(t) [(l-uv)s + (l-uw)r(t)]. 1-u

4. Replacement investment

Replacement investment is the second part of the generalized

accelerator mechanism, I I . 8 ., which must be specified before we have ?2 a complete theory of investment behavior. Jorgenson has shown

that for any initial age distribution of capital stock and for

(almost) any distribution of replacement investment over time, the

distribution of replacements approaches a constant fraction of capital

stock. A common assumption is that the distribution of replacements

over time is geometric. In this section we show that the geometric

distribution implies that replacement investment is a constant

fraction of capital stock.

^^Dale W, Jorgenson, "Capital Theory and Investment Behavior," Working paper No. 26, Committee on Econometrics and Mathematical Economics, Institute of Business and Economic Research, University of California at Berkeley, 1 September 1962, Chapter 4. 64

Let r.j = the proportion of investment goods acquired in period t and replaced in period t + i such that 0 < r-j < 1 for

i = 0 ,1 ,2 ,.,, and s r,* = 1. i=0 p Assume replacement investment in time period t , I£ , is a weighted average of past gross investment:

(11.38.) I* = r(L)It

p where: I - replacement investment in time period t . t I t = gross investment in time period t .

r(L) = a polynomial in the lag operator L. 2 r(L) ■ rQ + TjL + rgL + . . .

I f the distribution of replacements over time is geometric, the power series in the lag operator takes the form:

(11.39.) r(L) = 5L + 5(1-6)L2 + 6(1-6)2L3 + ... which reduces to: .

(11.40.) = 6Kt.

Not a ll of the authors whom we have associated with specific theories (Eisner-accelerator, Grunfeld-expected profits, Kuh-liquidity,

Jorgenson and Siebert-neoclassical) have been consistent in their specifications of replacement investment. For example, while Eisner 65 23 and Kuh assume that replacement Investment 1s proportional to gross capital stock, they define gross capital stock as an unweighted sum of past gross Investments net of retirements. "This assumption is inconsistent with a geometric mortality distribution for capital goods. and is thus inconsistent with their use of replacement investment as proportional to capital stock as was shown above. 25 Grunfeld assumes replacement investment to be proportional to net capital stock which he defines as gross capital stock less accumulated depreciation; however, he computes depreciation by the

"straight line method rather than the declining balance method implied 26 by the geometric mortality distribution for investment goods."

Since a ll of our models specify replacement investment as proportional to capital stock we must define capital stock as a geometrically weighted sum of past gross investments. The precise specification, which w ill be common to a ll our models, is presented in the measurement section of this study, IV.3.

5. Models of investment behavior

We may now combine the results of the previous sections in presenting the five models of investment behavior. Letting net

23 R. Eisner, Models of Income Determination. Vol. 28, and E. Kuh, Capital Stock Growth: A Micro-EcoFometr'ic Approach.

2^Dale W. Jorgenson, "Econometric Studies of Investment Behavior: A Survey/journal of Economic Literature. IX (December, 1971), 1139.

25Grunfeld, in Harberger, The Demand for Durable Goods.

26Jorgenson, Journal of Economic Literature. IX, 1139. 66

investment, I t - 6Kt , be a distributed lag function of changes In

desired or optimum capital stock we have:

Crr.4i.J rt - 6Kt = w( l) [k* - k^ ]

where: I t = gross investment.

6Kt = replacement investment.

k£ = desired capital stock.

W(L) = a rational lag function = j-j-

We can rewrite (11.41.) as:

(II.42.) 1^ - aK ^ + a-j + ... + am - b-jtl.j.-fiK^)^-!.

- SKt

Replacing AK* in each of five theories with an operating definition we have:

Neoclassical 1: aptft . aPt.,0,.! (11.43.) I t - H(L) t c j . t c ^ t - i J +

Neoclassical II:

ap.Q. apt tQ. . (n.44.) it - W(L, [ - -sfcLfel. 3 + iKt

Accelerator:

(11.45.) I t = W(L) [bqt - bqt . , ] + SKt 67

Liquidity:

(11.46.) It = Wd) [qLt - qLt.,] + SKt

Expected Profits:

(11.47.) It = M(L) [fVt - fVt.,] + «Kt CHAPTER I I I

ESTIMATION OF A RATIONAL LAG FUNCTION

Estimation of the parameters of the rational lag model,1

( I I I . l . ) Y* = A(L)xt + ut BTO

depends on the behavior of the disturbance term, ut. If Ut is a

normal, independently distributed random variable with mean zero and

common variance a^, [NID(0,a^)], then ordinary least-squares

estimates (OLS) will be consistent and asymptotically efficient but biased in small samples.^

Estimation is further complicated i f the autocorrelation

problem is added to the lagged dependent variables model. While

autocorrelation in the ordinary regression model produces inefficient

but unbiased estimates and while lagged dependent variables with a

random disturbance yield consistent estimates with small sample bias,

the combination of lagged dependent variables and autocorrelation

yields inconsistent least-squares estimates.**

^See Chapter I for a discussion of this model. 2 J. Johnston, Econometric Methods (2nd ed.; New York: McGraw-Hill, 1972), 303^ ------3 Johnston, Econometric Methods. 307. See also J. S. White, "Asymptotic Expansions for the Mean and Variance of the Serial Correlation Coefficient," Biometrika, XLVIII, 85-94; F.H.C. Marriott and J.A. Pope, "Bias in the Estimation of Autocorrelations," Biometrika, XXXI. 390-402; J.B. Copas, "Monte Carlo Results for Estimation 1n a Stable Markov Time Series." Journal of the Roval Statistical Society. A, 129, 110-116.

68 69

How, then, do we estimate the parameters in a rational

distributed lag function? Jorgenson** approached the estimation

problem by rewriting the non-stochastic rational lag function

( I I I . 2.) Yt = ACL)Xt B(L)

in final form and then adding a disturbance term,

( I I I . 3.) = 3^Xj.+a^ X^+. • bi Y -j-_ -j — • • • — bnY^_p+e^.. ( t - i , . * . , n }

Furthermore, he assumes et to be independent of a ll values of X and that p E(et ) = 0 and E(et,es) = 5st° • ( s »t » = 1 n) • Jorgenson then states* "...provided that the rational distributed lag function considered as an

equation in the dependent variable is stable, the ordinary least-squares 5 estimator of the parameters is best asymptotically normal."

Jorgenson's approach is equivalent to assuming

(III.4.) Yt -A(L)Xt + ut (t-l,....n)

b( lT

where: ut = 1 E. 0 B(L)* z and et = NIDtO.o*)

Thus, i f equation ( I I I . 4.) represents the original form of the rational

lag function then implicit in the final form model ( III.3.) is the

^Dale W. Jorgenson, "Rational Distributed Lag Functions," Econometrica, XXXIV, 143. See also the application of the rational lag 'function fcThls neoclassical investment theory in , for example. Dale W. Jorgenson and James A. Stephenson, "Investment Behavior 1n U.S. Manu­ facturing, 1947-1960," Econometrica, XXXV, 169-220. C Jorgenson, Econometrica. XXXIV, 143. 70

transformation of an nth order autoregressive residual. When put in

final form, this residual, et, is NID(0,a2). The implicit assumption

is that the coefficients of the lagged dependent variables and the coefficients of the autoregressive residual process are Identical.

The parameters of the lag structure cannot be separated from those of

the probability structure of the error term. As Dhrymes states,

"...th e re is no fundamental identification problem in the model under consideration, and this difficulty arises solely because we insist on estimating the parameters by ordinary least-squares."®

A further d iffic u lty with Jorgenson's approach is that the final form ( I I I . 3.) could have been determined by

(iri.5.) Yt = A*(L)Xt = I * B*(L) *et where A(L) = A*(L)B(L)

Thus, the final form does not necessarily imply the existence of an infinite (rational) distributed lag.^

Instead of assuming the model ( I I I . 4 .) , Jorgenson could have assumed

(HI.6.) Yt » + u+

6Phoebus J. Dhrymes, Distributed Lags: Problems of Estimation and Formulation. (San Francisco: "H'olden-Day, )97l)7 330.

?See Dhrymes, Distributed Lags, 330. where ut = NID(O.ct^)

Tn that case the model to be estimated would be the final form of

( I I I . 6 .) which, however, has an nth order autoregressive residual

(III.7.) = ^ t +al^t-l+,,,+^t-n f k|^t-l"**,“tyAt-n+ut

+ blUt-l+**’+ *hut-n

Of the problem of choosing between models such as ( I I I , 4.) and ( I II.6.), Griliches has said, "As far as I can see, there is no strong economic argument for making the uncorrelation assumption at O one end of the procedure rather than at the other."0 We therefor propose ordinary least-squares as one estimation technique to be used in this study. That is, we shall estimate the rational lag function on the working hypothesis that the underlying model is ( I I I . 4 .) .

I f , on the other hand, the true model is ( I I I . 6 .) , then we must use an estimation technique suitable for models with lagged dependent variables and auto-correlated residuals. Several methods are available.

We could, for example, assume a particular form for the residuals and estimate th eir parameters jo in tly with the others; however, this presents the added d iffic u lty of determining a specific form of the auto­ correlation dependence. As Griliches has pointed out, "for operational purposes we have to assume that the dependence is of the form of a f ir s t order or at most second or third order autoregressive process."

®Zvi Griliches, "Distributed Lags: A Survey," Econometrica, XXXV, 38.

9Gr1liches, Econometrica. XXXV, 41. 72

Another approach to the problem conies from noting the similar­ ity between the simultaneous equation problem and the problem of

lagged dependent variables being correlated with the true disturbance.

Griliches^0 suggests estimating Yt in unconstrained form by adding as many lagged X's as long as they reduce the residual variance. Then, substitute the estimated lagged Y's into the final form ( I I I .7 .) and re-estimate the equation. The lagged, estimated, dependent variables are uncorrelated with the disturbances and yield consistent parameter estimates.

Another approach to the lagged dependent variables problem comes from noting the sim ilarity between the final form of the rational lag model ( I I I . 3 .) with an independently and identically distributed random disturbance with mean zero and constant variance,

IID(0,o^), and the ordinary regression model with a stationary autoregressive residual.

Assume the model

(III .8.) Yt = B^X]t +•••+ BqXq-j- + ut (t—l,...,n) where ut+ aput-p =et (t=l ,...-1,0,1,...) and et = IID(0,cr^)

On multiplying successive terms of ( I I I . 8.) by !,a j,...,& p and adding, we obtain

*°Zvi Griliches, Econometrica. XXXV, 41. 73

( I I I . 9.) Yj. + + •».+ 9pYj|.p = B-|Xi|. +,..+ BqXqt

+ aiBiXlt-1 +.* .+ apBqXqt+et

Equation ( I I I . 9.) is of the same form as the final rational lag

form (£11.3.).

If we apply least-squares to ( III.9.) we will get optimum

estimates when e -|,...,e n are normally distributed;^ however, the

estimating equations are highly non-linear and therefor d iffic u lt to 12 solve.' Various solutions to the non-linearity problem have been

proposed. Generally, however, they involve successive approxima­

tions by an iterative technique which requires in itia l consistent 11 estimates.

^James Durbin, "Estimation of Parameters in Time-Series Reqression Models," Journal of the Royal S tatistical Society, Series B, XXII, 139-153. 12see Dhrymes, Distributed Lags. 239, for the normal equations of the rational lag funetion- obtained from the first-o rd er (necessary) conditions of minimizing the sum of squared residuals with respect to the parameter estimates.

13por example, see K. Steigletz and L. E. McBride, "A Technique for the Identification of Linear Systems," IEEE Trans­ actions on Automatic Control, AC-10, 1965, 461-464, and "Iterative Methods' "for Systems Identification," Technical Report No. 15, Department of Electrical Engineering, Princeton University, Princeton, N .J ., June, 1966; D. Cochrane and G. H. Orcutt, "Application of Least-Squares Regression to Relationships Containing Autocorrelated Error Terms," Journal of the American S tatistical Association. XLIV, 32-61; D. G. Champernowne, "Sampling Theory Applied to Autoregressive Sequences," Journal of the Royal Statistical Society, B, X, 204-231. 74

Durbin^ has proposed a two-stage estimating procedure for

the parameters in ( I II .3.) which is asymptotically efficient and

consistent. The steps to be followed are:

(a) Let -b i» ,..,- b n be the ordinary least-squares estimates

of Yt„i Yt_n in (III.3.).

(b) Define vt = Yt + Yt_-| + ...+ &nYt_n and

wit = X.t + b1Xit_1 +...+ bnXit.n where (i = l,...,n ).

(c) The coefficients 0f the variables in (H I.3.)

are the least-squares coefficients of vt on w ^ . - . w ^ .

Durbin has shown that parameter estimates using his two-

stage approach have asymptotically the same mean vector and 15 variance matrix as the ordinary least-squares estimator. The

problem of small sample bias s t ill exists* however, no other estimation procedure exists for lagged dependent variables which is

unbiased in small samples as well as being asymptotically efficie n t

and consistent.

In this study we shall estimate parameters of the rational lag

function in two ways: (a) We sh all, following Jorgenson, assume model

( I I I . 4.) and estimate the parameters by ordinary least-squares;

(b) We shall also apply Durbin's two-stage estimator to model ( I I I . 6.}.

^Durbin, Journal of the Royal S tatistical Society, XXII, 150-153.

^5Durbin, Journal of the Royal Statistical Society, XXII, 152. CHAPTER IV

EMPIRICAL: INVESTMENT THEORY PERFORMANCE

1. Introduction

The empirical results of this study are now presented. The sample of firms used in this study along with some of th eir charac­ teristics are discussed in IV .2. Sources of data and measurement of variables is the subject of IV .3. Various performance measures pertaining to the firms and theories are presented and discussed in

IV .4. These same measures are applied to the aggregates in IV .5.

2. The sample The sample consisted of twenty-seven large manufacturing corporations. These firms were chosen purposively rather than randomly for two main reasons: (a) The general Pascal lag function assumes continuity of investment projects over time. This is less lik e ly in smaller corporations where investment projects are discon­ tinuous "one only" situations; (b) Data was available only for large corporations with publicly traded stock. The non-random character of the sample is not inconsistent with the purposes of this study as they are stated in the Introduction. The firms were selected from Fortune's directory of the 500 largest U.S. corporations^ and are members of six OBE-SEC industry

^Fortune Magazine, Directory of the 500 Largest Industrial Corporations. 75 76 groups. Table 2 lists the firms alphabetically, their ranking by

various c rite ria , and th eir industry group.

Although the firms ranked within the top 100 U.S. manufacturing

corporations by sales, they differed both among a ll firms and within

groups. For example, average investment (constant 1954 dollars) for

the period 1946-1968 ranged from a low of 20 million dollars (General

Tire.and Rubber) to a high of 870 m illion dollars (General Motors).

Capital stock as of 1961 (in 1954 dollars) ranged from a low of 110 million dollars (General Tire and Rubber) to a high of 6 b illio n dollars (Standard Oil of New Jersey).

Concentration of firms within groups also varied widely. The difference between the proportion of total investment of the smallest j and largest firms within a group ranged from 19.2% (Chemicals) to

67.2% (Autos). Table 3 lis ts average investment and capital stock by each firm and their size distribution within their industry group.

3. Measurement

To make the fiv e theories of investment behavior developed in

Chapter II operational, we must find empirical counterparts to the theoretical variables. Of the approximately 600 time series collected for this study much of i t came from two sources . . . Moody's Industrial

Manual^ and The Commercial and Financial Chronicle.3 i t is the

o Moody's Industrial Manual (New York: Moody's Investors Service, *Inc.), various annual issues.

3The Commercial and Financial Chronicle (New York: William B. Dana Co.), various issues. Table 2

Sample firms, their ranking by sales, assets, and net income, and their QBE-SEC industry group Ranking by Sales Assets Net Income OBE-SEC Firm______(as of 31 December 1968) ______Industry Group ______Armco Steel 66 48 53 Primary Iron and Steel Bethlehem Steel 23 19 23 Primary Iron and Steel Chrysler 5 13 14 Motor Vehicles and Equipment Continental Oil 34 24 28 Petroleum and Coal Products Dow Chemical 50 29 32 Chemicals and Allied Products Dupont (E .I.) de Nemours 15 17 10 Chemicals and Allied Products Eastman Kodak 28 23 9 Chemicals and Allied Products Firestone Tire and Rubber 36 37 35 Rubber Products General Electric 4 11 11 Electrical Machinery and Equipment General Motors 1 2 1 Motor Vehicles and Equipment General Tire and Rubber 97 103 124 Rubber Products Goodrich (B.F.) 82 82 119 Rubber Products Goodyear Tire and Rubber 22 26 30 Rubber Products Gulf Oil 9 5 6 Petroleum and Coal Products Inland Steel 91 70 58 Primary Iron and Steel Monsanto 41 36 42 Chemicals and Allied Products National Steel 83 59 66 Primary Iron and Steel Republic Steel 63 50 61 Primary Iron and Steel Shell Oil 16 14 12 Petroleum and Coal Products Standard Oil of California 13 10 7 Petroleum and Coal Products Standard Oil of Indiana 19 12 13 Petroleum and Coal Products Standard Oil of New Jersey 2 1 2 Petroleum and Coal Products Texaco 8 4 4 Petroleum and Coal Products "j Union Carbide 26 18 24 Chemicals and Allied Products ~-4 Uni royal 60 75 93 Rubber Products U.S. Steel 10 8 16 Primary Iron and Steel Westinghouse Electric 17 31 33 Electrical Machinery and Equipment 78

Table 3

Size distribution of firms within groups by average investment and capital stock

Average Capi tal Investment stock (1946-1968) Percentage (1961) Percentage (billions of of (billions of of Firm 1954 dollars) Group 1954 dollars) Group

GROUP: Autos

Chrysler .1705 16.4 .4522 12.7 General Motors .8699 83.6 3.1225 87.3

TOTALS 1.0404 3.5747

GROUP: Chemicals

Dow .1013 17.1 .6375 17.4 Dupont .1832 30.9 .9405 25.7 Eastman Kodak .0695 11.7 .3608 9.9 Monsanto .0813 13.7 .7004 19.2 Union Carbide .1583 26.7 1.0147 27.8

TOTALS .5936 3.6539

GROUP: Electrical

General Electric .1683 73.6 .7254 65.4 Westinghouse Electric .0589 26.4 .3841 34.6

TOTALS .2272 1.1095 79 Table 3 (Continued)

Average Capital Investment stock (1946-1963) Percentage (1961) Percentage (billions of of (billions of of Firm 1954 dollars) Group 1954 dollars) Group

GROUP Oil Continental .1207 5.4 .7693 4.4 Gulf .3151 14.1 2.2783 13.1 Shell .2404 10.8 1.2242 7.1 Standard of Cal. .2439 10.9 2,1351 12.3 Standard of Ind. .2599 11.6 2.1673 12.5 Standard of N.J. .7075 31.6 6.3560 36.6 Texaco .3487 15.6 2,4179 13.9

TOTALS 2.2362 17.3481

GROUP: Rubber

Firestone .0577 26.6 .2858 25.1 General .0200 9.2 .1101 9.7 Goodrich .0339 15.6 .1942 17.0 Goodyear .0717 33.0 .3616 31.7 Uni royal .0338 15.6 .1891 16.6

TOTALS .2171 1.1408

GROUP: Steel

Armco .0571 8.1 .4400 7.1 Bethlehem .1525 21.5 1.0074 16.3 Inland .0544 7.7 .5194 8.4 National .0619 8.7 .5978 9.7 Republic .0673 9.5 .6665 10.8 U.S. Steel .3146 44.4 2.9436 47.7

TOTALS .7078 6.1747 80

transformations of these series into the empirical counterparts of 4 the theoretical variables which are discussed in this section.

a) Investment - The investment figure used in each of the five

theories is gross investment in property, plant, and equipment in

1954 dollars, It . The current dollar figure is reprinted in Moody's

Industrial Manual from reports filed by the corporations with the

Securities and Exchange Commission. The im plicit fixed Investment C nonresidential (structures and equipment) price deflator was used.

b) Replacement investment - Each of the five theories of investment

behavior use capital stock, K, as an explanatory variable. In

addition, they use lagged values of net investment, I -6 K (see IV .5.)

where

measures of capital stock and replacement investment.

I t was pointed out in section IV .4. that for any in itia l age

distribution of capital stock and for (almost) any distribution of

replacement investment over time, the distribution of replacements

approaches a constant fraction of capital stock. This constant

relation is:

^The measurement of variables presented here follows that described in the statistical appendix to Dale W. Jorgenson and Calvin 0 . Siebert, "A Comparison of Alternative Theories of Corporate Investment Behavior," (Working Paper No. 116), Center for Research in Management, Institute of Business and Economic Research, University of California, Berkeley, September, 1967. This appendix is referred to in th e ir pub­ lished paper, "A Comparison of Alternative Theories of Corporate Investment Behavior," American Economic Review, L V III (September, 1968), 681-712.

DEconomic Report of the President with the Annual Report of the Council of Economic Advisers (Washinqton. D.C.: United States Government Printing O fficeTTO l),' 200, rf we let K0 be an in itia l value of capital stock and Kt a terminal

value, we can expand ( IV .1.) to:

(IV.2.) Kt = (1-6)% + (l- 6 ) t_ 1 I 0 + (l-a)1"2^ +...+It

This equation can then be solved for 6 .^ The resulting values of6

are presented in Table 4. The in itia l and terminal values of capital stock were for the

years 1937 and 1961 respectively. 1937 was chosen because i t was the

earliest value available in the corporate reports filed with the

Securities and Exchange Commission, 1961 was chosen because i t was

the latest available year for the capital stock deflator.^

°Many numerical techniques exist for finding roots of such equations, I used iny own successive approximation technique as follows: I f K0 is the in itia l value of capital stock (1937), I 0, I - | , . . . are values of gross investment in periods 1937, 1 9 3 8 ,..., 6 is the rate of replace­ ment starting with a specified value, 6 0, then, define Xt as Xt = (1-5)% + ( 1 - 6 ) I q + ( 1-6)t_2 I-j + ...+ I t in equation IV .2.

Arbitrarily substitute 6 = So = .5 in the equation. If (Kt-X)>0 then substitute 5] = Sq/2 in IV ,2. I f (Kt~X)<0 then substitute 61 =(5o+l)/2 above. Each successive is found by (6 .+ 6 ^ 1)/2 unless 5 is approaching le ft or right end-points in which case S-j/2 or (5-j+l)/2 is substituted respectively. Repeat successive iterations until (Kt-X) equals a specified decimal value. (.000001), This procedure applied to equation IV .2. converged quite rapidly. i, ^The deflator was obtained and put into a 1954 base from the following sources: D. Creamer, "Capital Expansion and Capacity in Postwar Manufacturing," Studies in Business Economics, No. 72 (New York: National Industrial Conference Board, 1961)', table G-4, D. Creamer, "Recent Changes in Manufacturing Capacity," Studies in Business Economics, No. 79 (New York: National Industrial Conference Board, 1962), Table A-4, D. Creamer, S. Dobrovolsky, and I . Borenstein, Capital in Manufacturing and Mining: Its Formation and Financing (Princeton: Princeton University Press, I960), Table A -ll. Table 4

Calculated replacement coefficients for twenty-seven firms and six industry groups.

Firm Calculated Firm Calculated or Replacement or Replacement Group Coefficient Group Coefficient

Chrysler .28665 Continental .10649 General Motors .26141 Gulf .10516 Autos .26522 Shell .17396 Standard of California .08741 Dow .13631 Standard of Indiana .10155 Dupont .15973 Standard of New Jersey .08680 Eastman Kodak .13436 Texaco .12390 Monsanto .04590 Oil .10322 Union Carbide .13474 Chemi cals .12881 Fi restone .16289 General .13626 General Electric .16402 Goodrich .12565 Westinghouse .13003 Goodyear .14624 Electrical .15312 Uni royal .13216 Rubber .14344

Armco .08934 Bethlehem .10919 Inland .07159 National .09717 Republic .08212 U.S. Steel .10451 Steel .09919 83

Once 6 is found we can use the recursive equation (IV .1.) to

generate a ll K values intermediate between the in itia l and terminal

values. Net investment, I-6 K, can then be found by generating

replacement investment and subtracting i t from gross investment. c) Desired capital: Accelerator theory - The empirical counterpart

of real output, Q, in the accelerator theory (equation II.9.) is sales

plus inventory changes® divided by the wholesale price index (1954 base) of the firm 's industrial commodity group . 9 d) Desired capital: Liquidity theory - The liquidity variable, L, in equation 1 1 . 1 0 . was computed by adding profits a fte r taxes and depreciation charged to income less dividends paid. This figure, derived from Moody's Income Accounts, was deflated Into 1954 dollars using the investment goods deflator J® e) Desired capital: Expected profits theory - The market value of the firm, V, as a proxy for expected p ro fits , was presented in equation 11.11. The empirical measure assigned to V was the market

8 "Sales" are net of freight charges, discounts, returns, and allowances, as listed in Moody's Industrial Manual under Income Accounts. Inventory changes are computed as the difference between current end of year inventories and previous end of year Inventories as listed in Moody's Balance Sheets.

9Twelve industrial commodity groups plus a miscellaneous and total wholesale industrial index are available. See Economic Report of the President, 1971, 252-253.

^Economic Report of the President. 1971, 200. value of stock shares outstanding plus the book value of debt. This total was deflated into 1954 dollars by the GNP implicit price deflator. 17 The market value of the stocks outstanding was computed by multiplying the arithmetic average of the December (current year) high and low price and January (following year) high and low price times the number of shares outstanding as of December 31 (current year). These values were summed over the common and various preferred stocks. Stock prices were taken from the Commercial and

Financial Chronicle and the number of shares outstanding from

Moody*s (Financial and Operating Data). f) Desired c a p ita l: Neoclassical theories - The two versions of the neoclassical theory discussed in I I . 3 .d. have the same numerator, pQ.

The current dollar value of output, pQ, is the same as that used in the accelerator theory, net sales plus inventory change, as reported in Moody‘s.

The user cost of capital, c, for the neoclassical I model was defined in equation (11.31.) as:

(IV.3.) c = ( q-j)[(1-uv ) 6 + (l-uw)r - (l-ux ) $ 3

1-u 3

The investment goods deflator as used in the liq u id ity theory was used to measure q-j. This index was also used to measure the rate of capital loss due to price changes, (-q^). The income tax rate, u, was computed

7Vhe GNP implicit price deflator was computed by dividing total GNP in current dollars by total GNP in 1958 dollars. This index was then converted to a 1954 base. Data was from the Economic Report of the President, 1971, 202-203, 85

from Moody' s Income Accounts as the ratio of the difference between

profits before and after taxes to profits before taxes. The propor­

tion of depredation deducted from income for tax purposes* v , was

measured as the ratio of reported (1 n Moody's) depreciation to the

computed depreciation, 6 K (as explained in IV .3 .b .).

The cost of financial capital, r, will be measured as

f ol 1 ows:

CIV.4.) r = TrA+CCA-q]6 K-q1K^+q2 K2 +q3 K3 VTA

where: tt^ = profits after taxes.

CCA = capital consumption allowances as listed in the Supplementary Profit and Loss schedule under the Income Accounts in Moody's.

6 K = computed replacement.

Kj - depreciable assets.

K2 = depletable assets.

Kg = inventory assets.

q-j = investment goods deflator.

= the change in q^.

q2 - cL = the change in the wholesale price index in the industry of which the firm is a member.

VTA = the value of total assets. Measured by market value of all securities.

Generally, depletable assets were not listed separately in which case they were grouped in K j. 8 6

The neoclassical II model assumes capital gains, t 0 be zero. Therefor, the user cost of (physical) capital services, c, is:

(IV.5.) c = ( q_ )[(l-uv)s + (l-uw)r] V-u

where all the variables are measured as in neoclassical 1 except for r. Since capital gains are zero, we have:

(IV. 6 .) r = tta + CCA - q-jdK VTH------where variables are as defined above.

4, Micro performance 12 Ordinary least-squares regressions' of the five investment models discussed in Chapter IV were run for each of the twenty-seven firms listed in Table 1. Annual data for the period 1949-1968 was used (as described in section IV .3 .).

To choose the rational lag function for each firm and theory certain constraints were imposed. Specifically, the polynomials A(L) and B(L) (see section 1 .9 .) were limited to a maximum of second degree.

There is no theory which determines the maximum allowable degree of the polynomials; however, one can appeal to an authority who says:

^Durbin two-stage coefficient estimates are presented and compared with ordinary least-squares estimates in section IV. 6 , 87 "In practice one w ill not be interested in a B(L) polynomial of higher order than two or three. Higher order polynomials imply the estimation of four or five or more coefficients for as many lagged values of Y. At this point one may as well go back to the W(L) form directly and approximate i t by four or five lagged values of X."13

The above quotation is especially relevant since we shall be using

annual data. An advantage of the rational lag function is that it

conserves degrees of freedom while being able to approximate an

arbitrary lag function uniformly well to any desired degree of

accuracy. 1^,15

A structure was chosen for each firm and theory by choosing the "best" equation from the twenty-eight possible combinations of

polynomials A(L), B(L). The criterion of minimum residual variance was adopted to select the "best" equation.^ This

*3Zvi Griliches, "Distributed Lags: A Survey," Econometrica, XXXV, (January, 1967), 27. We have changed Griliches symbols to correspond to those of our equation ( I I I . 4 .).

14$ee the discussion in section 1.9.

1 5We recognize another problem common to practically all applications of distributed lags, including our own. That is, that there is no theory of how a lag structure is imposed on an economic theory... "In many ways, the work of Jorgenson and his co-workers is a perfect paradigm of the ad hoc superposition of a lag structure on what is essentially static theory and the resulting difficulties in the interpretation of the empirical lag structures." Marc Nerlove, "Lags in Economic Behavior," Econometrica, XL (March, 1972), 223. 1 fi Choice of the order of the polynomials A(L) and B(L) is a multiple decision problem and was not applied in this study; however, we did test for a reduction in R2 by choosing A(L) and B(L) as maxi­ mum 3d degree. The reduction in the unexplained variance was generally very small. See T.W. Anderson, The Statistical Analysis of Time Series (New York: Wiley, 1971) section 6.4. 8 8

criterion w ill, on the average, lead to the correct model

specification.^

The coefficient estimates for each firm and theory, using

the above procedure, are presented in Appendix A. This Appendix

presents the "unconstrained" estimates referring to the fact that

the coefficients of the rational lag function are not constrained

to conform to those of a general Pascal lag function as discussed

in section 1.10. Constrained estimates are discussed In

section V.2.b.

The column labelled X1 in Appendix A lists the constant

term. Columns X2, X^, and X4 list the coefficients of changes in

desired capital stock lagged zero, one, and two years respectively.

Columns X5 and Xg lis t the coefficients of net investment lagged one

and two periods respectively. Column X^ lis ts the coefficient of

capital stock. The numbers in parentheses are the standard errors

of the coefficients.

As a basis for comparing the micro performance of the five

investment theories a naive model was used. Gross investment for

each firm was regressed on its own lagged values up to a maximum of

^Henri T h eil, Economic Forecasts and Policy (Amsterdam: North-Holi and, 1961); "Specifications Errors and the Estimation of Economic Relationships," Review of the International Statistical Institute. XXV (January, 1957), 41-51. 89 three. A lagged investment variable was allowed to enter the equation i f i t reduced the residual variance. ^ 8

Goodness of f i t statistics for the selected regressions by ig ? theory and firm are presented in Table 5. In that table, R is the coefficient of determination un-adjusted for degrees of freedom and s is the minimum standard error of the regression equation 20 (adjusted for degrees of freedom).

I t is immediately obvious from Table 5 that each theory is clearly superior in explanatory power (s) to a purely autoregressive model. Indeed, the neoclassical I theory had a lower standard error than the naive model for each of the twenty-seven firms. The accelerator and neoclassical I I theories had lower standard errors than the naive model for twenty-six of the twenty-seven firms. These were followed by twenty-five for the liquidity theory and twenty-four for the expected profits theory. The five theories also performed well in terms of explanatory O power as measured by R . The null hypothesis that the vector of

18As mentioned above, the sole reason for introducing the naive model was to provide a basis fo r comparing the investment theories. Consequently, the coefficients of this model are not presented.

^9Two tests for autocorrelation are presented and discussed • in section VI. 6 . G.

2 0 R2 unadjusted for degrees of freedom cannot be reduced by additional explanatory variables while R2 (adjusted R2) can if the additional explanatory power of the added variable is insufficent to outweigh the decrease in R2 caused by a reduction of degrees of freedom. See Carl F. Christ, Econometric Models and Methods (New York: John Wiley and Sons, Inc., 1966), £09-516. Table 5

Goodness of fit statistics

GROUP: Autos

Chrysler General Motors R2 s R2 s

Accelerator .69 .06672 .82 .18465 Expected Profits .81 .05395 .83 .17912 Liquidity .63 .07295 .71 .22478 Neoclassical I .61 .07431 .93 .11108 Neoclassical I I .61 .07500 .91 .12486 Naive .53 .07467 .69 .21934

GROUP: Electrical

General Electric Westinghouse Electric

R2 s R2 s

Accelerator .93 .03287 83 .01425 Expected Profits . 8 6 .04446 80 .01600 Liquidity .87 .04363 67 .01979 Neoclassical I .90 .03946 70 .01901 Neoclassical I I .91 .03775 70 .01887 Naive .85 .04374 64 .01934 Table 5 (Continued)

GROUP: Chemicals

Dow Chemical Dupont Eastman Kodak R2 s R2 s - R2 s

Accelerator .79 .03065 .81 .03927 97 .01070 Expected Profits .55 .04327 .73 .04513 98 .00887 Liquidity .60 .04069 .75 .04372 98 .00887 Neoclassical I .55 .04329 .73 .04529 96 .01119 Neoclassical 11 .67 .03815 .74 .04436 96 .01181 Naive .45 .04635 .72 .04587 94 .01350

Monsanto Union Carbide R2 s R2 s

Accelerator . 8 6 .02677 .79 .03468 Expected Profits .87 .02513 .63 .04369 Liquidity .87 .02517 . 6 6 .04296 Neoclassical I .71 .03576 . 6 8 .04306 Neoclassical I I .82 .02930 .84 .03043 Naive . 6 6 .03659 .60 .04666 Table 5 (Continued)

GROUP: Oil

Standard Oil Continental Oil Gulf Shell of California

R2 S R2 s R2 s R2 s

Accelerator .94 .01919 .91 .05273 .90 .03707 .80 .03496 Expected Profits .87 .02704 .83 .07651 .81 .04786 .83 .03499 Liquidity . 8 8 .02620 .91 .05298 . 8 6 .04471 .85 .03159 Neoclassical 1 .90 .02498 .87 .06666 .89 .03799 .83 .03228 Neoclassical I I .91 .02249 .85 .07105 .91 .03496 .81 .03395 Naive . 8 8 .02655 .74 .08225 .79 .04765 .72 .04267

Standard Oil Standard Oil of Indiana of New Jersey Texaco

R2 S R2 s R2 s

Accelerator . 6 6 .04823 .81 .12684 .81 .06695 Expected Profits .75 .04107 .79 .13375 .80 .06940 Liquidity .58 .05149 .83 .12159 .85 .06054 Neoclassical I .76 .04182 .83 .12997 .84 .06392 Neoclassical I I .70 .04362 . 8 8 .11088 .82 .06761 Naive .35 .06232 .69 .14943 .79 .07023 Table 5 (Continued)

GROUP: Rubber Fi restone General Tire Goodri ch ,2 ,2 R2 s s 5

Accelerator .84 .01303 89 .00543 82 .00915 Expected Profits .90 .0 1 1 0 1 80 .00709 8 6 .00835 Liquidity . 8 8 .01250 84 .00650 81 .00935 Neoclassical I . 8 6 .01237 8 6 .00603 81 .00941 Neoclassical II .83 .01330 8 8 .00570 84 .00931 Naive .79 .01537 79 .00729 73 .01098

Goodyear Uni royal R s R2 s

Accelerator . 8 6 .01692 .83 .00712 Expected Profits .85 .01736 .84 .00699 Liquidity .85 .01732 .87 .00643 Neoclassical I .84 .01776 .81 .00772 Neoclassical II .87 .01704 .81 .00767 Naive .74 .02194 .78 .00805 Table 5 (Continued)

GROUP: Steel

Armco Steel Bethlehem Steel Inland Steel

R2 s R2 s R2 s

Accelerator .87 .01366 .75 .04711 .60 .02118 Expected Profits .84 .01447 .72 .05016 .65 .02005 Liquidity .87 .01440 .75 .04874 .60 .02136 Neoclassical I .89 .01226 .70 .05142 .62 .02067 Neoclassical I I .91 .01150 .72 .04983 .61 . 0 2 1 0 0 Naive .77 .01707 .64 .05503 .49 . 0 2 2 0 0

National Steel Republic Steel U.S. Steel

R2 s R2 s R2 s

Accelerator .71 .01717 .64 .02719 .45 .08518 Expected Profits . 6 6 .01915 .64 .02627 .60 .07530 Liquidity .56 .02107 .69 .02428 . 6 6 .07190 Neoclassical I .60 . 0 2 0 1 2 .64 .02626 .62 .07309 Neoclassical I I .63 .01929 .63 .02638 .63 .07249 Naive .47 .02233 .47 .03066 .42 .08180 95 slope parameters equalled zero was tested by the F

statistic:*^

(IV .7 .) F -- R2 / ( k -l) (l-R ^ A n -k )

p where R is the coefficient of determination unadjusted for degrees

of freedom, k is the number of parameters (including the constant),

and n is the number of observations (twenty throughout this dis­

cussion). The null hypothesis was rejected for every theory and

firm (except accelerator-standard Oil of New Jersey) at the five

percent level of significance. At the one percent level of signi­ ficance the null hypothesis was rejected, from a possible twenty-

seven: twenty-six times for the accelerator theory; twenty-five

times for the expected profits, liquidity, and neoclassical I

theories; twenty-seven times for the neoclassical I I theory;

twenty-four times for the naive theory.

To discriminate among the theories we compare the explana­

tory power of each theory to that of each other theory for the

twenty-seven firms. The results are recorded in Table 6 . Read

row-wise, Table 6 tells us how many times, out of twenty-seven,

that that theory in the row had a lower standard error relative

to the theory 1n the column heading. For example, the expected

profits theory (row) had a lower standard error for eleven of the

twenty-seven firms than the accelerator theory (column).

21see J, Johnston, Econometric Methods (2nd ed.; New York: McGraw-Hill, Inc., 1972), 1337 9 6 Table 6

Number of times, out of twenty-seven, a theory (row) had a lower standard error than competing theories (column)

in 4-> 4—1 ‘r- t—t 1—4 *4- O r— r — s- S~ (O «o o CL O u 4-> :>> ■r— *(— fO 4-J tn in S- a> •r— in in u CL t r o o •i— o X ■r— a) a) (C lu —1 !Z z z

Accelerator X 16 18 18 17 26 Expected Profits 11 X 1 0 14 11 24 Liquidity 9 17 X 16 1 2 25 Neoclassical I 9 13 11 X 1 0 27 Neoclassical II 1 0 16 15 17 X 26 Naive 1 3 2 0 1 X

No clear conclusions concerning the superiority of one theory over another can be determined from Table 6 . While i t is obvious that the naive model was out-performed by each of the other theories, the same cannot be said for intra-theory comparisons. The accelerator theory appears to be the best but with a maximum of 18 firms out of

27 firms with a lower standard error than competing theories this could not be classified as superior. It is interesting to compare our findings with those of Jorgenson and Siebert2 2 who, using fifteen

220ale W. Jorgenson and Calvin D. S iebert, "A Comparison o f A lte rn ative Theories o f Corporate Investment Behavior," American Economic Review. LVIII (September, 1968), 704-707. firms, found that the neoclassical I model had a lower standard error than the accelerator, expected profits, and liquidity models for 1 2 ,

12, and 14 firms respectively. They conclude by ranking the models, on the basis of goodness of f i t tests, as neoclassical I (best), neoclassical II, expected profits, accelerator, and liquidity. Using the data in Table 6 , we would rank the models as: accelerator (best), neoclassical II, liquidity, expected profits, and neoclassical I.

Since the firms were not the same in both studies nor were the samples of firms randomly selected we cannot make probabilistic statements concerning the acceptability of each theory.

The five competing theories differ in their definition of desired capital stock and a test which discriminates between the theories on the basis of these definitions is appropriate. F irs t, the number of times a desired capital stock variable was included in a regression was counted for a ll twenty-seven firms by theory. That is , we counted the number of times columns X£, X3 , and X4 in

Appendix A had non-zero coefficients and added these values over twenty-seven firms for each theory. The numbers, presented in

Table 7 under the column labelled "number of desired capital stock coefficients," ranged from a low of 40 (neoclassical I) to a high of

49 (accelerator). No one theory was superior in terms of number of included desired capital stock variables. I f , using Appendix A, we count only those desired capital stock variables with coefficients at least twice th eir standard errors, we get more discrimination among the theories. These results are presented in the second column of Table 7. The ranking is as follows: 98

Accelerator (best); neoclassical II; neoclassical I; liquidity; expected profits. On the basis of this tes t, Jorgenson and Slebert pQ ranked the theories: ^ * 3 Neoclassical I (best); neoclassical II; expected profits; accelerator; liquidity.

Table 7

Number of desired capital stock coefficients and number at least twice th eir standard errors for twenty- seven companies

Number of desired capital stock Number of desired coefficients at capital stock least twice their Theory coefficients standard errors

Accelerator - 49 28 Expected Profits 43 1 2 Liquidity 46 16 Neoclassical I 40 17 Neoclassical II 42 2 1

5. Macro performance

The data for the individual firms was aggregated into group data using the procedure described in section 1.5. That is , the analogous aggregation procedure was adopted using equal weighted linear aggregation of the variables.

The “best" equation was selected from the rational lag class by minimization of the residual variance. The numerator and

oq Jorgenson and S iebert, American Economic Review. L V III, 707. 99 denominator polynomials in the lag operator were limited to a maximum of second degree as explained in section IV .4. The estimated aggre­ gate coefficients are presented In Appendix A.

Goodness of f i t statistics are presented in Table 8 for the six aggregates: Autos; chemicals; electrical; o il; rubber; steel.

As before, a ll theories out-perform the naive model although in the case of the liquidity and neoclassical II theories it was in five out of six cases.

Using the F statistic (equation IV.7.) to test the null hypothesis that the slope parameters are zero i t was found that this hypothesis was rejected in a ll cases at the one percent level of significance.

The next test on the aggregates was to determine the number of times out of six that a theory had a lower standard error than competing theories. Referring to Table 9 we find the ranking:

Accelerator; expected profits; neoclassical I; liquidity; neoclassical

I I . Compare this with our findings for the micro-equations where the ranking was: Accelerator; neoclassical II; liquidity; expected profits; neoclassical I. When we compare the theories by the number of desired capital stock variables included in the six regressions, we find the accelera­ tor theory again at the top as in the micro equations. We also find the two neoclassical theories at the bottom of the lis t which is in opposition to that found by Jorgenson and S ie b e rt.^ The ranking in

^Jorgenson and S iebert, American Economic Review, L V III, 707. Table 8

Goodness of f it statistics

GROUP: Aggregates Autos Chemicals Electrical

R2 S R2 s R2 s

Accelerator . 8 6 .20207 .94 .07467 .94 .03718 Expected Profits .87 .19503 .94 .07930 .89 .05218 Liquidity .75 .26109 .93 .07941 .87 .05444 Neoclassical I .78 .25395 .93 .08231 .91 .04576 Neoclassical I I .72 .27347 .93 .08098 .91 .04583 Naive .72 .26131 . 8 8 . 1 0 0 2 2 .85 .05363

Oil Rubber Steel in R2 S R2 S

Accelerator .96 .18719 .95 .02744 .65 .16461 Expected Profits .92 .26656 .95 .02996 .76 .14018 Liquidity .95 .20039 .97 .02404 .63 .17450 Neoclassical I .94 .24279 .94 .03050 .72 .15198 Neoclassical I I .94 .22915 .94 .03101 .71 .15463 Naive . 8 8 .29480 .93 .03290 .56 .17755 1 0 1 terms of the largest number of included desired capital stock variables with coefficients at least twice their standard error is reported in Table 10. The ranking is: Accelerator; liq u id ity ; expected profits; neoclassical I and II (tied). Notice that the accelerator theory is s till the best but that the neoclassical theories which were second and th ird , respectively in the micro equations are now last in the aggregate equations.

Table 9

Number of times, out of six, a theory (row) had a lower standard error than competing theories (column)

■4-» r—r • p— >—< ►—i 4 - O s - 5 - ro rtf o a . u O 4-> > > •r— •t— rtf ■a 4-> o l V} S- a t ■r— Vt CO Q) 4-» • o « rtf r— i O •r- f— i ■■ a ) a t 3 U o u CL c r o o o X •i—

Accelerator X 4 5 5 5 6 Expected Profits 2 X 4 4 4 6 Liquidity 1 2 X 3 4 5 Neoclassical I 1 2 3 X 4 6 Neoclassical II 1 2 2 2 X 5 Naive 0 0 1 0 1 X 102

Table 10

Number of desired capital stock coefficients and number at least twice their standard errors for six aggregates

Number of desired capital stock Number of desired coefficients at capital stock least twice their Theory coefficients standard errors

Accelerator 1 2 8 Expected Profits 1 0 3 Liquidity 1 0 5 Neoclassical I 9 2 Neoclassical II 9 2 CHAPTER V

EMPIRICAL: AGGREGATION BIAS

1. Bias in the unconstrained macro coefficients

A. Introduction.

Our interest in empirical measures of aggregation bias centers

around three effects: (a) The effect of aggregation on the unconstr­

ained aggregate coefficient estimates of the rational lag function;

(b) The effect of aggregation on the lag structure and on the average

lag; (b) the effect of aggregation on estimates of the aggregate residual variance. Each of these effects w ill be analyzed in separate sub-sections.

This sub-section (V .l.) is entitled "bias in the unconstrained macro coefficients" and serves to answer the following questions:

(a) What are the empirical values of the component parts of the est­

imated aggregate coefficients? The values presented below follow the outline of Table 1, i .e . , a true value, an aggregation bias, a specif­ ication bias, an implied sampling error, and-bias attributable to corresponding and non-corresponding micro-parameters (section V .I.B .);

(b) is aggregation bias present in our estimated macro coefficients?

(section V.I.C.); (c) if bias is present, does it vary by invest­ ment theory, industry, or macro coefficient? (section V.l.D .);

(d) what are the sizes of the aggregation bias components (aggregation

103 104 bias, specification bias, implied sampling error) and do these differ significantly from one another? (section V.l.E); (e) is there a significant difference between the corresponding and non-corresponding sources of bias? (section V.I.F.); (f) is there a significant diff­ erence in bias under the ordinary least-squares and Durbin two-stage least-squares estimators? (section V.I.G .).

B. Empirical measures of the aggregate coefficients and their com­ ponents.

We developed the theory of aggregation bias in rational lag functions in Chapter I. In that Chapter the estimated aggregate coe­ ffic ie n t (E) was shown to be the sum of a true value (T ), an aggregat­ ion bias (AB), a specification bias (SB), and an implied sampling error (SE). Both the AB and SB were further divided into a bias attributable to corresponding micro-parameters (C) and non-correspon­ ding micro-parameters (NC). This analyis was summarized as a set of formulas in Table 1.

Empirical measures of T, AB, SB, SE, C, NC, and E are pres­ ented in Appendix B. This data is the raw material on which we shall base our analysis in sections V .l.C .-F . Each of the empirical meas­ ures is presented for each of six industry groups, five investment theories, and seven coefficients. BQ is the constant term. B-|, B2, and B3 are the coefficients of the fir s t difference in desired capital stock lagged zero, one, and two periods respectively. B4 and B5 are the coefficients of net investment lagged one and two periods respect­ ively. B6 is the coefficient of actual capital stock.

A zero (0.0) in an entire column (coefficient) of Appendix B 105 signifies that that coefficient was zero in all of the micro equations in that industry group and in the industry equation.

The aggregation and specification biases for the constant term,

Bq, attributable to corresponding micro-parameters is zero by defin­ itio n. This can be seen by referring to equation (1.25.) where the constant term (there called B-j) is written as the sum of the micro constant terms plus certain covariance terms. These covariance terms are functions of the slope parameters alone and therefor any bias in the constant term is attributable to these (non-corresponding) micro slope parameters.

C. A test for the presence of aggregation bias.

We have shown, in section I . 3 .,the necessary and sufficient conditions for consistent aggregation of J firm functions in K indep­ endent variables into one function in K independent variables. It was further shown that i f the micro and macro functions and the aggregates are linear ( I .6 .- I . 9.) the necessary and sufficient cond­ itions for consistent aggregation are that the individual functions be linear with identical slopes.

The underlying slope parameters of each micro equation are unknown; however, estimates of these parameters have been obtained by least-squares. The relevant question then becomes whether, subject to sampling fluctuations, the parameters o f each firm within a group are "not significantly different."

Formally, the problem can be set up by writing the J indiv­ idual firm equations: where Yj is a (Txl) vector of observations for the jth firm and Xj is a (TxK) matrix of observations on the K independent variables. Bj is the (Kxl) vector of coefficients to be estimated and Uj is a (Txl) vector of random disturbances with identical variance-covariance matri­ ces equal too ^Iy. We may write equations (V .l.) as:

(V .2.) V 7<1 0 0 V V Y2 0 x2 u2 B 2

« . 0 • • • * « • • « • • • a o .. 0 0 BJ UJ

• • • • • • • • • ■

Y j 0 0 0 •J UJ

We shall call (V .2.) the individual regressions model.

I f we assume as our null hypothesis

(V .3 .) H0: B = Bj (j a 1 * ••• »J) we may write equation (V .2.) as: 107

We shall call (V .4.) the master regression model.

I f the errors of the J firms, Up Ug, . . . ,U j, . . . ,llj, are independent across equations and have a common variance-covariance matrix a lp then we may test the null hypothesis (V .3.) using the F statistic:^

fV.5.1 F = (h/K

where: q3 = Qr Q2

Qt = Y'Y - B'X'Y

$2 = ^ej' ej J ej = the (Txl) vector of computed residuals for the jth equation.

We applied this F sta tis tic to the individual regressions models for the five investment theories summarized in section I I . 5. and to the master regression for the corresponding industry group.^

The results are summarized in Table 11

We see from Table 11 that the null hypothesis of parameter homogeneity is rejected for all five theories of investment behavior

V o r the two equation case see Gregory C. Chow, "Test of equal­ ity between Sets of Coefficients in Two Linear Regressions," Economet- ric a , XXVIII (July, 1960). 591-605. We are here extending the Chow test to the J equation case.

Recalling that structures were chosen from the rational lag class by minimizing the residual variance, it follows that the number of included variables, K, w ill not necessarily be the same in the individual and master models. I chose, for the numerator degrees of freedom, the maximum number of included variables from the individual and master regression models. This has the effect of "biasing" the results towards acceptance of H . That is , the burden of proof is put on showing that aggregation bias is present. 108 Table 11

Test results of parameter vector homogeneity by theory and group

Theory

•r“ •r--o 3 Group •i—O" Accelerator Expected ProfitsExpected Neoclassical Neoclassical I Neoclassical Neoclassical I I

Autos F (computed) 2 . 0 0 3.45 2.78 16.88 3.99 F.01 9.38 7.23 9.38 3.70 3.47 Biased=B BB Degrees of freedom (5,30) (6,29) (5,31) (5,31) (6,30)

Chemicals F (computed) 93.02 45.67 2 1 . 2 1 7.16 80.24 F.01 3.07 3.04 3.04 2.87 3.07 Biased=B BBB BB Degrees of freedom (6,73) (6,76) (6,77) (7,78) (6,75)

Electrical F (computed) 6.67 5.88 1.92 1.89 1.17 F.01 7.23 7.23 9.38 7.23 3.50 Biased=B Bat.05 Bat.05 Degrees of freedom (6,28) (6.29) (5,30) (6,29) (6,29)

Oil F (computed) 13.38 10.73 17.66 14.32 16.72 F.01 2.99 2.82 2.82 2.82 2.82 Biased=B B B BBB Degrees of freedom (6,109) (7,109) (7,108) (7,103) (7,107)

Rubber F (computed) 4.81 9.96 5.45 8.30 7.74 F.01 3.04 3.04 3.04 3.25 4.03 Biased=B BB BBB Degrees of freedom (6,79) (6,76) (6,76) (5,79) (6,75)

Steel F (computed) 6.17 6.50 12.72 6.90 5.63 F.01 3.04 2.87 2.87 3.04 3.04 Biased=B B B B BB Degrees of freedom (6 , 8 8 ) (7,87) (7,86) (6,89) (7,89) 109 in four industry groups . . . chemicals, o il, rubber, and steel. We conclude that the micro and macro results are inconsistent and that the macro parameter estimates are biased. We shall pursue the extent of this bias below.

The null hypothesis of parameter homogeneity across firms was accepted for three theories in each of two industry groups . . . autos and electrical.

The interpretation of the test should be clear. If the corr­ esponding coefficients in each individual regressions model are not significantly different from one another, a master regression model is implied. The F statistic tests to see if the difference between the sum of squared residuals from the master regression model and the individual regression models (adjusted for d .f.) is significantly different from the individual regression residuals.

D. The assumptions for the analysis of variance.

We shall use the analysis of variance (ANOVA) technique to make inferences concerning the means of various classifications. For example, we w ill wish to test the null hypothesis of no significant difference in the error of the estimated (E) and true (T) aggregate coefficients, T/E, between investment theory and bias component

(aggregation bias, specification bias, implied sampling error ),

(see Table 14.); however, before using the ANOVA technique, we must be clear on its assumptions and whether they are lik ely to be satis­ fied. To do this, we introduce the following rectangular array of r rows and c columns where a typical cell observation is X.jjt n o

(i = 1, . . . ,r ; j * 1, . . . ,c ). Also designate the true mean values of a cell observation as m^j, of a row as m^ , of a column as m^j, and of a ll the observations as m . • •

Column 1 2 3 ... j « t« C Row Means ■ . . . X-jj . . . X'jg 1 X 11 X12 x13 xl. X CM CM

2 X ... Xgc X21 ro CO ••• X2j X2 ,

• • •

• • •

Row i . . . X^j ... X-Jq xii xi 2 xi3 xi. • • •

• • ♦

r Xr i . . . xrj- . . . Xyg xr 2 xr3 xr. nn X Colu x.l X . 2 x.3 — X. j ••• x.c Means

O Eisenhart lis ts four necessary and sufficient assumptions for the validity of the ANOVA procedures:^

Assumption 1 (Random Variables): The numbers • are random variables distributed about the true means, m.., (i = 1 , ... ,r; j = 1 , ... ,c), that are fixed constants. J

This is an assumption previously made for the estimated coeff-

3Churchill Eisenhart, "The Assumptions Underlying the Analysis of Variance," Biometrics, I I I (March, 1947), 1-21.

4These assumptions apply to Class I ANOVA problems, i . e . , the detection and estimation of fixed (constant) relations among the means of sub-sets of the population. See Eisenhart, Biometrics, I I I , 3-5. I l l

icients of our five investment theories.® That is . the estimated

coefficients are random drawings from a population with coefficients

which are fixed parameters. Thus, for example, i f we wish to test for

differences in T/E by investment theory and coefficient (Table 14) then

the various cell values w ill be random variables since E is a random

variable.

Assumption 2 (A dditivity): The parameters m.-j are related to the means nu , m j , and m

which, i f ignored, w ill have the effect of increasing the residual sum

of squares and biasing the results towards acceptance of the null

hypothesis . 6 Consequently, we shall compute an interaction term and

test its significance in all of our ANOVA tables.

Assumption 3 (Equal Variances and Zero Correlations): The random variables X^j are homoscedastic and mutually uncorrelated.

One can test for homoscedasticity by B artlett's test;^ however,

this test is sensitive to the kurtosis coefficient. In particular, if

the kurtosis coefficient is negative, Bartlett's test is biased

towards homoscedasticity, while i f i t is positive, the test is biased

®See Chapter I I I .

6See Henry Scheffd', The Analysis of Variance (New York: Wiley, 1959), Chapter 4. Also, see W. G. Cochran, "Some Consequences when the Assumptions for the Analysis of Variance are not Satisfied," Biometrics, I I I (March, 1947), 35-37.

?M. S. Bartlett, "Properties of Sufficiency and Statistical Tests," Proceedings of the Royal Society. London. Series A, 160, 268-282. 112

towards heteroscedasticity . 8 Fortunately, the ANOVA technique is

robust with respect to inferences about means in our case of equal cell frequencies.

Positive correlation in the observations, X^j, has the effect of increasing the probability of a Type I error, while negative corr­ elation has the opposite effect. 8 The type of data we shall be analy­ zing such as different coefficients within an equation, components of bias ( i . e . , aggregation bias, specification bias, implied sampling error), and different theories, would seem to yield uncorrelated X-jj.

Assumption 4 (Normality): The X^j are jointly distributed in a multivariate normal (Gaussian) distribution.

Since the ANOVA technique is robust against departures from normality in the case of inferences about meansj 8 violation of this assumption w ill have l i t t l e effect on our re s u lts .^

I t should be clear from the above discussion that the relevant question using the ANOVA technique is not "are the assumptions sat­ isfied?" but rather, "how closely are they satisfied?" Perhaps, even better, "how badly are they violated?" because we have seen that i t is

8 G. E. P. Box, "Non-Normality and Tests on Variances," Biometrika, XL, 318-335.

8See Scheffe, The Analysis of Variance, 360 and Cochran, Biometrics, III, 32-35.

T°Scheffe", The Analysis of Variance, 334-351, and 362.

^We can summarize the above four necessary and sufficient assumptions for the application of the ANOVA technique as: X... - m + (m.j -m ) + (m j-m. ) + Z jj , (i - 1 , ... ,r; j « 1 , . . . ,c) where the Z jj’arfe'normally ancl independently distributed about zero with a common variance, a2. See Eisenhart, Biometrics, I I I , 14. 113

the robustness of our inferences in the face of violations of the 12 assumptions which are important.

D. Size of the aggregation bias.

Having established that aggregation bias exists in our macro-

parameter estimates we shall now measure its size. To do th is, we divide the true value (T) by the estimated value (E) in Appendix B.

Ideally, T/E should equal unity. If it is greater (less) than unity our estimates are downward (upward) biased.

Appendix C presents our values of T/E by coefficient, theory, and industry. Visual inspection of these figures reveals that only Bg comes close to the correct figure of 1.00. The other coefficients • range from a low of -51.64 (Bg, accelerator, autos) to a high of 11.22

(Bg, neoclassical I I , autos). One encouraging result is that all the

T/E ratios for the slope parameters are of the correct sign, i.e ., positive.

The average of the absolute values of the relative error,

|(E-T)/E| , over the six industries is presented in Table 12. This table suggests that the coefficient with the least error is Bg. The average for all values of |(E-T)/E| over the six industries is \2%.

The coefficients 8 4 and Bg of lagged net investment have the next

^Consider the following quotes; "Since an experimenter could rarely, i f ever, convince himself that a ll the assumptions were exactly satisfied in his data, the technique must be regarded as approximative rather than exact." Cochran, Biometrics, I I I , 37; "Among the under­ lying assumptions made in deriving statistical methods are usually some that are apt to be violated in applications and are introduced only to ease the mathematics of the derivation." Scheff£. The Analysis of Variance. 360. Table 12

Average size of | ^ - | over six industries by theory and coefficient E

Theory B0 B1 B2 B3 B4 B5 B6

Accelerator 1.40 .32 .36 .41 .45 .54 .1 0

Expected Profits .70 .16 .30 .34 .41 .36 .11

Liquidity 2.98 .42 .43 . 2 0 .45 .24 .15

Neoclassical I 1.97 .40 .42 .55 .43 .44 .11

Neoclassical 11 1.76 .33 3.90 .48 .44 .34 .14

Average 1.76 .33 1.08 .40 .44 .38 .12 115

smallest error averaging 44% and 38% respectively; however, these

values are more disperse than those of Bg. Of the desired capital

stock coefficients, B-j, B2, and B 3 , two of them, B-j and B3 ,, have

average errors (33%, 48%) close to those of B4 and B5 ; however, they

are much more disperse. The other desired capital stock coefficient,

B2, has an average error of 108%. The worst coefficient in terms of

|(E-T)/E| is Bo, the constant term, with an error of 176%.

The above results are not unexpected. The classification of

the coefficients into four groups follows the classification suggested

by the investment theories in Chapter I I . That is, a constant term,

coefficients of desired capital stock, coefficients of lagged net

investment, and the coefficient of capital stock. Now, the five

investment theories differed only in their determinants of desired

capital stock. The dependent variable, gross investment, and the

independent variables, lagged net investment and capital stock, were

the same in each theory. We should then expect the coefficients, B 4 ,

B5 , B@, of these variables to be the same or very sim ilar. Thus, these

coefficients should come closest to satisfying the requirement for

consistent linear aggregation, i.e ., the corresponding micro slope

parameters must be identical. The same is not true for the desired capital stock coefficients. These coefficients are quite dissimilar even for a particular theory and firms within a given industry.

The above impressions can be verified by performing a two-way analysis..of variance. We tested the two hypotheses that: (a) There is no significant difference between coefficients in their effects on 116

f(E-T)/E1 averaged over six industries; (b) there is no significant difference between theories in their effects on |(E-T)/E| averaged over six industries. Our results are presented in Table 13. We conclude that the average |(E-T)/E| does vary significantly between coefficients but not between theories.

Since the object of our aggregation study is to measure bias created by aggregating from the firm to the industry level we should run separate tests for bias by industry group. Using the data in

Appendix C we test the null hypotheses: (a) There is no significant difference in T/E between theories; (b) there is no significant difference in T/E between coefficients. The results of our two-way analysis of variance are presented in Table 14 by industry group.

We accept the null hypothesis of no significant difference in

T/E betv/een theories in five out of six cases. We accept the null hypothesis of no significant difference in T/E between coefficients in four out of six cases.

E. Components of the bias.

We have shown in Chapter I that an aggregate estimated coeff­ icient (E) is composed of a true value (T), an aggregation bias (AB), a specification bias (SB), and an implied sampling error (SE). That is , E - T + AB + SB + SE.

In Appendix D we present the three bias components, AB, SB, and

SE, as a proportion of the error (E-T). We should like to use these values to test two null hypotheses: (a) There is no significant difference in |E—T| averaged over five theories between components Table 13

Analysis of variance of average | ~ | over six industries by theory and coefficient

Source of Sum of Degrees of Mean F variation squares freedom square ratio

Between theories 1.98 4 .495 1.04

Between coefficients 9.42 6 1.570 3.30

Residual 11.43 24 .476 f .95(4,24)

Total 22.83 34 f .95(6.24) 118 Table 14

Analysis of variance of T/E by theory and coefficient

Degrees Sum of of Mean F Source of variation squares freedom square ratio

Autos

Between theories 8.853 4 2.213 .65 Between coefficients 28.327 6 4.721 1 39 Residual 81.289 24 3.387 f ’ 9 5 (4,Z4) = 2.78 Total 118.469 34 F:95(6,24) * 2.51

Chemicals

Between theories 8.736 4 2.184 .15 Between coefficients 13.598 6 2.266 .16 Residual 346.856 24 14.452 F.95(4,24) = 2.78 Total 369,190 34 F:95(6,24) = 2.51

Electrical

Between theories 1.270 4 .318 .74 Between coefficients 3.063 6 .511 1.19 Residual 10,273 24 .428 F.g5(4,24) = 2.78 Total 14.606 34 F;95(6.24) = 2.51

Oil

Between theories .698 4 .175 .75 Between coefficients 5.822 6 .970 4.16 Residual 5.589 24 .233 F.95(4*24) = 2.78 Total 12.109 34 F;95(6.24} = 2.51

Rubber

Between theories .184 4 .046 5.11 Between coefficients 7.080 6 1.18 131.11 Residual .216 24 .009 F.95(4*24) * 2.78 Total 7.4804 34 ^95(6.24} = 2.51

Steel

Between theories 27.281 4 6.820 .95 Between coefficients 4.218 6 .703 .1 0 Residual 171.988 24 7.166 F.95C4.24) = 2.78 Total 203.487 34 F.95^6*24) = 2.51 119

AB, SB, and 5E; (b) there is no difference in |E-T| averaged over five theories between coefficients.

The results of our two-way analysis of variance on |E-T| 13 between components and coefficients are presented in Table 15. In that table, we accept the null hypothesis of no significant difference between component effects in five out of six cases. We reject the null hypothesis of no significant difference between coefficients in five out of six cases. This is merely a reflection of the fact that the seven coefficients Bq to Bg are of different orders of magnitude. For example, Bg is an estimate of the replacement rate, and should approach the values in Table 4; whereas the coefficients of lagged net invest­ ment, B4 and B5 , tend to be about .7 and -.5 respectively. The const­ ant term, Bq, also varies quite widely.

F. The corresponding and non-corresponding bias components.

Another hypothesis suggested by Appendix C is that the error between the true and estimated aggregate coefficients is different for corresponding (C) and non-corresponding (NC) components. To test for this we establish the null hypotheses: (a) There is no significant difference between C and NC components of (E-T); (b) there is no significant difference between coefficient's effects on (E-T).

A two-way analysis of variance was run for each industry and the results reported in Table 16. Each theory was assumed to be a replication of the experiment to test for differences between means of

^There is no interaction term in this table because there is only one observation per cell, i.e ., the average jE-Tj. 120 Table 15

Analysis of variance of average |E-T| over five theories by bias component and coefficient

Degrees Sum of of Mean F Source of variation squares freedom square ratio

Autos

Between components 48.809 2 24.504 4.29 Between coefficients 87.067 6 14.511 2.55 Residual 63.306 1 2 5.692 F.95(2,12) = 3.89 Total 204.182 20 F,95(6*12) = 3.00 Chemical

Between components .805 2 .403 .09 Between coefficients 83.662 6 13.944 3.22 Residual 52.043 1 2 4.337 F 9 5 ( 2 , 1 2 ) = 3.89 Total 136.510 2 0 F,9516,12) = 3.00 Electrical

Between components 87.493 2 43.747 1.42 Between coefficients 1,622.504 6 270.417 8.77 Residual 370.013 1 2 30.834 F.95(2,12) = 3.89 Total 2,080.010 2 0 F,95(6*12) » 3.00 Oil

Between components 29.436 2 64.943 1 .1 1 Between coefficients 3B9.659 6 13.202 4.92 Residual 153.413 12 F.95(2»12) = 3.89 Total 577.513 2 0 F,9 5 ( 6 ,12) = 3.00

Rubber

Between components 21,927.673 2 10,963.837 1.04 Between coefficients 394,349.461 6 65,724.910 6.23 Residual 126,581.403 12 10,548.450 F,9 5 ( 2 ,12) = 3.89 Total 542,858.537 20 F.9 5 (6,12) « 3.00

Steel

Between components 583.454 2 291.727 2,06 Between coefficients 10,178.480 6 1,696.413 11.98 Residual 1,699.531 12 141.628 F 9 5 ( 2 , 1 2 ) = 3.89 Total 12,461.465 20 F.95(6,12) = 3.00 121 Table 16

Analysis of variance of E-T by corresponding-noncorresponding components and coefficient

Sum of of Mean F Source of variance squares freedom square ratio

Autos

Between C and NC .008220 1 .008220 . 1 0 Between coefficients .083813 5 .0167626 .2 1 Interaction 2.299399 5 .4598798 5.81 Subtotal 2.391432 11 .217403 f \ 9 5 (M 8 ) = 4.08 Residual 3.798483 48 .079135 F.95(5,48) - 2.45 Total 6.189915 59 Chemicals

Between C and NC .017545 1 .017545 1.32 Between coefficients .046339 5 .009268 .70 Interaction .096425 5 .019285 1.46 Subtotal .160309 11 .014574 1 . 1 0 Residual .635854 48 .013247 F. 95(1.'48) = 4.08 Total .796163 59 F.95(5t48) = 2.45

Electrical

Between C and NC . 0 0 0 0 2 2 1 . 0 0 0 0 2 2 .0 1 Between coefficients .085242 5 .017048 6.54 Interaction .015825 5 .003165 1 .2 1 Subtotal .101089 11 .009190 3.52 .125201 48 .002608 Residual F.95P*48> * 4 - 0 8 Total .226290 59 F.9 5 (M 8 } - 2.45

Oil

Between C and NC .000390 1 .000390 .03 Between coefficients .278299 5 .055660 4.93 Interaction .040295 5 .008059 .71 Subtotal .318984 11 .028999 2.57 Residual ,542251 48 .011297 F.950.48) = 4.08 Total .861235 59 F.95C5.48) = 2.45 122 Table 16 (Continued)

Degrees Sum of of Mean F Source of variation squares freedom square ratio

Rubber

Between C and NC .016238 1 .018238 2.17 Between coefficients .007254 5 .001451 .17 Interacti on .091453 5 .018291 2.17 Subtotal .116945 11 .010631 1.26 Residual .404196 48 .008421 F qcO »48) = 4.08 Total .521141 59 F;95(5.48) = 2.45

Steel

Between C and NC .078504 1 .078504 6 . 2 0 Between coefficients .082339 5 .016468 1.30 Interaction .131413 5 .026283 2.08 Subtotal .292259 11 .026569 2 . 1 0 Residual .607635 48 .012659 F.95p *48) - 4.08 Total .899894 59 F,95(5,48) * 2.45 123 (E-T) due to C and NC and due to coefficients. The hypothesis of no

significant difference between C and NC components of (E-T) was accep­

ted in five out of six cases. The hypothesis of no significant

difference between coefficient effect on (E-T) was accepted in four

out of six cases.

G. The effect of estimator on aggregation bias.

In Chapter I I I we decided on two estimators for the coeffic­

ients of the rational lag function. The ordinary least-squares

estimator (OLS) was to be used if the residuals were not-autocorrelated,

whereas the Durbin two-stage least-squares estimator (D-TSLS) was to

be used i f they were.

We first tested for the presence of auto-correlation in the

five investment theories plus the naive model by the use of the

Durbin-Watson (DW) s ta tis tic .^ The results are presented in Tables

17 and 18 under the heading "D.W.". Three results are possible when

testing the null hypothesis of no auto-correlation using the DW

sta tis tic . . . acceptance of the hypothesis, rejection of the hypothesis,

and inconclusiveness. The results of our DW test are summarized by

theory, level of aggregation, and decision in Table 19. Of the six

theories tested, both for the twenty-seven firms and six industries,

not one had residuals which tested auto-correlated. Of the 198 applications of the DW s ta tis tic , the null hypothesis of no auto-corr- elation was accepted in 120 of these. The remaining 78 cases tested

n ^The Durbin-Watson s ta tis tic , DW, is defined as: DW = i ( e t - e ^ i ) / B^et* See Johnston, Econometric Methods. 251-252. Table 17

Autocorrelation statistics

GROUP: Autos

* Chrysler General Motors D.W. h D.W. h

Accelerator 1.61 2.47 - 1.71 Expected Profits 2.33 *** 2 . 1 0 -.46 Liquidity 2.06 -2.07 1.96 .29 Neoclassical 1 2 .2 1 *** 2.24 — Neoclassical I I 2.19 *** 2.33 — Naive 1.87 .42 2.03 -.07

GROUP: Electrical

General Electric Westinghouse Electric D.W. h D.W. h

Accelerator 2.24 -2.28 • 1.97 .13 Expected Profits 1.87 *** 1.92 *** Liquidity 2.18 *** 1.83 *** Neoclassical I 1.97 *** 1.99 *** Neoclassical I I 1.95 *** 1.94 *** *** Naive 1.96 .2 1 1.75 Table 17 (Continued)

GROUP: Chemicals Dow Chemical Dupont Eastman Kodak D.W. h D.W. h D.W. h

_ _— Accelerator 1 . 8 6 1.81 — 2.00 Expected Profits 1.61 *** 1.58 — 1.99 .12 Liquidity 1.67 *** 1.61 — 1.57 *** Neoclassical I 1.64 *★* 1.64 — 1.97 *** Neoclassical I I 1.85 1.40 1.70 — 1.32 *** Naive 1.98 *** 1.64 2.22 1.82 ***

Monsanto Union Carbide D.W. h D.W. h

Accelerator 2.03 -.41 2.05 — Expected Profits 1.96 1.52 — Liquidity 1.58 — 1.62 — Neoclassical I 1.91 *** 1.98 *** Neoclassical I I 1.72 — 1.91 .37 Naive 1.76 .6 6 1.90 ***

ro ( j i Table 17 (Continued)

GROUP: Oil Standard Oil Continental Oil Gulf Shell of California D.W. h D.W. h D.W. h D.W. h

Accelerator 1.69 5.61 1.82 ___ 2.15 - 1.19 1.80 ___ *** Expected Profits 1 . 8 8 *** 1.69 1 . 6 8 2.16 — Liquidity 1.84 *** 2 . 0 0 — 2.19 *** 1.95 — Neoclassical I 1.61 *** 1.51 *** 2.06 - 1.50 2 .0 1 ___ Neoclassical I I 1.57 — 1.96 *** 2.25 *** 1.55 ___ ■kirk Naive 2.05 -.34 1 . 8 6 .42 1.79 .55 1.95

Standard Oil Standard Oil of Indiana of New Jersey Texaco O.W. h O.W. h D.W. h

Accelerator 1.30 _ _ _ 2.15 *** 1.64 _ _ _ Expected Profits 1.47 — 2 .0 1 *■** 1.50 --- Liquidity 1.19 2.09 *** 1.38 —p. Neoclassical I 1.59 2.13 *** 1.60 ----- Neoclassical I I 1 . 2 2 ___ 2.24 -8.52 1.55 ----- Naive 1.53 *** 1.63 1.23 1.76 1.16

rocn Table 17 (Continued)

GROUP: Rubber

Firestone General Tire Goodri ch D.W. h D.W. h D.W. h *** *** Accelerator 2 . 0 0 — 2.15 1.96 Expected Profits 1.87 *** 2.41 ... 2.09 *** Liquidity 1.76 *** 1.83 *** 1.82 ... Neoclassical I 1.56 — 2.34 — 1.33 ... Neoclassical I I 1.65 — 2.33 — 1.89 *** Naive 2.03 *** 2.08 -.50 2.14 ★**

Goodyear Uni royal D.W. h D.W. h

Accelerator 2.09 *** 1.90 .73 Expected Profits 1,26 — 1.89 *** Liquidity 1.94 1.89 .52 Neoclassical I 1.81 —- 1.80 *** Neoclassical I I 2 . 1 2 *** 1 . 8 6 Naive 2.08 ★** 1.89 *** Table 17 (Continued)

GROUP: Steel

.Armco Steel Bethlehem Steel Inland Steel D.W. h D.W. h D.W. h

_ _ _ Accelerator 1.87 .50 2 . 0 2 2.15 *** *★* •kirk Expected Profits 1.84 . 6 8 1.57 2.09 Liquidity 1.89 .90 2.09 *** 2.13 kkk Neoclassical I 2 . 1 2 -.37 1.61 *** 2.15 *** *** Neoclassical I I 1 . 8 8 .36 1.58 2 .1 1 *** Naive 2.46 -1.95 2 .2 1 -4.23 1.62 1.37

National Steel Republic Steel U.S. Steel D.W. h D.W. h D.W. h

kkk Accelerator 2 . 0 0 - . 0 1 1.58 2.39 1.62 Expected Profits 1.90 .45 1.90 .40 1.32 3.38 *** Liquidity 1.67 1 . 6 8 1.85 .65 1.94 Neoclassical 1 1.97 .1 1 1.73 1 . 1 0 1.49 2.50 Neoclassical I I 1.59 .92 1.80 .82 1.59 3.64 Naive 2.29 *** 1.96 *** 1.58 9.75 Table 18

Autocorrelation statistics

GROUP; Aggregates

Autos Chemicals Electrical D.W. h D.W. h D.W. h

Accelerator 2.32 -.99 2.27 -1.63 1.92 1.28 Expected Profits 2.18 - . 6 6 1 . 8 8 *** 1.80 1.70 Liquidity 1 . 8 8 .61 1.73 4.51 1.73 2.25 Neoclassical I 2.36 *** 1.62 4.99 2.04 -.17 Neoclassical I I 1.90 .67 1.77 *** 1.79 .79 Naive 1 . 8 6 .38 2.27 -1.87 1.75 1.14

Oil Rubber Steel D.W. h D.W. h D.W. h

Accelerator 1.80 .73 1.53 *** 1.89 1.49 Expected Profits 1.70 *** 1.78 *** 1.76 1.36 Liquidity 2.31 -.98 1.83 *** 1.42 *** Neoclassical I 1.28 *** 1.67 **★ 1.82 .95 Neoclassical 11 1.60 *** 1.64 *** 1.90 .53 Naive 1.49 1.28 2.14 **★ 1.97 ***

U?N Table 19

A summary o f results using the Durbin-Watson (D.W.) s ta tis tic by theory and level o f aggregation

Acc. Exp. Prof. liq . Neo I Neo I I Naive Totals

FIRMS Autocorrelated 0 0 0 0 0 0 0

Inconclusive 9 11 11 13 13 0 57

Not Autocorrelated 18 16 16 14 14 27 105

INDUSTRIES 0 0 0 0 0 0 0 Autocorrelated

Inconclusive 3 5 4 4 5 0 21

Not Autocorrelated 3 1 2 2 1 6 15

TOTALS Autocorrelated 0 0 0 0 0 0 0

Inconclusive 12 16 15 17 18 0 78

Not Autocorrelated 21 17 18 16 15 33 1 2 0 w 198 131

inconclusive J 5

As is well known, the DW sta tis tic is biased towards randomness

in the presence of lagged dependent variablesJ6 S tric tly speaking,

our investment theories do not incorporate lagged dependent variables

since the dependent variable is gross investment and the lagged

“dependent"variables are net investment; however, we shall proceed as

though lagged dependent variables do appear in the equation. Never­

theless, there are three reasons for including the DW statistic:

(a) I t is the basis for computing the h s ta tis tic ^ (discussed below)

which is a test for auto-correlation in the presence of lagged

dependent variables; (b) of the two tests, DW and h, only DW exists

for all of our regressions. This point is discussed below; (c)fin­

a lly , we use DW in spite of a bias towards randomness in the presence

of lagged dependent variables because "this bias affects a ll the

distributed lag functions equally so that values of the Durbin-

Watson s ta tis tic provide useful information about the relative presence 18 or absence of auto-correlation."

The naive model is , of course, purely auto-regressive. I t is ,

therefor, tempting to label the results of the DW s ta tis tic as useless.

l^See the discussion on the applicability of the DW test below.

1 6 J. Durbin, "Testing for Serial Correlation in Least-Squares Regression when some of the Regressors are Lagged Dependent Variables," Econometrica, XXXVIII (May, 1970), 410-421.

^Durbin, Econometrica, XXXVIII, 410-421.

^8Dale W. Jorgenson and Calvin D. Siebert, "A Comparison o f A lte rn a tive Theories o f Corporate Investment Behavior," American Economic Review, LV III (September, 1968), 701. 132

Thus, one could say that the results of the naive model In Table 19

(33 out of 33 cases tested non-auto-correlated) are a result of the bias towards randomness in this case. One would then be at a loss in explaining the results of the h statistic for the naive model in

Table 20.(6 not auto-correlated, 3 auto-correlated).

The asymptotic bias in the DW statistic for the model Yt 3 BYt-1

+ Vt where v^ = rv t-l + et and e^ = NID(0,o2) is^ 9 plimDW-DW* =

2r(l-B 2 )/(l+B r) where DW* = the (unknown) true DW. Thus, the bias in

DW depends on B and r. I f r = 0 the bias is zero. In our models an approximate value of B is B = .8 ; therefor, the bias in DW is

.7 2 r/(l+ .8 r). We have reason to believe that r-0 since the investment series is essentially a first difference series; however, even if r is as high as .5, the bias is only .22. We conclude that the DW is not severly biased in our case.

Also, i t has been shown that "the test on the OLS residuals was quite powerful [in the above model]. Considering only experiments with r > .6 in absolute value, the Durbin-Watson test detected the presence of autocorrelation in from 80 to 1 0 0 per cent of the samples from most experiments. " 2 0 I t should also be pointed out that models with exogenous variables as well as lagged dependent variables have an even smaller asymptotic bias.2^

^Johnston, Econometric Methods, 309-310.

2 0 L. D. Taylor and T. A. Wilson, "Three-Pass Least-Squares: A Method for Estimating Models with a Lagged Dependent Variable," Review of Economics and S tatistics, XLVI, 329-346.

2 ^Malinvaud. Statistical Methods of Econometrics, 462-465. 133

To test for auto-correlation In equations with lagged depen­ dent variables we introduce Durbin's h s ta tis tic : 2 2

(V.6 . ) h = (1 - . 5DW)[ n /(l - nV(B4 )i] 1 / 2 where; DW = the Durbin-Watson s ta tis tic .

n = the number of observations = 2 0 .

V(B4) = an estimate of the variance of B4, the first lagged value of the dependent variable.

The h statistic is normally distributed with mean zero and unit variance. A 95% confidence level implies values of ±1.96. I f our .calculated h s ta tis tic fa lls within ±1.96 we accept the null hypothesis of no autocorrelation* otherwise, we reject the null hypothesis.

The calculated values of h are presented in Tables 17 and 18.

It will be noted in these tables that four possibilities may occur:

(a) The test is applicable and we accept the null hypothesis of no auto-correlation; (b) the test is applicable and we reject the null hypothesis; (c) the test is inapplicable because there is no dependent variable lagged one period in the equation being tested. This case is signified by (d) the test is inapplicable because the value of nV(B4) is larger than unity in which case h is undefined. This case is signified by

We summarize our results of the h .s ta tis tic in Table 20. Of

22Durbin, Econometrica. XXXVIII, 410-421. Table 20

A summary of results using Durbin's h statistic by theory and level of aggregation

Acc. Exp. Prof. Liq. Neo I Neo I I Naive Totals

FIRMS Autocorrelated 4 2 2 2 3 3 16

Not Autocorrelated 7 5 5 4 5 12 38

INDUSTRIES Autocorrelated 0 0 2 1 0 0 3

Not Autocorrelated 5 3 2 2 3 4 19

TOTALS Autocorrelated 4 2 4 3 3 3 19

Not Autocorrelated 12 8 7 6 8 16 57

to 135 the 76 applications of the h s ta tis tic 2 3 19 cases tested auto-corre­ lated and 57 cases tested non-auto-correlated. Of the twenty applic­ ations of the h statistic to the neoclassical I and II models, four­ teen cases tested non-auto-correlated.

These results are not unexpected. While the dependent var­ iable, gross investment, may be auto-correlated due to continuity of investments, trend, etc. this does not imply that the disturbances w ill be auto-correlated. Four reasons may be advanced to support this statement:2^ (a) Deflation of the variables eliminates any price trend; tb) net investment is a f ir s t difference series; (c) capital stock, used as an explanatory variable, will remove trend; (d) the use of lagged net investment as explanatory variables eliminates any further portion of the trend in the dependent variable.

While the above evidence argues against the presence of auto­ correlation in our equations i t was decided to apply Durbin's two- stage least-squares procedure (D-TSLS) to see i f a different estimator affected aggregation bias. The results of applying OLS and D-TSLS to the neoclassical I and I I models are presented in Appendix E.

23There was a possible 198 applications of h. Of these, 45 cases were inapplicable because there was no dependent variable lagged oneAperiod in the equation. Another 77 cases were inapplicable beacuse nV(I3 4 ) was greater than unity. Compare our finding with that of potluri Rao and Roger LeRoy M ille r, Applied Econometrics (Belmont, California: Wadsworth, 1971), 124 who, when referring to the possib­ ility of ntf(§4 ) being greater than unity say, "this case is not frequent in applied econometrics. "

2 ^C. D. Siebert, "A Micro-Econometric Study of Fixed Capital Investment Behavior," unpublished doctoral dissertation, University of California, Berkeley, 1966, 111. 136

Our Interest is In differences in T/E between the two est­

imators OLS and D-TSLS. Therefor, following the procedure used to

produce Appendix C, we present T/E, AB/E, SB/E, and SE/E for D-TSLS

in Appendix F.

We can see in Appendix F that the coefficient of capital ' stock, Bg, comes closest to the unbiased value of unity for T/E.

Next comes B4 and B5 where the estimated values are generally over­ estimated. The desired capital stock coefficients are the most variable, having T/E ratios ranging from a low of zero to a high of

30.08. All the ratios, T/E, are of the correct (positive) sign.

We next test two null hypotheses concerning the ratios T/E for the OLS and D-TSLS estimators: (a) There is no significant difference in the T/E values over the six industries between the OLS and D-TSLS estiamtors; (b) there is no significant difference in the

T/E values over the six industries between coefficients. These hypotheses were tested for the neoclassical I and II theories.

To test the above hypotheses we performed a two-way analysis of variance with replication. The replications were the six industry tria ls . Our results are presented in Table 21. We conclude that there is no significant difference in aggregation bias between the estiamtors OLS and D-TSLS. This is true for both the neoclassical I and I I theories. We reject the null hypothesis of no significant difference in T/E between coefficients in the neoclassical I theory and accept i t for the neoclassical I I theory. In no case is the interaction term significant. Table 21

Analysis of variance of T/E by estimator (OLS and D-TSLS) and coefficient

Degrees Sum of of Mean F Source of variation squares freedom square ratio

Neoclassical I

Between OLS and D-TSLS .016502 1 .016502 .0 2 Between coefficients 9.806523 3 3.268841 4.62 Interaction .807858 3 .269286 .38 . Subtotal 10.614381 7 1.516340 2.14 Residual 28.297719 40 .707443 F qc(l,40)=4.08 Total 38.912100 47 f!95(3,40)=2.84

Neoclassical I I

Between OLS and D-TSLS .177634 1 .177634 . 0 2 Between coefficients 38.133184 3 12.711061 1.51 Interaction 3.595382 3 1.198461 .14 Subtotal 41.906200 7 5.986600 .71 Residual 336.214567 40 8.405364 F gc(l,40)=4.08 Total F*95(3,40)=2.84

t o •v! 138 i

2. Bias in the lag structure and average lag

A. Introduction.

The purpose of this section is to determine the effect of aggregation on the estimated lag structure and average lag. Towards this end we will follow an outline: Empirical measures of the constrained coefficients for the twenty-seven firms and six industries are discussed in section B; next, in section C,we discuss empirical measures of the components of the estimated aggregate constrained coefficients i .e . , AB, SB, SE, C and NC. The true and estimated lag structures and average lags are discussed in section D.

B. The constrained coefficients.

We have shown, (equation 1.46.), that an infinite distributed lag model, Yj. = W(L)Xt , may be deduced from a rational lag model

= A(L)Xt/B(L) where W(L) = wo + wiL + W2 L^ + ...» A(L) = aQ + a-jL +

. . . + amLm, B(L) = 1 + bjL + . . . + bnLn, L is the lag operator.

It was also shown, in section I . 10., that if the sequence of coefficients {wsJwas constrained to correspond to the probability distribution of a non-negative, integer-valued, random variable we would have the general Pascal distributed lag function. The resulting coefficients for the twenty-seven firms and six industry groups are presented in Appendix G.for the neoclassical I and II theories.

C. Empirical measures of the aggregate constrained coefficients and their components.

From the theory of aggregation bias developed in Chapter I and 139

summarized as a set of formulas in Table 1 we obtain measures of the

aggregation bias components. The components of the estimated coeff­

icients (E) are: A true value (T); an aggregation bias (AB); a spec­

ificatio n bias (SB); an implied sampling error; bias attributable to

corresponding (C) and non-corresponding (NC) micro-parameters. These

values are presented in Appendix H. Bq is the constant term. B j, B2 »

and B3 are the coefficients of changes in desired capital stock

lagged zero, one, and two periods respectively. B4 and B5 are the

coefficients of net investment lagged one and two periods respectively.

Bg is the capital stock coefficient.

One can at best gain only an impressionistic feeling for the

data in Appendix H. I t w ill be our purpose to analyze the data and

its effects on the lag structure and average lag in subsequent sub­

sections; however, a few points can, I think, be made at present:

(a) The estimated coefficient of capital stock, Bg, appears to be

" fa irly close" to the true value. Also, in every case, the signs of

both T and E are correct; (b) the T and E coefficients of lagged net

investment, B4 and B5 , are of the same sign; however, they are not as close as that of Bg, appearing to have an average error of 50%. It will be shown in section D the importance of this error on the lag structure and average lag; (c) the E coefficients of the desired capital stock, B-j, Bg, and Bg are, as for the unconstrained coeffic­ ients, poor estimates of the T values, frequently being in error by

1 0 0 % or more. As a measure of the overall bias between our true and e s ti- 140 mated coefficients we compute the ratio T/E (Appendix I ) . This ratio has the value of unity i f E is unbiased. I f T/E is less (more) than unity then the estimated coefficient is upward (downward) biased.

The T/E ratio for the capital stock coefficient, Bg, comes closest to the unbiased value of unity; nevertheless, the average of the absolute values of the relative error, |(E -T )/E |, for a ll six industries is 15.5% and 19.5% for the neoclassical I and I I models respectively. The average |(E-T)/E |for B4 and B5 is 41.5% and 54.7% respectively for neoclassical I and 42.5% and 50.2% respectively for neoclassical I I . The desired capital stock coefficients,B] and Bgphad an average |(E-T)/E| of 53.7% and 1.5% respectively for neoclassical I and 44.6% and 543.5% respectively for neoclassical II. | (E-T)/E| was not computed for Bg because all the table values were zero. The constant term, Bq, had an average |(E-T)/E| of 174.2% and 216.5% for neoclassical I and I I respectively.

The bias components relative to E . . . AB/E, SB/E, SE/E . . . are also presented in Appendix I. Of interest is whether one of these components contributes more to the overall bias as measured by T/E.

To determine i f this is the case we fir s t establish the null hypoth­ esis; There is no significant difference between AB/E, SB/E, and

SE/E. We also establish the hypothesis; There is no significant difference in T/E between coefficients.

The technique of analysis of variance was used to test the above two hypotheses. A two-way analyis of variance in which each industry served as a replication was compute*!. Table 22.records the Table 22

Analysis of variance of T/E by bias component and constrained coefficient.

Degrees Sum of of Mean F Source of variation squares freedom square ratio

Neoclassical I

Between components 11.98 2 5.99 5.81 Between coefficients 1.26 4 .32 .31 Interaction 15.50 8 1.94 1 . 8 8 Subtotal 28.74 14 2.05 1.99 Residual 77.25 75 1.03 F' ,(2,75) = 3.15 Total 105.99 89 F-||(4,7S) = 2.53 P"(8,75) = 2.10

Neoclassical I I

Between components 7.42 2 3.71 .40 Between coefficients 68.84 4 17.21 1.84 Interaction 47.94 8 5.99 .64 Subtotal 124.20 14 8.87 .95 Residual 702.34 75 9.36 F gc(2,75) = 3.15 Total 826.54 89 F-qc(4.75) = 2.53 F$(8,75) =■ 2.10 142 results. It w ill be seen in that table that we accept the null hyp­ othesis of no significant difference between aggregation bias for both theories. We reject the null hypothesis of no significant difference in T/E between coefficients for the neoclassical I theory and accept it for the neoclassical I I theory.

D. Effect of the bias on the lag structure and average lag.

The true and estimated lag structures were computed using the recursive equations 1.81. and are recorded in Table 23. In that table,

{w> represents the sequence of estimated aggregate probabilities using the constrained coefficient estimates of Appendix G. {w} is the sequ­ ence of true probabilities computed from the true coefficients in

Appendix H. The column labelled "Bias" is the estimated probability less the true probability. E is the expected value of the respective distribution and was computed via formula 1.91. STD. DEV. is the standard deviation of the respective distribution computed by equation

1.94. REM. is the sum of a ll remaining probabilities from the sixth year on.

Two points may be made concerning the twelve cases ( 6 indus­ tries fo r each of two theories) in Table 23: (a) The estimated react­ ion speed of investment to changes in desired capital stock is larger than the true reaction speed. That is , the sum of the probabilities over, fo r example, 0 , 1 , and 2 years lag is estimated to be larger than the true value; (b) the estimated average lag is larger than the true average lag. This la tte r point is a result of bias in the est­ imated coefficients B4 and 8 5 . Recall that in section V.l.G. we 143 Table 23

Estimated and true lag structures and th eir bias by industry group. Neoclassical I .

Lag______{w> Tw> Bias Lag ~ {w> Bias

Autos Oil

0 .40 .41 -. 0 1 , 0 .32 .46 -.14 1 .29 .44 -.15 1 .28 .27 .0 1 2 .16 .13 .03 2 .18 . 2 0 - . 0 2 3 .08 . 0 2 .06 3 .1 0 .06 .04 4 .04 .0 0 .04 4 .06 .0 1 .05 5 . 0 2 . 0 0 .0 2 5 .03 . 0 0 .03 REM. .0 1 . 0 0 REM. .03 . 0 0 E 1.15 .74 E 1.52 .87 STD. DEV. 1.35 .70 STD. DEV. 1.63 .94

Chemicals Rubber

0 .14 .27 -.13 0 .48 .42 .06 1 .45 .47 - . 0 2 1 .29 .40 - . 1 1 2 .25 .17 .08 2 .13 .16 -.03 3 .1 1 .06 .05 3 .05 .0 1 .04 4 .04 . 0 2 .0 2 4 .0 2 . 0 0 . 0 2 5 .0 1 .0 1 . 0 0 5 .01 . 0 0 .0 1 REM. . 0 0 . 0 0 REM. . 0 2 .0 1 E 1.55 1 .1 1 E . 8 8 .75 STD. DEV. 1.16 1 . 0 0 STD. DEV. 1.12 .73

Electrical Steel

0 .09 . 1 2 -.03 0 .32 .41 -.09 1 . 1 2 . 2 0 -.08 1 .34 .36 - . 0 2 2 . .13 .19 -.06 2 .19 .15 .04 3 .1 2 .15 -.03 3 .09 .05 .04 4 .1 1 .1 1 .0 0 4 .04 . 0 2 . 0 2 5 .09 .08 .01 5 .0 1 . 0 0 .0 1 REM. .34 .15 REM. .0 1 .0 1 E 4.78 2.99 E 1.26 .91 STD. DEV. 4.01 2.54- STD. DEV. 1.28 1 .0 1 144

Table 23 (Continued)

Estimated and true lag structure and their bias by industry group.Neoclassical I I ,

Lag TO TO Bias Lag TO TO Bias

Autos Oil

0 . 0 0 .35 -.35 0 . 0 0 .82 -.82 1 .37 .49 - . 1 2 1 . 0 0 .23 -.23 2 .23 .15 .08 2 .13 - . 0 2 .15 3 .15 .0 2 .13 3 .17 -.03 .2 0 4 .09 . 0 0 .09 4 .16 - . 0 1 .17 5 .06 . 0 0 .06 5 .14 . 0 0 .14 REM. .1 0 . 0 0 REM. .40 E 2 . 6 8 .83 E 5.46 .09 STD. DEV. 2.12 .71 STD. DEV. 3.07 .39

Chemicals Rubber

0 .19 .51 -.32 0 .43 .29 .14 1 .39 .33 .06 1 .30 .34 -.04 2 .23 .14 .09 2 .15 .29 -.14 3 .1 1 .03 .08 3 .07 .07 . 0 0 4 .05 . 0 0 .05 4 .03 .0 1 .0 2 5 .0 2 . 0 0 .0 2 5 .0 1 . 0 0 .01 REM. .01 . 0 0 REM. .01 . 0 0 E 1.56 .67 E 1.05 1.14 STD.DEV. 1.30 .77 STD. DEV. 1.26 .93 Electrical Steel

0 .11 .13 - . 0 2 0 .48 .44 .04 1 .15 .23 -.08 1 .29 .39 - . 1 0 2 .15 . 2 0 -.05 2 .14 .13 .0 1 3 .13 .15 - . 0 2 3 .06 .03 .03 4 .11 .1 1 . 0 0 4 .0 2 .01 .01 5 .09 .07 . 0 2 5 .0 1 . 0 0 .01 REM. .26 .1 1 REM. . 0 0 . 0 0 E 4.01 2.69 E . 8 8 .77 STD.DEV. 3.47 2.27 STD. DEV. 1.13 .84 145 showed that the estimated B4 and Bg were upward biased. This fact combined with the expected value equation (1.91.) implies an upward bias in the estimated average lag.

An error of 50% in an estimated coefficient is of greater concern when these coefficients are B 4 and Bg. Small changes in B4 and B5 imply quite large changes in the lag structure and average lag.

To demonstrate this point we compute a 95% confidence ellipse for B 4 and Bg centered around th eir estimated values.2** These ellipses, for the neoclassical I theory, are shown in Figure 1. Within each ellipse are twelve points representing different values of 4 B and Bg. All of these points are consistent with the general Pascal lag function.

Point #2 is the estimated set ofB 4 and Bg. Point #1 is the true values of B4 and Bg. Points #3-12 are various other values ofB 4 and

Bg.

For each of the twelve points in each ellipse we computed two measures: (a) The average lag of the implied general Pascal d is trib ­ ution; (b) the proportion of total investment completed after two years. These two measures fo r the various values of B4 and Bg are presented in Table 24.

The range of values of B4 and B5 within a 95% confidence ellipse is quite large and in every case encompasses the true values.

Two aspects of the average lags are worth mentioning: (a) There is a wide range of average lag values consistent with a 95% confidence

25Arthur S. Goldberger. Econometric Theory (New York: Wiley, 1964), 178. Figure 1

95% confidence ellipse from constrained estimates of and Bj.. Neoclassical I

AUTOS CHEMICALS

a ' 1.0 - LO

t-t-l-Srl'.l-.l.-t Figure 1 (Continued)

95% confidence ellipse from constrained estimates of and B 5 . .Neoclassical I

Electrical OIL

. J.5

u ~ u -tf-y -f-.7 tt-zs - 3 -1

1 Figure 1 (Continued)

95 % confidence ellipse from constrained estimates of B 4 and Bg. Neoclassical I.

RUBBER STEEL

i- r 1.1L #.# u\ 10

1 - s - 1 -3 -1 - . /

00 Table 24 Average lags and proportion of investment completed after two years for various values of and B5 consistent with a general Pascal lag.

Data Autos Chemicals Electrical Oil _Rubber X prop. X prop. X prop. Set B4 B5 X prop. X prop.

1 True .74 .986 1 .1 1 .915 2.99 .516 .87 .934 .75 .991

2 Estimated 1.15 .858 1.55 .831 4.78 .340 1.52 .783 . 8 8 .913

3 0 0 0 1 . 0 0 0 .73 1 . 0 0 0 0 1 . 0 0 0 0 1.000 0 1.000

4 . 8 -.16 1.33 .821 2.06 .695 1.33 .821 1.33 .821 1.33 .821

5 1.4 -.49 4.66 .348 5.39 .252 4.66 .348 4.66 .348 4.66 .348

6 1.4 -.41 5.72 .040 5.78 .028 5.72 .040 5.72 .040 5.72 .040

7 .9 0 7.47 .271 8.08 .2 1 2 7.47 .270 7.47 .271 7.47 .271

8 0 .9 8.73 .190 8 . 8 6 .124 8.73 .190 8.73 .190 8.73 .190

9 0 .5 2 . 0 0 .750 2.73 .567 2 . 0 0 .750 2 . 0 0 .750 2 . 0 0 .750

10 .4 .4 5.63 .392 6.31 .311 5.63 .392 5.63 .392 5.63 .392

11 . 8 0 3.97 .488 4.69 .394 3.97 .488 3.97 .488 3.97 .488

12 1 . 0 - . 1 7.03 .290 7.67 .224 7.03 .290 7.03 .290 7.03 .290 150

ellipse for B4 and B5. These values range from a low of .00 to 9,16

years; (b) the average lags between industries but for the same values

of B4 and Bg are very similar. This is a result of the denominator

coefficients, B(L), in equation I I 1 . 4 . dominating W(L) even though

the numerator coefficients, A(L), are different for each industry.

The results of the proportion of investment completed after

two years have the same characteristics as the average lags, i .e . ,

they show a wide range of values within industries depending on the

parameter set B4 and 85, but show more homogeneity between firms for

the same parameter set B4 and B5 . 2 6

26While the sensitivity of the lag structure and average lag to changes in B4 and B5 is bad enough, this problem is compounded by a second problem. The probability of obtaining a general Pascal lag function, GPLF, from the unconstrained coefficients of the rational lag function, RLF, is very low to begin with. This author has com­ puted posterior probabilities of obtaining a GPLF, P(GPLF), from the unconstrained coefficient estimates of the RLF for some firms in our sample and found the following:

Corporation P(GPLF) Corporation P(GPLF)

Armco Steel . 0 0 0 Inland Steel .084 Bethlehem Steel .013 Republic Steel . 0 0 0 Chrysler .227 Shell .075 Dow Chemical .0 0 1 Standard Oil (N.J.) . 0 0 0 Eastman Kodak . 0 0 0 Union Carbide .107 General Electric .377 Uni royal . 0 0 0 Goodrich .025 U. S. Steel .119 Goodyear .043 Westinghouse .300 Gulf .066

See Henry G. Rennie, "Bayesian Probability and the General Pascal Distribution," American Statistical Association, Proceedings of the Business and Economic Statistics Section, 1972. For an analysis of the various reasons for not satisfying the necessary constraints for a GPLF see Henry G. Rennie, "Bayesian Infer­ ence in the Rational Lag Function with Applications to a Neoclassical Investment Theory," invited paper presented to the European Meetings of the Econometric Society, Budapest, Hungary (September, 1972). 151

3. Bias in the aggregate residual variance

A. Introduction.

The final effect of aggregation bias in which we are inter­ ested is that on the estimated aggregate residual variance. We f ir s t present empirical measures of the estimated aggregate residual varia­ nce arid its components in section B. Next we inspect the error between the true and estimated residual variance by sign and size in section C.

The components of the bias are discussed in section D. Finally, the effect of the bias on theory selection is discussed in section E.

B. Empirical measures of the estimated aggregate residual variance and its components.

We have shown in 1.14. that the estimated aggregate residual variance, T*2, may be written as the sum of various variances and covar­ iances. Specifically, in equation I.118L, we have shown that S 2 equals the sum of the true residual variance, U'U/T, an aggregation bias component, V'V/T, a specification bias component, Z’ Z/T, an implied sampling error component, E'E/T, and covariance components between each of these biases ... 2U'V/T, 2U'Z/T, 2U'E/T, 2V'Z/T, 2V'E/T,

2Z'E/T.

We have computed measures of '§2 and each of these components by theory and industry. The results are recorded in Appendix J. This data forms the raw material for the analysis in this section.

C. Size and sign of the aggregation bias.

A quick inspection of Appendix J reveals that in every case the true and the estimated residual variances are unequal. Thus, bias 152

ts present. I t is meaningful to inquire i f this bias varies by theory

and/or industry. Therefor, we compute the sign of the difference ,

3J2 - U'U/T. These signs are recorded in Table 25. In that table, for example, the sign of (-) for Accel erator-Autos refers to the fact that

S2 - U'U/T = (30623.55 - 35955.02) <0. One needn't use sophisticated

statistical tools to see that there is no dominant effect of either

theory or industry on sign. One industry, Chemicals, has all negative signs for each theory, and one industry has all positive signs. The accelerator theory has negative signs for five out of the six industr­ ies. On the other hand, two theories, expected profits and neoclass­ ical I I , have positive signs for four out of the six theories. Of the thirty cases ( 5 theories and 6 industries), "5^ - U'U/T had 14 posit­

ive signs and 16 negative signs.

The ratio of the estimated to the true aggregate residual variance, J2/(U'U/T), is presented in Table 26 by theory and industry.

Except for two unusually high ratios (Autos-neoclassical I and II) we find the ratios ranging from a low of .61 to a high of 1.27 (excluding the above two large values). The average 2 S/(U'U/T) (excluding the above two cases) for the 28 cases is .967.

We inquire whether these ratios vary significantly by theory and industry by setting up a two-way analysis of variance and testing the two hypotheses: (a) There is no significant difference in

Sign of the differences between the estimated and true aggregate residual variances by theory and industry.

Totals Autos Chemicals Electrical Oil Rubber Steel +

Accelerator - - - -- + 1 5

Expected Profits + •» + - + + 4 2

Liquidity - - + -- + 2 4

Neoclassical 1 + - - + - + 3 3

Neoclassical 11 + - + + - + 4 2

Totals + 3 0 3 2 1 5 14

■B 2 5 2 3 4 0 16

U1 u Table 26 —2 Ratio of the estimated to the true aggregate residual variance, S /{U'U /T), by theory and industry.

Autos Chemicals Electrical Oil Rubber Steel 0 0 LO

Accelerator .85 • ,96 .69 .74 1.39

Expected Profits 1 .0 1 .71 .99 .96 1.14 1.07

Liquidity .99 .61 1.03 .6 8 .80 1.48

Neoclassical I 3.45 .77 .97 1.16 .82 1.24

Neoclassical I I 3.33 .82 1.06 1 . 0 2 .99 1.27 155

Table 27

Analysis of variance of S^/CU'U/T) by theory and industry

Degrees Source of variation Sum of squares of Mean F ratio freedom square

Between theories 1.58 4 .395 1.31 Between industries 4.62 5 .924 3.07 Residual 6 .0 1 20 .301 F 0 5 (4 *2 0 )= 2.87 Total 1 2 .2 1 29 F 9 5 ( 5 , 2 0 )= 2.71

between theories. We reject the null hypothesis that there is no significant difference in ^/(U'U/T) between industries. This latter result appears to have been caused by the two unusually large values of S^/(U'U/T) fo r the neoclassical I and I I theories in the auto ind­ ustry.

D. Components of the bias.

If we divide each of the variance and co-variance components of the estimated residual variance in equation 1.118. by !T2, the values sum to unity. This implies that the difference between U'U/T/S^ and unity is the sum of the ratios of the bias components to this makes visual comparison of our bias components easier than looking at the raw data in Appendix J.

We present each of the bias components as a proportion of S2 in Table 28. Three observations may be made concerning these values;

(a) The majority of the estimated aggregate residual variance is composed of the true value, U'U/T. Indeed, the average of the Table 28

Ratio of the bias components to the estimated aggregate residual variance.

Autos Chemicals Electrical Neo. Neo. Neo. Neo. Neo. Neo. Acc. E(ir) Liq. I I I Acc. E(ir) Liq. I II Acc. E(tt) Liq. I II

U'U/T 1.17 .99 1 .0 1 .29 .30 1.18 1.40 1.63 1.30 1 .2 1 1.04 1 .0 1 .97 1.03 .95

2U'V/T 0 .04 0 .16 .11 .09 .07 -.07 -.09 -.05 - . 0 2 .04 .07 - . 0 1 - . 0 1

2U'Z/T -.17 -.03 0 .09 .08 -.25 -.06 -.36 - . 1 0 - . 1 0 - . 0 1 - . 0 2 - . 0 2 -.03 -.06

2U'E/T -.07 - . 0 2 -.03 -.03 - . 0 1 -.64 - 1 . 1 0 -.95 -.55 -.44 -.04 - . 1 2 -.16 - . 2 0 -.08

V'V/T 0 .01 0 .30 .2 0 . 1 0 .07 .08 .01 .17 0 .0 1 .03 0 0

2V'Z/T 0 0 0 -.17 - . 1 0 - . 1 0 -.03 .0 1 - . 0 1 -.09 0 0 0 0 0

2V'E/T 0 0 0 . 0 2 - . 0 2 - . 1 0 -.08 -.05 - . 0 2 0 0 - . 0 1 .0 1 0 0

Z'Z/T - .03 .0 1 .0 1 .35 .42 .36 .14 . 2 2 .16 .07 .01 .03 .04 .1 1 .16

ZZ'E/T 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

E'E/T .03 .0 1 .0 1 .0 2 .01 .37 .59 .50 .28 . 2 2 . 0 2 .06 .08 .1 0 .04 on CTl Table 28 (Continued)

Oil Rubber Steel Neo. Neo. Neo. Neo. Neo. Neo. Acc. E(u) Liq. I I I Acc. E(tt) Liq. I I I Acc. E(ttJ Liq. II II [x[

U'U/T 1.46 1.04 1.48 .8 6 .99 1.35 .87 1.29 1 . 2 2 1 .0 1 .72 .93 .67 .81 .79 1 . 0 0 2

2U1V/T .1 2 - . 0 2 -.18 .01 .03 -.04 .30 - . 1 1 - . 0 2 .18 . 1 2 .15 .25 .2 0 .2 2 .093

2U'Z/T .0 2 .1 0 -.16 .1 1 .01 -.17 - . 1 0 . 1 2 - . 1 1 - . 1 1 .03 -.03 .0 1 0 -.06 .084

2U'E/T -1.52 -.37 -.79 -.26 -.43 - . 6 6 -.38 -.63 -.40 -.34 -.04 -.30 ■ .05 - . 1 2 - . 0 2 .357

V'V/T .05 .0 2 .04 .09 .08 .09 .15 .1 1 . 0 2 .07 .03 . 0 2 .13 .03 .05 .065

2V'Z/T -.09 .0 1 .0 2 - . 0 2 - . 0 1 - . 0 2 -.04 -.13 0 -.03 - . 0 2 0 .0 1 - . 0 2 -.03 .032

2V'E/T -.06 0 .01 - . 0 1 0 -.04 -.14 -.08 0 - . 0 1 0 - . 0 1 .0 1 -.04 - . 0 1 .024

Z'Z/T .24 .04 .17 .08 . 1 2 .14 .06 .13 .09 .06 .15 .09 .0 1 .07 .05 .1 2 1

2Z'E/T 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0

E'E/T .79 .18 .36 .13 . 2 2 .35 .26 .36 .2 0 .17 . 0 2 .16 .03 .08 . 0 2 .189 Cfl 'vl 158 absolute values of U'U/S over a ll 30 cases is 1.002; (b) the second largest component of S2 is 2U'E/T, the covariance between the true residuals and the implied sampling errors; however, this is not a separate piece of information since the variance of the implied samp­ ling errors is always twice the covariance of E and U and of opposite sign; (c) the third largest component of S 2 is the specification bias,

Z'Z/T. The average of the absolute values of 2Z'Z/T/?^ is .121.

The average of the absolute values of the ratios of the o n components of Sc to S are presented in Table 29.

Table 29

Average of the absolute values of the ratios of the components to “S’2

U'U/T 2U'V/T 2U'Z/T 2U’E/T V'V/T 2V'Z/T 2V'E/T Z'Z/T 2Z'E/T E'E/T

1.002 .093 .084 .357 .065 .032 .024 .121 .000 .189

4. Bias and theory selection

One empirical criterion for selecting one theory over competing theories is by comparing residual variances (or R2 ,s). The theory with the smallest residual variance is, ceteris paribus, selected as "best.”

An explanation of why this selection procedure works was given in section 1.14.

We have compared our five competing investment theories with one another on the basis of aggregate residual variance in Table 9; however, this comparison was based on the estimated aggregate residual 159

variance. I f the comparison is made on the basis of the true residual

variance w ill i t change our results? We make a comparison between the

true and estimated residual variance in Table 30. This table is read

Table 30

Number of times, out of six, a theory (row) had a lower standard error than competing theories (column) by true and estimated residual variance

+J K-« ♦(—' M M 4 - f~ i-O i— rtf >— rU O Q _ U O +J >j *r- -I— ra - a +-> w w t~ a> *i— in in Of -M X3 ro QJ *r- 3 i— u r— u o a . cr o o o x *r- tu a) < U — I Z "ZL

Accelerator True X 3 4 3 3 Estimated X 4 5 5 5

Expected Profits True 3 X 4 3 2 Estimated 2 X 4 3 4

Liquidity True 2 2 X 1 1 Estimated 1 2 X 3 4

Neoclassical I True 3 3 5 X 3 Estimated 1 3 3 X 4

Neoclassical I I True 3 4 5 3 X Estimated 1 2 2 2 X

the same as Table 9. That is , read row-wise we learn that, for example, the expected profits theory (row) had a lower standard error

for four of the six industries than the neoclassical II theory 160

(column) on the basis of estimated residual variance; however, on the

basis of the true residual variance the expected profits theory (row)

had a lower residual variance for only two of the six industries than

the neoclassical theory. Generally the true and estimated values 1n

this table are different, indicating different performance ratings

depending on whether the true or estimated residual variances are

used. To be more specific, we ranked each theory within a specific

industry on the basis of residual variance. This was done on the

basis of both the true and estimated residual variances. The results

are presented in Table 31. A "1" in that table means that that theory

Table 31

Investment theories ranked by true (T) and estimated (E) residual variance

«/> (T3 •i—U o S- 01 4-> XI OJ -M 3 j= r—■ •r— 3 4-J ■s: <_j> UlI O cc 00 T E T E T E T E T E T E

Accelerator 4 2 1 1 1 1 2 1 3 2 5 4

Expected Profits 3 1 3 2 4 4 5 5 2 4 1 1

Liquidity 5 4 5 3 5 5 4 2 1 1 4 5

Neoclassical I 1 3 4 5 3 2 1 4 5 3 2 2

Neoclassical II 2 5 2 4 2 3 3 3 4 5 3 3 161 had the lowest residual variance of the competing theories in that specific industry. A (5) means that that theory had the highest residual variance. We see that the rankings do not coincide for the true and estimated values within an industry.

To compare how closely the T and E rankings were correlated we computed the Kendall t coefficient2^ for each industry. These results are presented in Table 32. The r coefficients ranged from

Table 32

Kendall's t correlation of the true and estimated rankings of five investment theories

Autos Chemicals Electrical Oil Rubber Steel

.1 .4 . 8 . 2 .4 . 8

a low of .1 indicating a very poor correlation between the true and estimated rankings based on the true residual variance and the estimated residual variance to a high of .8 .

5. The empirical literature: A comparison of findings

A. Introduction.

The bulk of the aggregation lite ra tu re is theoretical?® This

2 7 J. U. Yule and M. G. Kendall, An Introduction to Statistics (London: Hafner, 1961), 262-264.

2®See the Bibliography for a good sampling of the aggregation literature. 162

literature is primarily concerned with the conditions under which 29 aggregation is consistent. When we enquire as to what happens when

the (severe) consistency conditions are not satisfied in practice we

are essentially asking an empirical question as to the size of the

aggregation bias. Of the relatively few empirical studies of aggre­

gation bias, most have been concerned with coefficient bias in an

input-output framework.

Our purpose in this section is to compare our findings with

"comparable" empirical studies. By "comparable" we shall mean those

studies using single equations (rather than simultaneous equations),

aggregation over individuals, and estimation by ordinary least-squares.

There are no such empirical studies specifically concerned with aggre­

gation bias within a distributed lag framework.

We organize this comparison around individual studies. Some

of these studies are primarily concerned with aggregation bias in

coefficient estimates; others are concerned witha ggregation bias in

the estimated residual variance (or R2 's); some of the studies are

concerned with both the coefficient bias and residual variance bias.

B. Boot-DeWit.

Boot-DeWit^ere primarily concerned with providing an empir-

29See the discussion in Chapter I.

3 0 J. C. G. Boot and G. M. DeWit, "Investment Demand; An Empir­ ical Contribution to the Aggregation Problem," IhtefnatiPhal Economic Review, I Glanuary, 1960), 3-30. 163

ical example of Theil's analogy approach to aggregation.3^ As a

vehicle for using Theil‘ s approach they chose Grunfeld's expected

profits investment theory . 3 2 They applied annual data for the period

1935-1954 to ten large U. S. corporations. This data was aggregated

into one large "industry."

The (non-distributed lag) equation which they used was:

(V.7.) It ■ B„ + + B2 Ct

where: 1 ^ - gross investment in period t.

Ft- 1 = the market value of the firm in period t -1 .

Ct = the existing stock of capital in period t.

For comparison purposes, their findings can be summarized 1n

Table 33.

Table 33

Ratio of the components of the aggregate estimated coefficients (E) to E. Boot-DeWit. CO m B-j/E b2/ e o

T .643 .919 .789 AB .380 .148 . 1 2 0 SE -.023 -.069 .092

Source: Adapted from Boot-^DeWit, International Economic Review, I (January, 1960), Table 3.

31 See the discussion of T heir s analogy approach in 1.4.-5. 32 ^ See the discussion of the expected profits theory in I I . 3 .c. 164

Their finding of a 36% bias for the constant term, while large,

is smaller than our finding of 176% (Table 12). Boot-DeWit found that

the true slope parameters, Bj and B2 , were 92% and 79% respectively

of the estimated values. That is, an error |(E-T)/E| of 8 % and 21%

for Bi and B2 respectively. Since capital stock was the only variable

always included in our regressions then Bg could be compared with

th eir B] and B2 . Recall (Table 12) that Bg had an average error of

12%.

Boot-DeWit1s aggregation bias (AB) relative to their estimated

coefficients was 15% and 12% for B-j and B2 respectively. While they

did not perform any tests for significant differences in their bias

components, i t appears that the sampling error (SE) is not too d iffe r­

ent (ignoring sign) from AB. SE was -7% and 9% of E for B-j ,and B2

respectively.

Boot-DeWit's ratio of the true to the estimated aggregate

residual variance was 1.02. Our value was 1.00 (Table 29); however,

i t must be remembered that we had 30 case studies to their one and

that our values ranged from .30 to 1.63. We both found that the

largest component of the estimated aggregate residual variance was the

true residual variance.

The largest source of bias between the estimated and true

residual variance was found, by Boot-DeWit, to be the covariance

between the true residual and the implied sampling error. This value was 11.5% of E. We also found this covariance to be the largest component of the bias in E but averaging 35.7% of E. Our values 165

ranged from a low of -1 1 0 % to a high of - 1 %.

The aggregation bias was found to be 8 . 6 % of E by Boot-DeWit.

Our study found the aggregation bias to be 8 . 6 % of E; however, the wide range of values in Table 28 should not be ignored.

C. Gupta.

Gupta3 3 applied Theil's analogy approach to five industries of the non-manufacturing sector of the British economy . . . mining and quarrying (excluding coal); building and construction; gas, electric­ ity and water; transport and communication (excluding railways,

London transport and the British Road Services).

The model used by Gupta was:

(V.8 .) Yt = B0 + B-,Xlt + B2 X2t + B3 X3t + B4 X4t + Ut where: Y = the wage rate.

X] = unemployment less unfilled vacancies.

X2 = rate of change of nominal output.

X3 = lagged rate of change of prices.

X4 = proportion of the labor force unionized,

t = time. 1950-1959.

This model and its variables are in reference to each of the industries, named above. I t is also applied to the aggregate or non-manufacturing sector of the British economy in which the linear industry relations were translated into a linear non-manufacturing relation by means of

3 3 K. L. Gupta, Aggregation in Economics: A Theoretical and Empirical Study (Rotterdam: Rotterdam University press, 1969), 52. 166 linear aggregates.

We have computed the ratio of the true values (T), aggregat­

ion biases (AB), and sampling errors (SE) to the estimated aggregate parameters Bg, . . . , 8 4 of equation V.8 .:

Table 34

Ratio of the components of the aggregate estimated coefficients (E) to E. Gupta.

Bo/E B-j/E b2/ e b3/ e B4 /E

.719 4.854 .250 -.529 .071 .044 -3.869 1.146 1.089 .894 .214 .0 0 0 -.358 .455 .050

Source: Adapted from K. L. Gupta, Aggregation in Economics: A Theoretical and Empirical Study {Rotterdam: Rotterdam University Press, 1969), 52.

We see from the above table that the estimated values of the parameters bear l i t t l e resemblance to the true values. Indeed, the true and estimated values of B3 are of opposite sign and the true B-j is over 4.5 times the estimated value. The aggregation bias accounts for most of the difference between the true and estimated coefficients; however, the sampling error is s t ill substantial.

D. Kuh.

Kuh3^ studied twelve equation forms to explain the fixed

3^Edwin Kuh, Capital Stock Growth: A Micro-Econometric Approach (Amsterdam: hlopth-HollaniJ, 1963). ' 1 ^ 167 capital investment behavior of sixty firms during the period 1935 to

1955 (excluding 1942 to 1945).35 He used Theil's analogy approach to

aggregation to group these firms into one large aggregate.

The twelve equation forms used by Kuh are listed in Table 35.

Table 35

Kuh's equation forms.

I = gross investment S = sales C = capital intensity index P - gross retained profits K = capital stock t = time

+ a i C + a2AKt + a3ASt Al a 0 (3.9S) = + + Al a 0 a] C 32AKt + a3APt (3.9P) = •f + Al a 0 ajC a2 AKt + a3 ASt a4 APt (3.9SP) = + + I a 0 ajC a2Kt + a3st (3.1 OS)

I = a-|C + 2 4 a 0 a Kt + a Pt (3.10P)

I - + a-jC + a2 Kt + a3 S£ + (3.10SP) a 0 lPt + + I a 0 a^C a2 Kt + a3 st-l (3.11S) I - a-jC + (3 .IIP ) a 0 a2^t + a4pt-l

I = + a-| C + 2 3 + (3 .11SP) a 0 a Kt + a St_-| a4pt- l I = + + a 0 a-jC a2 Kt + a3f(St St- 1 >/2] (3.12S)

•f + 2 4 + I a 0 3-| C a Kt + a [(Pt pt .l)/2 3 (3.12P) = + I a 0 a-j C a2Kt + a3 t(st St«i)/2] + a4 ECPt + Pt-l)/2 ] (3.12SP)

Source: Edwin Kuh, Capital Stock Growth: A_Hicr 6 "Ecdnometric Approach (Amsterdam: North-Holi and, 1963), 71 .

35Not a ll of Kuh's 19 equation forms were used in his time- series analyis. Of those that were, 12 were used in aggregation. 168 We present, in Table 36, the ratio of the true to the estim-

Table 36

Ratio of the true to the estimated aggregate coefficients. Kuh.

Capital Stock Sales Profit Equation Slope Slope Slope

(3.9S) -2.258 .661 (3.9P) -2.909 .427 (3.9SP) -2.522 .771 .847

(3.10S) .491 1.039 (3.10P) .691 .560 (3.1QSP) 5.161 1.350 -1.558

(3.1 IS) 1.669 1.143 (3 .up) .829 m ^ p* .601 (3 .lisp) .400 3.439 .154 (3.12S) .382 .929 3.12P) — .547 (3.12SP) .933 1.463 .116

Source: Adapted from Edwin Kuh, Capital Stock Growth: A Micro Econometric Approach (Amsterdam: North-Holland, 1963), l9£.

ated aggregate coefficients. It will be seen in that table that of the twenty-seven ratios, four had the incorrect (negative) sign. The values of the ratios (excluding the negative ratios) ranged from a low of .116 to a high of 5.161. The average of the ratios (excluding the negative ratios) was; .120 for the capital stock slope; 1.207 for the sales slope; ,465 for the p ro fit slope. CHAPTER VI

SUMMARY AND CONCLUSIONS

1. Summary

The primary-objective of this study was measurement of errors caused by aggregation. Three errors of aggregation were of partic­ ular concern; (a) Bias in aggregate coefficient estimates; (b) bias in aggregate lag structures and average lags; (.c) bias in estimates of the residual variance in aggregate equations.

The process of aggregation, i .e . , reducing a set of quantities and/or relations to a smaller set of quantities and/or relations, generally results in the aggregation problem of inconsistent results.

A discussion of the concept of consistency ( I . 1 .-2 .) and of the necessary and su fficient conditions for attaining consistency (1 .3 .) resulted in a dismal conclusion . . . consistency requires identical slope parameters when translating linear micro relations into a linear macro relation by means of linear aggregates.

Since consistent aggregation is unlikely to be attained in practice, the relevant question becomes "what is the size of the bias created by inconsistent aggregation?" We proceeded to answer this question by adopting Theil's analogy approach to aggregation ( I . 4 .-5 .); however, his theory ( 1 . 6 .) applies to non-distributed lag equations

169 170

and this study used distributed lags.

The concept of a distributed lag was introduced in 1.8. The

rational distributed lag function used in this study was introduced

in 1.9. This function has the desirable property of subsuming the

commonly used fin ite , arithmetic, geometric, and Pascal lag functions

as special cases. A claim that i t can approximate an arbitrary lag

function to any desired degree of accuracy has recently been disputed.

The relation between the estimated and the implied macro­

parameters in a rational lag function was developed in 1.12. I t was

shown that the estimated aggregate parameters of the rational lag

function are composed of a true value (T), an aggregation bias (AB),

a specification bias (SB), and an implied sampling error (SE). AB and

SB are each composed of a bias due to corresponding micro-parameters

(C) and non-corresponding micro-parameters (NC). The formulas for the

estimated aggregate parameters and their components are summarized

in Table 1.

The theory of aggregation bias in rational lag functions was

extended to include the effect of aggregation on the lag structure of

an economic reaction, the average lag, and the variance of the lag

(1.1 3.).

The relationship between the estimated aggregate residual

variance of an equation and the true residual variance was developed

in 1.14. The aggregate estimated residual variance, S2, was shown to be the sum of the true residual variance, U'U/T, an aggregation bias component, Z'Z/T, a specification bias component, V'V/T, an implied 171

sampling error component, E'E/T, and a ll the covariance components . . .

2U'V/T, 2U'Z/T, 2U'E/T, 2V'Z/T, 2V'E/T, and 2Z'E/T.

Five theories of investment behavior were chosen as a vehicle

for applying our aggregation theory. These investment theories,

develuped in Chapter I I and summarized in I I . 5. are called accelerator,

expected profits, liquidity, neoclassical I, and neoclassical II.

Parameter estimation of the rational lag function was dis­

cussed in Chapter III. Two alternative specifications of the

rational lag function were shown to be possible: (a) If we start

with the rational lag model Yt = [A(L)/B(L)]Xt + where

Ut » [I/B (L )]e t and ej. is NIDCO.o2), then the final form of the model

has a disturbance term, ej.» with the properties E(et ) * 0 and

E(et .es) = crst 2 where (s ,t « 1 , ... ,n). Ordinary least-squares

estimation of the parameters is best, asymptotically normal under

this formulation; (b) If, instead, the rational lag function has

Ut = NID(0,c£) then the final form has an nth order autoregressive

residual. Durbin's two-stage least-squares procedure applied to

this formulation yields paramater estimates which, asymptotically,

have the same mean vector and variance matrix as the ordinary least-

squares estimator.

The sample used in this study consisted of twenty-seven large

U.S. manufacturung corporations (.Table 2). These corporations were aggregated into six industry groups ... autos, chemicals, electrical, o il, rubber, steel. Annual data for the period 1949-1968 was applied to each of the five investment theories. The numerator and denominator polynomials of the rational lag function were limited to a maximum of second degree. The specific structure was chosen for each firm , industry, and theory by minimizing the residual variance of the regression equation.

The five theories were all superior to a purely autoregressive model in which lagged values, up to a maximum of three, were allowed to enter the equation i f they reduced the residual variance (Tables

17 and 18). The five theories also performed well in terms of R2

(Tables 17 and 18). The null hypothesis that the vector of slope parameters equalled zero was rejected, at the 5$ significance level, for all theories and firms but one.

A ranking of the models on the basis of the number of desired capital stock coefficients at least twice their standard errors

(Table 7) yielded: Accelerator (best); neoclassical II; neoclassical I liquidity; expected profits. None of the theories was clearly superior to the others. The ranking of the aggregate equations on the same basis (Table 10) was: Accelerator (best); liquidity; expected profits; neoclassical I and II (tied).

2. Conclusions

Empirical measures of our aggregation bias components are discussed in Chapter V. Since these results and their statistical analysis are so closely interrelated we shall integrate the summary of these sections with our conclusions.

The effect of aggregation on the unconstrained coefficient estimates of the rational lag function was the subject of section V .l. 173

The estimated aggregate coefficients (E) and their components, T, AB,

SB, SE, C, and NC, were presented by theory and industry in Appendix B.

We tested for the presence of aggregation bias in V.l.C . and concluded that it was present in twenty-four of our thirty cases (5 theories with 6 industries).

The size of the Mas as measured by | (E-T)/E| averaged over a ll theories and industries was found to be 1.76 for the constant term,

Bq; .33, 1.08, and .40 for the coefficients of the change in desired capital stock, B-j, Bg* and B3, lagged zero, one, and two periods respectively; .44 and .38 for the coefficients of net investment, B 4 and B5 , lagged one and two years respectively; .12 for the coefficient of capital stock, Bg. We concluded from a two-way analysis of variance that the ratio , |(E-T)/E| , averaged over six industries does vary significantly between coefficients but not between theories.

We also performed a two-way analysis of variance for the ratio

T/E for each industry group. We found that there was no significant difference in T/E between theories for five of the six industries. We also found that there was no significant difference in T/E between coefficients for four of the six industries.

The three bias components, AB, SB, and SE (Appendix D) were found not to be significantly different for five of the six industries.

Also, the hypothesis of no significant difference between the C and NC bias components was accepted for five of the six industries.

We applied Durbin's h statistic to test for the presence of autocorrelation. Of the 76 applications of this test, 57 cases tested 174 non-autocorrelated. Of the twenty applications of the h statistic to the neoclassical t and II models, fourteen tested non-autocorrelated.

Four a priori reasons were given for expecting the absence of auto­ correlation; however, i t was decided to apply Durbin's two-stage least-squares (D-TSLS) estimator to the neoclassical I and II models and test for difference in aggregation bias between estimators.

We tested for differences in T/E between the OLS and D-TSLS estimators. Using a two-way analysis of variance with six (industry) replications we concluded (Table 21) that there is no significant difference in T/E between estimators.

The effect of aggregation on the lag structure and average lag was studied in section V.2. We found (Table 23): (a) The estimated reaction speed of investment to changes in desired capital stock is larger than the true reaction speed; (b) the estimated average lag is larger than the true average lag.

It was found that the coefficients of lagged net investment,

B4 and B5 , had an average error, j (E-T)/E| , for all six industries of

41.5% and 54.7% respectively for neoclassical I and 42.5% and 50.2% respectively for neoclassical II (Appendix I). It was then demon­ strated that small changes in B4 and B5 imply large changes in the reaction speed of investment to changes in desired capital stock and in the average lag. We conclude from this that estimates of the lag structure and average lag of a rational lag function at the industry level of aggregation are very imprecise. 175 Our final enquiry concerned the estimated aggregate residual

variance, S2. We found that this was just as likely to be over

estimated as under estimated (Table 29).

The average ratio of the estimated to the true aggregate

residual variance, "^/(U'U/T), was found to be .967 for all theories

and industries. The null hypothesis that there is no significant

difference in S^/OJ'U/T) between theories was accepted.

The largest component of the estimated aggregate residual

variance was, as may be expected, the true value, U'U/T. The ratio

of the true to the estimated residual variance for a ll theories and

industries was 1.002. The second largest component of is 2U'E/T,

the covariance between the true residuals and the implied sampling

error. The ratio (2U'E/T) / ^ 2 was .357. The third largest component

of S2 was the specification bias with an average ratio of .121.

The result of the bias between the true and estimated residual

variances has interesting implications for model selection. We counted the number of times, out of six, that a theory had a lower

standard error than competing theories. These values (Table 29} were different when using the true versus the estimated residual var­

iance. Kendall's t coefficient of rank correlation between the true and estimated rankings of the five investment theories was low

(Table 31).

We have shown that aggregation bias affects coefficient estimates, lag structure and average lag, and the residual variance and hence model selection based on minimization of the residual variance. These results can not be comforting to the macro theorist 176 who builds his model by analogy with micro theory. More particularly, one can not accept the neoclassical model as superior to competing theories on the basis of goodness of f i t tests on an aggregate struc­ ture. The results can be easily reversed i f micro data were available.

I t is fa ir to say that the debate on the determinants of fixed capital investment is far from over. Indeed, i t is just beginning.

Our study did not u tiliz e a random sample of firms and there­ for can not be generalized to all firms aggregated into industries; however, judging by the literatu re comparisons of section V.5. our results were not unexpected. Aggregation bias is important, at least as important as sampling error, and should be given as much attention as historically has been given to the sampling error. Economists can no longer be satisfied with macro "theories" without a sound micro foundation. APPENDIX A Unconstrained regression coefficients and their standard errors, 1949-1968. Armco Steel.

Model x 2 X 1 X3 X4 X5 X 6 X7

Accelerator -.00632 .12052 .05553 .63722 -.84328 .17310 • (.04802) (.04992) (.17977) (.18036) (.03300)

Expected Profits -.00257 -.04272 .67568 -.94399 .18423 (.02343) (.18847) (.18468) (.03449)

Liquidity - . 0 0 2 0 1 -.35801 .41041 .45147 .57609 -.66750 .16275 (.33493) (.35985) (.40090) (.21442) (.22006) (.03804)

Neoclassical I -.00428 -.01735 .72151 -.84141 .18158 (.00535) (.15935) (.15424) (.02925)

Neoclassical 11 -.00333 -.00972 -.02536 .77569 -.97949 .19307 (.00642) (.00658) (.15268) (.15531) (.02769) Unconstrained regression coefficients and th e ir standard errors, 1949-1968. Bethlehem Steel.

Model X 1 X 2 X3 X4 X5 X 6 X7

Accelerator -.17110 .14656 .12251 -.35142 .33118 (.05714) (.05316) (.23369) (.05967)

Expected Profits -.13923 .05399 .39094 -.57436 .29827 (.03277) (.23247) (.23437) (.07060)

Liquidity -.16395 .50886 .85699 .16203 -.71582 .32705 (.42325) (.46006) (.24701) (.23174) (.07018)

Neoclassical I -.11504 .00571 .40390 -.51953 .27358 (.00419) (.23911) (.24879) (.07526)

Neoclassical 11 -.11119 .01106 .42518 -.53157 .26695 (.00644) (.23265) (.23649) (.07299) Unconstrained regression coefficients and their standard errors, 1949-1968. Chrysler.

Model x 2 X 1 X3 X4 x5 " X 6

Accelerator .04868 .05000 .09331 .04414 .20451 (.02411) (.02560) (.02562) (.06810)

Expected Profits .03387 .08568 .03731 .02987 .48086 .23091 (. 0 2 0 1 0 ) (.02920) (.02453) (.22939) (.06105)

Liquidity .04746 .62374 .58858 .43952 .22386 (.35483) (.36533) (.22320) (.07681)

Neoclassical I .03644 .00518 .65951 -.33999 .26663 (.00298) (.23690) (.30090) (.09444)

Neoclassical I I .03840 .00320 .67543 -.32634 .26143 (.00195) (.24242) (.30307) (.09486) Unconstrained regression coefficients and their standard errors, 1949-1968. Continental Oil.

Model

Accelerator .01746 .32309 .13991 .47647 -.36539 .08471 (.07570) (.09066) (.22185) (.23385) (.04667)

Expected Profits -.03321 -.04411 .37681 .20587 (.03949) (.25291) (.04034)

Liquidity -.01863 1.01798 .50140 .16643 (.66154) (.26270) (.04648)

Neoclassical I -.04737 .01128 .29397 -.35826 .23906 (.00556) (.24944) (.27908) (.04133)

Neoclassical II -.05382 .03742 .23202 (.01141) (.02104) Unconstrained regression coefficients and th e ir standard e rro rs, 1949-1968. Dow Chemical.

Model X 1 X 2 X3 X4 X5 X 6 X7

Accelerator .01671 .27049 .58070 .45787 -.17786 .02512 (.11747) (.17845) (.17673) (.17662) (.06214)

Expected Profits .01151 -.02751 .31826 -.56869 .20677 (.03675) (.24865) (.23297) (.06027)

Liquidity - . 0 0 1 2 0 1.88821 .41024 -.58616 .19813 (1.17126) (.24145) (.21468) (.05676)

Neoclassical I .00861 .01233 .30853 -.45668 .19581 (.01666) (.24549) (.24088) (.06141)

Neoclassical I I .00929 .06028 -.06462 .38015 -.54909 .19558 (.02959) (.03909) (.21679) (.19812) (.05856) Unconstrained regression coefficients and th e ir standard errors, 1949-1968. Dupont.

Model X 1 X 2 X3 X4 X5 X 6 X7

Accelerator -.07849 .08017 .16055 -.63963 .30567 (.06469) (.06187) (.26251) (.04226)

Expected Profits -.04344 -.00614 -.65897 .29992 (.00830) (.25876) (.04727)

Liquidity -.06075 .61182 -.63325 .31009 (.47899) (.24587) (.04646)

Neoclassical I -.04349 .00342 -.55163 .29151 (.00520) (.27540) (.04919)

Neoclassical I I -.05416 .01838 -.47990 .29446 (.01728) {.28230) (.04676) Unconstrained regression coefficients and their standard errors, 1949-1968. Eastman Kodak.

Model X1 X2 X3 X4 X& X6 X7

Accelerator -.03956 .04459 .10198 .02029 .59233 -1.05178 .28742 (.02729) (.03881) (.01836) (.39308) (.40848) (.07888)

Expected Profits -.03347 .01237 .00530 -.00881 .72402 .25223 (.00372) (.00416) (.00592) (.21586) (.06071)

Liquidity -.03021 .26083 .56490 .73731 -.40737 .25269 (.16724) (.17943) (.23875) (.24609) (.06340)

Neoclassical I -.04429 .00837 .41591 .30377 (.00285) (.25208) (.07189)

Neoclassical II -.03903 .00852 1.01424 -.90323 .29706 (.01066) (.30982) (.38560) (.08151) Unconstrained regression coefficients and their standard errors, 1949-1968. Firestone. X X

Model x 3 in X 1 X 2 X4 6 X7

Accelerator -.02431 .06574 .30582 (.03130) (.03977)

Expected Profits -.02217 .02977 -.07256 .39796 .29775 (.02184) (.02506) (.27035) (.04632)

Liquidity -.03238 .76287 .81355 .80404 -.42823 .34690 (.34970) (.35822) (.41877) (.30913) (.05169)

Neoclassical I -.03090 .00401 -.00741 .34634 (.00320) (.00312) (.03671)

Neoclassical I I -.03186 -.01317 .35535 (.00700) (.03908) Unconstrained regression coefficients and their standard errors, 1949-1968. General Electric.

Model xi X 2 X3 X4 X5 X 6 X7

Accelerator -.01383 .08424 .04851 .96304 -.35312 .17619 (.02159) (.03157) (.25743) (.26122) (.06046)

Expected Profits .01475 .01213 1.31035 -.71948 .16090 (.01188) (.22706) (.26037) (.07911)

Liquidity .02292 .32504 1.31354 -.74251 .15224 (.25229) (. 2 2 0 2 0 ) (.24853) (.07752)

Neoclassical I .02544 .00742 .00411 1.13785 -.44717 .13470 (.00353) (.00353) (.22945) (.27930) (.07150)

Neoclassical I I .00050 .01333 .01154 1.02933 -.32891 .15267 (.00542) (.00682) (.24669) (.28519) (.06738)

cn00 Unconstrained regression coefficients and th e ir standard errors, 1949-1968. General Motors.

Model x5 X X 1 X2 X3 X4 6 X7

Accelerator .10345 .04301 .09011 .53799 .22664 (.02796) (.02846) (.17653) (.03819)

Expected Profits .10348 .04388 .03251 .43343 .24193 (.01456) (.01662) (.19524) (.03824)

Liquidity .21170 .34047 .55540 .21235 (.31953) (.21375) (.04612)

Neoclassical 1 .20558 .06170 .05099 .20808 (.01007) (.01007) (.02294)

Neoclassical I I .10975 .07269 .08169 .22371 (.01633) (.01644) (.02505) Unconstrained regression coefficients and th e ir standard errors, 1949-1968. General T ire.

Model

Accelerator -.00007 .06739 -.65142 -.31618 .31361 (.01790) (.22874) (.23460) (.03940)

Expected Profits .00083 .04575 .02854 .21052 (.02521) (.02629) (.02720)

Liquidity ‘ .00200 .66663 -.29440 -.33603 .26968 (.28684) (.26243) (.28127) (.04528)

Neoclassical I .00045 .00666 .00288 .22421 (.00208) (.00253) (.02332)

Neoclassical II .00072 .00994 .00392 .00405 .22232 (.00267) (.00335) (.00328) (.02225) Unconstrained regression coefficients and th e ir standard errors, 1949-1968. Goodrich.

Model

Accelerator .01789 .06460 .40932 .25806 (.03285) (.27198) (.06067)

Expected Profits -.02852 .02872 .01840 .30889 -.41610 .33982 (.01212) (.01052) (.27017) (.31211) (.06923)

Liquidity .01987 .64950 .29389 (.36400) (.04302)

Neoclassical I -.02552 .00542 .00349 .32295 (.00301) (.00306) (.03984)

Neoclassical II -.02722 .01219 .00560 .30506 -.50907 .33074 (.00555) (.00481) (.27887) (.33658) (.07261) Unconstrained regression coefficients and their standard errors, 1949-1968. Goodyear Tire and Rubber.

Model '1

Accelerator -.02379 .08197 .34876 -.25784 .25453 (.03386) (.26347) (.25713) (.06666)

Expected Profits -.04274 .01461 -.30996 .34675 (.00723) (.26544) (.05041)

Liquidity -.02468 1.18773 .93727 .25871 (.67362) (.55156) (.03826)

Neoclassical I -.03284 .00304 -.00256 .30575 (.00238) (.00232) (.03412)

Neoclassical II -.03777 .01120 .01119 .29705 -.46936 .30985 (.00583) (.00558) (.26440) (.27316) (.06221) Unconstrained regression coefficients and their standard errors, 1949-1968. Gulf.

Model ll

Accelerator -.09091 .40062 .26685 .16764 .15169 (.10563) (.12515) (.11523) (.02059)

Expected Profits -.04575 .02560 .02692 .91862 -.44186 .15742 (.01689) (.01864) (.29037) (.32098) (.03159)

Liquidity -.08974 1 .55691 1.07232 .72579 .18777 (.33858) (.31846) (.32032) (.01919)

Neoclassical I -.04903 .03036 .02433 .77765 -.53087 .16374 (.01080) (.01201) (.24718) (.28677) (.02751)

Neoclassical II -.05730 .02983 .02710 .82995 -.52075 .16427 (.01335) (.01484) (.26641) (.30595) (.02933) Unconstrained regression coefficients and their standard errors, 1949-1968. Inland Steel. X X

Model CM X 1 3 X4 X5 X 6 X7

Accelerator .00578 .04495 .72566 -.42254 .09938 (.07505) (.24452) (.24446) (.03397)

Expected Profits - . 0 0 0 2 2 .07017 .73777 -.59163 .12036 (.04781) (.22685) (.26087) (.03451)

Liquidity .00625 .21498 .67741 -.39556 .09968 (.67293) (.24267) (.25225) (.03450)

Neoclassical I .00441 .00804 .66791 -.44924 .10743 (.00756) (.23251) (.24047) (.03348)

Neoclassical I I .00533 .00763 .65854 -.43777 .10467 (.00962) (.23861) (.24381) (.03377)

to PO Unconstrained regression coefficients and th e ir standard errors, 1949-1968. Monsanto.

Model x2 X 1 X3 X4 X5 . X 6 X7

Accelerator - . 0 0 1 0 1 .18164 .28693 .24861 .26612 .03016 (.07011) (.07986) (.08524) (.22033) (.02582)

Expected Profits -.00762 .10590 .05782 .07235 .12644 (.02319) (.02044) (.04408) (.01837)

Liquidity -.00706 1.03348 2.13375 1 .86918 .09540 (.46941) (.46963) (.49613) (.01609)

Neoclassical I .01628 .00374 .66702 .05106 (.00225) (.25111) (.03361)

Neoclassical I I -.00318 .02638 .01657 .01658 .11744 (.00621) (.00562) (.01049) (.02060)

VD CJ Unconstrained regression coefficients and their standard errors, 1949-1968. National Steel.

Model X 1 X 2 X3 X4 X5 X 6 X7

Accelerator .02752 -.15309 .58928 -.78712 .09642 (.05151) (.16619) (.15974) (.03259)

Expected Profits .01370 .08296 -.06893 .58080 -.47791 .10675 (.04017) (.05088) (.19230) (.19194) (.03690)

Liquidity .02473 -.35516 .32508 -.66851 .09159 (.38872) (.20069) (.18896) (.03994)

Neoclassical 1 .02232 .01792 .28420 -.40706 .09248 (.01167) (.19178) (.24334) (.03815)

Neoclassical 11 .02557 .02910 .01458 -.28608 .08450 (.01269) (.00985) (.22177) (.03679) Unconstrained regression coefficients and their standard errors, 1949-1968. Republic Steel.

Model x2 x7 X 1 X3 X4 X5 X 6

Accelerator -.00750 .06556 .10230 .47120 -.47633 .13667 (.05606) (.05573) (.20585) (.22431) (.05672)

Expected Profits -.01435 .07332 .45413 -.74756 .15679 (.03862) (.18814) (.18776) (.05536

Liquidity -.00749 -1.00641 .19369 -.75450 .15811 (.38641) (.19063) (.17285) (.05094)

Neoclassical I -.00686 .01030 .39488 -.69760 .14444 (.00542) (.18649) (.18480) (.05459)

Neoclassical I I - . 0 0 1 0 1 .01136 .39704 -.69016 .13263 (.00611) (.18725) (.18546) (.05482) Unconstrained regression coefficients and their standard errors, 1949-1968. Shell.

Model x2 xi X3 X4 X5 X 6 X7

Accelerator .02921 .13198 .24510 .82551 -.37035 .13339 (.07976) (.08059) (.21325) (.21088) (.03779)

Expected Profits .01106 .01714 .59909 .18551 (.01459) (.22739) (.04103)

Liquidity -.04074 1.04481 .72273 .86912 .43078 -.29868 .23503 (.58336) (.61936) (.60093) (.30934) (.27823) (.04764)

Neoclassical I .02363 .02655 .02488 .91545 -.49581 .15625 (.01137) (.01136) (.22276) (.23027) (.03533)

Neoclassical 11 .03263 .02655 .02488 .91545 -.49581 .15625 (.01137) (.01136) (.22276) (.23027) (.03533) Unconstrained regression coe fficients and th e ir standard errors, 1949-1968. Standard Oil o f C alifornia.

Model x2 X1 X3 X4 X5 X6 x7

Accelerator .04250 -.12196 .12251 (.09218) (.01484)

Expected Profits .03884 .02887 .03752 -.10562 .11572 (.01858) (.02571) (.25919) (.01622)

Liquidity .04246 .93450 .57974 .10686 (.39184) (.33122) (.01362)

Neoclassical I .03067 .01905 .11645 (.00852) (.01326)

Neoclassical 11 .03216 .01936 .11570 (.01138) (.01397) Unconstrained regression coefficients and th e ir standard errors, 1949-1968. Standard Oil of Indiana.

Model V1

Accelerator -.10736 .20349 .16096 .17508 (.13722) (.13446 (.03440)

Expected Profi ts -.06411 .05908 .11141 .15755 (.03377) (.03383) (.02988)

Liquidity -.08228 .30958 .17614 (.54488) (.03661)

Neoclassical I -.18132 .03250 .01570 .01827 .21547 (.01009) (.00983) (.00902) (.03210)

Neoclassical II -.12579 .02715 .19456 (.01016) (.03165) Unconstrained regression coefficients and their standard errors, 1949-1968. Standard Oil of New Jersey.

Model X 1 X 2 X3 X4 X5 X 6 X7

Accelerator -.07376 .11350 .85012 -.37724 .11483 (.06683) (.23410) (.25784) (.02411)

Expected Profits -.03126 -.01891 .79898 -.45030 .12403 (.01817) (.24428) (.26939) (.02435)

Liquidity -.09486 .65141 .88825 .82563 .10872 (.39304) (.34104) (.22917) (.02377)

Neoclassical I -.04238 .00819 -.00687 .00725 .87409 -.58600 .12356 (.00543) (.00574) (.00610) (.25015) (.29316) (.02485)

Neoclassical I I .00991 .00955 -.01848 .01846 1.05416 -.76950 .11327 (.00702) (.00733) (.00899) (.22315) (.27019) (.02129)

v o VO Unconstrained regression coefficients and th e ir standard errors, 1949-1968. Texaco.

Model X 1 X 2 X3 X4 X5 X 6 X7

Accelerator .01799 .20508 .15649 (.12429) (.02932)

Expected Profits -.01590 -.01196 .36206 .17535 (.02215) (.25990) (.02561)

Liquidity -.01341 1.25362 .54759 .14765 (.53885) (.23837) (.02478)

Neoclassical I -.01433 .02590 .01760 .27248 .16197 (.01364) (.01386) (.25235) (.02449)

Neoclassical I I -.02571 .02019 .33230 .17218 (.01870) (.25524) (.02476) Unconstrained regression coefficients and th e ir standard errors, 1949-1968. Union Carbide.

Model V1

Accelerator -.00003 .22647 .22663 .18351 .11889 (.07160) (.07507) (.08282) (.04103)

Expected Profits -.05173 .04068 .24550 (.02499) (.04613)

Liquidity -.01061 -.64104 .72330 .19943 (.53425) (.58865) (.04178)

Neoclassical I .00943 .00877 .48007 -.44931 .17150 (.00647) (.26990) (.30311) (.04928)

Neoclassical II .00406 .06769 .32038 -.29692 .15392 (.01565) (.18706) (.21781) (.03486) Unconstrained regression coefficients and their standard errors, 1949-1968. Uniroyal.

Model x 2 X 1 X3 X4 x5 x 6 X7

Accelerator -.00267 .05744 .72806 .15235 (. 0 2 1 1 0 ) (.21253) (.06435)

Expected Profits -.02027 -.02517 .96646 -.48459 .25662 (.01322) (.23046) (.33427) (.10480)

Liquidity -.01764 .57347 .85742 -.48519 .23927 (.21585) (.19863) (.29786) (.09496)

Neoclassical I -.03769 .00094 .81663 -.78696 .35186 (.00174) (.23790) (.35489) (.10368)

Neoclassical 11 -.03574 .00383 .83219 -.79109 .33998 (.00547) (.23819) (.34824) (.10299) Unconstrained regression coefficients and th e ir standard errors, 1949-1968. U.S. Steel.

Model X xi X 2 X3 X4 5 X 6 X7

Accelerator .20632 .04121 .79816 -.52814 .03723 (.05025) (.23821) (.22847) (.07605)

Expected Profits .14234 .03899 .04862 .76024 -.59058 .06126 (.02410) (.02248) (.19952) (.21324) (.06788)

Liquidity .50302 -.40110 -1.21354 -.64649 .42334 -.83165 -.06206 (.35607) (.40855) (.33068) (.25821) (.23640) (.07088)

Neoclassical I .26825 .01581 .01758 .55520 -.34237 .01165 (.00725) (.00724) (.19921) (.20232) (.06458)

Neoclassical 11 .30020 . 0 2 1 0 2 .01910 .44712 -.29332 - . 0 0 2 0 2 (.00825) (.00889) (.21623) (.20614) (.06452) 203 Unconstrained regression coefficients and their standard errors, 1949-1968. Westinghouse Electric.

Model x2 X1 X3 X4 X5 X6 h

Accelerator -.01212 .09170 .04752 .87119 .16139 (.02428) (.02541) (.17826) (.04813)

Expected Profits -.00237 .03643 .02314 .96468 -.60015 .15571 (.01275) (.01640) (.23225) (.25060) (.05493)

Liquidity -.00647 -.09308 1.08414 -.46541 .17420 (.20952) (.27795) (.30381) (.06656)

Neoclassical I -.00523 -.00304 1.14659 -.42998 .16949 (.00250) (.27266) (.28787) (.06403)

Neoclassical I I -.00366 -.00537 1.10880 -.36863 .16651 (.00409) (.26490) (.29099) (.06369) Unconstrained regression coefficients and th e ir standard errors, 1949-1968. Autos.

Model * 1 h X3 X4 X5 X6 X7

Accelerator .13939 .04387 .08954 .56463 . 2 2 1 2 1 (.02278) (.02334) (.15650) (.03508)

Expected Profits .10568 .05499 .02941 .51403 .24484 (.01405) (.01712) (.18293) (.03576)

Liquidity .24531 .42229 .62051 .21289 (.30595) (.20105) (.04521)

Neoclassical 1 .16859 .02036 .73056 -.41217 .25137 (.01059) (.22386) (.29124) (.05124)

Neoclassical I I .26165 .00364 .62713 .21050 (.00643) (.21066) (.04741) Unconstrained regression coefficients and th e ir standard errors, 1949-1968. Chemicals.

Model X 1 X 2 X3 X4 X5 X 6 X7

Accelerator .01814 .09496 .14470 .70729 -.64596 .14602 (.05680) (.05322) (.20783) (.17850) (.03263)

Expected Profits -.07386 .01917 .01850 .87761 -.57662 .18447 (. 0 1 0 1 0 ) (.00926) (.26124) (.21524) (.02835)

Liquidity -.02413 .80163 .77113 -.64385 .18034 (.35662) (.22164) (.18253) (.02884)

Neoclassical I -.00770 .00543 .58058 -.41645 .18107 (.00283) (.22040) (.21259) (.03011)

Neoclassical 11 -.03147 .01688 .01911 .65438 -.44481 .17077 (.01197) (.01253) (.24629) (.21231) (.03121) Unconstrained regression coefficients and th e ir standard errors, 1949-1968. E le c tric a l.

Model x 2 X 1 X3 X4 X5 X 6 X7

Accelerator -.02495 .08530 .04451 1.01451 -.34134 .17113 (.01956) (.02842) (.22151) (.23414) (.04468)

Expected Profits -.01433 .01742 .01493 1.28449 -.64993 .17574 (.01150) (.01227) (.21550) (.24519) (.06170)

Liquidity .00718 .17785 1 .33666 -.76815 .16841 (.24664) (.21535) (.23907) (.06405)

Neoclassical I -.01283 .00555 -.00399 1.41056 -.71959 .18086 (.00288) (.00316) (.19203) (.20274) (.05545)

Neoclassical I I -.00419 .01183 1.33478 -.70030 .16453 (.00450) (.18056) (.20267) (.05394) Unconstrained regression coefficients and th e ir standard errors, 1949-1968. O il.

Model X-, X2 X3 X4 Xg Xg Xy

Accelerator -.13292 .25487 .15991 .10936 .54579 .09448 (.06386) (.07090) (.07889) (.17680) (.01584)

Expected Profits -.21467 .02344 .98697 -.47681 .14034 (.01803) (.23719) (.26306) (.01770)

Liquidity -.27552 .99345 .93627 .86864 .12018 (.31735) (.26672) (.15831) (.01497)

Neoclassical I -.31229 .01251 .00943 .86332 -.41137 .14579 (.00624) (.00604) (.22587) (.25925) (.01635)

Neoclassical I I -.07271 -.01731 .02232 1.26737 -.75645 .13165 (.00955) (.01070) (.22772) (.26575) (.01560) Unconstrained regression coefficients and their standard errors, 1949-1968. Rubber.

Model

Accelerator -.06161 .04406 .84096 -.43437 .22969 (.01851) (.22885) (.22500) (.04414)

Expected Profits -.08507 .01246 .76555 -.50659 .26516 (.00816) (.24527) (.24717) (.04433)

Liquidity -.06132 .71537 .89183 .39956 .39862 .22142 (.23374) (.25251) (.30520) (.24895) (.03610)

Neoclassical I -.09147 .00236 .61025 -.41101 .27575 (.00179) (.24769) (.25265) (.04497)

Neoclassical II -.09232 .00430 .68785 -.49465 .27471 (.00396) (.24679) (.25580) (.04570) Unconstrained regression coefficients and th e ir standard errors, 1949-1968. Steel.

Model X 1 X 2 X3 X4 X5 X 6 x7

Accelerator -.17529 .13859 .13439 .59441 .14221 (.05109) (.04793) (.22067) (.04127)

Expected Profits -.09782 .05827 -.04934 .87590 -.65522 .14397 (. 0 2 2 1 1 ) (.02295) (.20530) (.19155) (.03559)

Liquidity .02726 -.48218 -.47344 .36143 -.69691 .14103 (.40414) (.39447) (.27138) (.25518) (.04359)

Neoclassical 1 -.12071 .01061 -.00794 .65271 -.39958 .14545 (.00497) (.00496) (.20136) (.21050) (.03902)

Neoclassical 11 -.04443 .01266 -.00879 .61334 -.42142 .13330 (.00659) (.00734) (.20458) (.21547) (.03941) APPENDIX B COMPONENTS OF ESTIMATED C OtFF IC I ENTS - AUTOS

30 31 82 B3 B4 35 B6

ACCELERATOR T 0.152130 0.C46505 0.091710 9.022070 0.263995 0.0 0.215575 AB -0.001127 - 0,001466 -0.000494 0.022753 0.259331 0.0 0.008177 C 0.0 -0.001473 -0.000727 0.022070 0.268995 0.0 0.007529 NC -0.001127 0.900012 0.039234 0.000633 0.000836 0.0 0.000648 SB 0.004505 -0.001466 -0.009817 -0.044323 -0.051773 0.0 -0.001812 C 0.0 0.0 0.0 -0.044323 -0.110307 0.0 0.0 NC 0.004508 0.001466 -0.009317 0.0 0.059034 0.0 -0.001812 SE -0.003313-0.002635 0.008141 0.0 0.077576 0.0 -0.000730 E 0.139390 0.043570 0.089540 0.0 0.564630 0.0 0.221210

EXPECTED PROFITS T 0.137350 0.064780 0.034910 0.014935 0.457145 0.0 0.236420 AB -0.003670 -0.015032 -0.003237 0.010179 -0.019087 0.0 0.006572 C 0.0 -0.015537 -0.001825 0.014935 -0.015649 0.0 0.003347 NC -0.003670 0.000454 -0.001411 -0.004757 -0.002439 0.0 0.002725 SB -0.004112 0.001341 -0.000246 -0.025113 0.020222 0.0 0.001448 C 0.0 0.0 0.0 -0.025113 0.0 0.0 0.0 NC -0.004112 0.001341 -0.000246 0.0 0.020222 0.0 0.001448 SE 0.007820 0.003952 -0.002017 0.0 0.054751 0.0 0.000400 E 0.105690 0.054990 0.029410 0.0 0.514030 0.0 0.244840

LI GUI 01TY T 0.259160 0.0 0.482105 0.2942^0 0.497460 0.0 0.218105 AB -0.000866 0.0 -0.086803 0.261907 0.034130 0.0 -0.003402 C 0.0 0.0 -0.092321 0.294293 0.032360 0.0 -0.003943 NC -0.000866 0.0 0.005518 -0.032383 0.001770 0.0 0.000541 SB 0.000612 0.0 -0.078991 -0.556197 0.033780 0.0 -0.000793 C 0.0 0.0 0.0 -0.556197 0.0 0.0 0.0 NC 0.000612 0.0 -0.078991 0.0 0.033780 0.0 -0.000793 2 SE 0.000370 0.0 0.105979 0.0 0.055140 0.0 -0.001020 ™ E 0.245310 0.0 0.422290 0.0 0.620510 0.0 0.212890 COMPONENTS i> ESTIMATED TGEFFICIfcNTS - AUTOS

BO B1 B2 B3 B4 B5 B6

NEOCLASSICAL I T 0.242020 0.033440 0.025495 0.0 0.329755 -0.169995 0.237340 AB 0.011408 -0.020504 0.024439 0.0 1.204429 -1.002408 -0.010170 C 0.0 -0.020291 0.025495 0.0 0.329755 -0.169995 -0.020716 NC 0.011408 -0.000213 -0.001056 0.0 0.374674 -0.832413 0.010546 SB -0.032854 0.003468-0.04 9934 0.0 -0.874614 0.786647 0.020127 C 0.0 0.0 -0.049934 0.0 -1.201437 0.944982 0.0 NC -0.032854 0.003463 0.0 0.0 0.326923 -0.158335 0.020127 SE 0.021494 0.003956 0.0 0.0 0.070990 -0.026414 0.004073 E 0.168590 0.020360 0.0 0.0 0.730560 -0.412170 0.251370

NEOCLASSICAL I I T 0.148150 0.037945 0.040845 0.0 0.337715-0.163170 0.242570 AB 0.030199 -0.033094 0.051866 0.0 0.889283 -0.836907 -0.008900 C 0.0 -0.033019 0.040845 0.0 0.337715 -0.163170 -0.013386 NC 0.030199 -0.000075 0.011021 0.0 0.551568 -0.673737 0.004486 SB 0.038289 -0. 004851 -0.089769 0.0 -0.679338 1.000077 -0.024310 C 0.0 -0.004351 -0.093276 0.0 -0.975934 1.000077 0.0 NC 0.038289 0.0 0.003506 0.0 0.296596 0.0 -0.024310 SE -0.068300 0.0 0.000698 0.0 0.079470 0.0 0.001140 E 0.261650 0.0 0.003640 0.0 0.627130 0.0 0.210500

t\3 U COMPONF NT S OF E STI MATFD COEFFICIENTS -Chemicals

BO B1 32 .83 B4 B5 B6

ACCELERATOR T -0.102380 0.160672 0.271368 0.182056 0.171690 -0.373854 0.153452 AB 0.011043 -0.008407 -0.061341 -0.034674 -0.143179 -0.409735 0.009880 C 0.0 -0.017168 -0.059710 -0.062392 0.045191 -0.042195 -0.013323 NC 0.011043 0.008761 -0.001631 0.027717 -0.188370 -0.367541 0.023203 SB 0.005624 -0,038658 -0.038432 -0.147382 0.257640 0.280260 0.006371 C 0.0 0.0 0.0 -0.147382 -0.022485 0.244951 0.0 NC 0.005624 -0.038658 -0.033432 0.0 0.280125 0.035309 0.006371 SE -0.016628 -0.018647 -0.026386 0.0 0.421139 -0.142630 -0.023683 E 0.018140 0.094960 0.144700 0.0 0.707290 -0.645960 0.146020

EXPECTED PROFITS T -0.124750 0.023654 0.020760 0.005978 0.208456 -0.245532 0.226172 AB 0.016755 0.012833 -0.003161 -0.013106 -0.081066 -0.470047 -0.010749 C 0.0 0.011457 0.005131 -0.012969 0.120744 -0.369275 -0.011517 NC 0.016755 0.001381 -0.008292 -0.000137 -0.201810 -0.100771 0.000768 SB -0.009622 -0.025618 -0.006935 0.007128 0.098585 0.276614 0.012039 _C O.o -0.025351 -0.010731 0.007123 -0.068589 0.269371 0.0 NC ' ' -0.009622 -0.000267 0.003796 0.0 0.167174 0.006743 0.012039 SE -0.007182 0.008296 0.007836 0.0 0.651635 -0.137655 -0.042992 E -0.073860 0.019170 0.018500 0.0 0.877610 -0.576620 0.184470

LIQUIDITY T -0.109330 0.130654 1.039736 0.518496 0.229510 -0.325356 0.21114B AB' 0.005290 -0. 343731 -0. 366094 0.205709 -0.172550 -0.230221 0.012005 C 0.0 -0.200299 -0.042864 0.091998 -0.004005 -0.256221 -0.013900 NC 0.005290 -0.143432 -0.343230 0.113712 -0.168545 0.026000 0.025905 SB 0.002479 0.213077 -0.03243? -0.724205 0.132977 0.053308 0.003783 C 0.0 0.213077 -0.207133 -0,724205 -0.022071 0.165128 0.0 NC 0.002479 0.0 0.124701 0.0 0.155048 -0.111819 0.003783 SE -0.007772 0.0 0.230420 0.0 0.581192 -0.141582 -0.046597 E -0.024130 0.0 0.801630 0.0 0.771130 -0.643850 0.180340 COMPONENTS OF F ST IMATFD COE FF IC I ENTS -Chemicals

BO 91 B2 93 B4 B5 B6

NEOCLASSICAL I T -0.053460 0,004176 0,003150 0.0 0.374306 -0.291.524 0.202730 A9 0.012829 0.000777 0.003545 0.0 0.033116 -0.173674 -0.017826 C 0.0 0.001245 0.001964 0.0 0.065691 -0.194063 -0.024923 NC 0.012B29 -0.000468 0.00158? 0.0 -0.027575 0.020389 0.007097 SB -0.008698-0.004953 -0.004510 0.0 -0.151386 0.023506 0.027602 C 0.0 -0.004953 -0.004628 0.0 -0.108622 0.061177 0.0 NC -0.008698 0.0 0.000113 0.0 -0.042764 -0.037671 0.027602 SE -0.004128 0.0 0.003244 0.0 0.319545 0.025242 -0.031436 E -0.007700 0.0 0.005430 0.0 0.590580 -0.416450 0.181070

NEOCLASSICAL I I . . _ T -0.095610 0.030870 0.006990 -0.012154 0.347620 -0.417578 0.220626 A8 0.008275 -0.001581 0.010998 0.022288 0.122797 0.009669 -0.022395 C 0.0 0.002531 0.010449 0.019319 0.039197-0.062544-0.023529 NC 0.008275 -0.004113 0.000548 0.002471 0.083600 0.072213 0.001144 SB -0.000258 -0.011665 -0.002°36 -0.010134 -0.153556 0.065209 -0.001959 „C _ 0.0 -0.010000 -0.003051 -0.010134 -0.176890 0.026586 0.0 . .. NC -0.000258 -0.001665 0.000115 0.0 0.023334 0.038624 -0.001959 SE -0.008029 -0.000744 0.004058 0.0 0.337519 -0.102110 -0.025512 E -0.031470 0.016890 0.019110 0.0 0.654380 -0.444810 0.170770

ro tn COMPONENTS Hr ESTIMATED C06 FFI C I ENTS-ELECTRICAL

BO B1 B2 B3 B4 B5 B6

ACCELERATOR T -0.025950 0.087970 0.048015 0.0 0.9L7115 -0.176560 0.168790 AB -0.001425 -0.0007°0 0.002721 0.0 0.016459-0.156839 0.004396 C 0.0 -0.002007 0.000275 0.0 0.022152 -0.176560 0.002613 NC -0.001425 0.001218 0.002445 0.0 -0.005693 0.019721 0.001782 SB 0.003201 -0.001190 -0.003192 0.0 0.001257 0.062599 -0.004102 C 0.0 0.0 0.0 0.0 0.0 0.062599 0.0 NC 0.003201 -0.001190 -0.003192 0.0 0.001257 0.0 -0.004102 SE -0.001788 -0.000690 -0.003033 0.0 0.079679 -0.070539 0.002047 E -0.024950 0.035300 0.044510 0.0 1.014510 -0.341340 0.171130

EXPECTED PROFITS _ . T 0.012380 0.01P215 0.006065 0.011570 1.137515 -0.659815 0.158305 AB -0.001957 0.019982 0.006659 0.012553 0.102599 -0.037043 0.003907 C 0.0 0.018215 0.006065 0.011570 0.112069 -0.029311 0.000920 NC -0.001957 0.001767 0.000594 0.000988 -0.009471 -0.007732 0.002987 SB -0.003438 -0.030307 -0.003675 -0.024123 0.049518 -0.013105 0.007029 C _ 0.0 -0.030111 -0.002304 T0 . 024128 0.0 0.0 0.0 NC -0.003438 -0.000696 -0.001372 0.0 0.049518 -0.013105 0.007029 SE 0.005374 0.010030 0.005881 0.0 -0.005142 0.060033 0.006499 E -0.014330 0.017420 0.014930 0.0 1.284490 -0.649930 0.175740

LIQUIDITY T _ 0.016450 0.115980 0.0 0.0 1.198840 -0.603960 0.163220 AB -0.000230 0.082305 0.0 0.0 0.069391 -0.092885 -0.002578 C 0.0 0.098804 0.0 0.0 0.075236 -0.070892 -0.003935 NC -0.000230 -0.006499 0.0 0.0 -0.005845 -0.021993 0.001357 SB -0.000470 -0.198285 -0.075521 0.0 0.036177 -0.031148 0.002976 C 0.0 -0.198285 0.0 0.0 0.0 0.0 0.0 _NC -0.000470 0.0 -0.075521 0.0 0.036177 -0.031148 0.002976 SE 0.000699 0.0 0.253371 0.0 0.032252 -0.040156 0.004792 E 0.007180 0.0 0.177850 0.0 1.336660 -0.768150 0.168410 COMPONENTS OF E ST [MATED COE FF IC I ENTS -ELECTRICAL

BO B] B2 B3 B4 B5 B6

NEOCLASSICAL I T 0.020210 0.003710 0.002055 -0.001520 1.142220 -0.438575 0.152095 AB 0.000194 0.003833 0.002052 -0.001374 -0.008551 -0.009975 -0.006382 C 0.0 0.003710 0.002055 -0.001520 -0.002632 -0.004172 -0.006138 NC 0.000194 0.000123 -0.000003 0.000146 -0.005919 -0.005802 -0.000244 SB -0.009337-0.002040-0.004107 0.001933 0.118426-0.190234 0.022674 C 0.0 -0.002564 -0.004107 0.001643 0.0 0.0 0.0 NC -0.009337 0.000525 0.0 0.000293 0.118426-0.190234 0.022674 SE 0.009120 0.000047 0.0 -0.003030 0.158465 -0.080806 0.012474 E -0.012830 0.005550 0.0 -0.003990 1.410560 -0.719590 0.180860

NEOCLASSICAL I I T -0.003160 0.006665 0.005770-0.002685 1.069065-0.348770 0.15°5°0 AB 0.000404 0.006580 0.005564 -0.002509 -0.021805 0.004595 -0.002857 C 0.0 0.006665 0.005770 -0.002685 -0.022419 0.008366 -0.002424 NC 0.000404 -0.000085 -0.000206 0.000176 0.000614 -0.003771 -0.000433 SB 0.000502 -0.003402 -0.011334 0.005194 0.134661 -0.278985 0.009467 C_ 0.0 -0.004686 -0.011334 0.005194 0.0 0.0 0.0 NC 0.000502 0.001284 0.0 0.0 0.184661 -0.278985 0.009467 SE -0.000909 0.001988 0.0 0.0 0.102859 -0.077139 -0.001670 E -0.004190 0.011830 0.0 0.0 1.334780-0.700300 0.164530

ro • j COMPONENTS 0= c STI MATFD Cut FFI Cl ENTS-OIL

BO . 31 32 B3 84 B5 B6

ACr E LER AT OR T -0.164870 0.138456 0.116476 0.064534 0.307443 -0.158997 0.134100 AB -0.000384 0.002550 0.14654? 0.0^7643 0.493800 -0.143554 -0.001598 C 0.0 0.022146 0.095673 0.098966 0.524802 -0.217429 -0.000553 NC -0.000384-0.019596 0.050369 0.008677-0.041002 0.073874-0.001045 SB -0.009257 -0.038753 -0.139994 -0.073786 -0.511607 0.302551 -0.001241 C 0.0 -0.027904 -0.185634 -0.123936 -0.425389 0.302551 0.0 NC -0.009257 -0.010349 0.046640 0.050150 -0.086218 0.0 -0.001241 SE 0.000532 0.152618 0.035337 0.020969 0.256153 0.0 -0.036781 E -0.132920 0.254370 0.159910 0.109360 0.545790 0.0 0.094480

EX?ECTFD PROFITS T -0.140330 0.010230 -0.002271 0.025121 0.394714 -0.090817 0.160207 AB -0.005140 0.001378 -0.007761 0.005789 0.466944 -0.135310 -0.010735 C 0.0 0.012346 -0.010723 0.010157 0.382633 -0.198048 -0.012023 NC -0.005140 -0.01096° 0.002962 -0.004367 0.084311 0.012738 0.001287 SB -0.009705 -0.011603 0.010033 -0.026917 -0.175061 0.033920 -0.000531 _C 0.0 -0.011609 0.010033 -0.021321 -0.218116 0.035378 0.0 NC -0.009705 0.0 0.0 -0.005596 0.043055 -0.001458 -0.000531 SE 0.014651 0.0 0.0 0.019447 0.310372-0.234603-0.008601 E -0.214670 0.0 0.0 0.023440 0.986970 -0.476810 0.140340

LIQUIDITY T -0.297100 0.743659 0.427554 0.489753 0.251116 0.035559 0.161229 AB 0.004850 0.016692 0.235786 -0.193303 0.762941 0.051797-0.019781 C 0.0 0.139720 0.377730 0.298577 0.454719 0.128413 -0.017198 NC 0.004850 -0.173023 -0.141944 -0.492390 0.308222 -0.076616 -0.002583 SB -0.001492 -0.099719 -0.145161 -0.295949 -0.393324 -0.087355 -0.005162 C 0.0 -0.131503 -0.154483 -0.295949 -0.496472 -0.087355 0.0 _ NC „ -0.001492 0.031739 0.009322 0.0 0.103148 0.0 -0.005162 SE -0.003592 0.332818 0.413090 0.0 0.247907 0.0 -0.016105 E -0.275520 0.993450 0.936270 0.0 0.868640 0.0 0.120180 COMPONENTS of e s t im a t e d COEFFICIENTS-OIL

BO 31 B2 B3 B4 B5 B6

NEOCLASSICAL I T -0.271130 0.021976 0.001261 0.013190 0.403737 -0.242637 0.168071 AB 0.005714 -0.009164 -0.007978 -0.001445 0.461675 -0.125734 -0.017444 C 0.0 -0.006725 -0.006174 -0.000363 0.393719 -0.192465 -0.017081 NC 0.005714 -0.002439 -0.001804 -0.001085 0.067956 0.066731 -0.000363 SB -0.008544 -0.001401 0.006717 -0.001365 -0.276057 0.066008 0.001908 C 0,0 0.0 0.006717 -0.001746 -0.199478 0.066092 0.0 NC -0.008544 -0.001401 0.0 -0.000123 -0.076579 -0.000084 0.001908 SE 0.002557 0.001099 0.0 -0.000443 0.263965 -0.109006 -0.006745 6 -0.312290 0.012510 0.0 0.009430 0.863320 -0.411370 0.145790

NEOCLASSICAL I I T -0.187340 0.025271 -0.002640 0.012423 0.418890 -0.221203 0.162129 AB -0.005165 -0.009963 -0.019999 0.006772 0.613588 -0.273352 -0.010218 C 0.0 -0.008053 -0.015840 0.009876 0.566090 -0.285253 -0.015830 NC -0.005165 -0.001909 -0.004159 -0.003104 0.047499 0.011902 0.005612 SB 0.015011 -0.015309 0.010202 -0.002481 -0.146199 -0.052907-0.008286 C 0.0 -0.015309 0.009456 -0.004354 -0.341362 0.082465 0.0 NC 0.015011 0.0 0.000747 0.001873 0.195163 -0.135372 -0.008286 SE -0.009893 0.0 -0.004373 0.005606 0.381090 -0.208986 -0.011974 E -0.072710 0.0 -0.017310 0.022320 1.267370 -0.756450 0.131650 CDKDONtNTS GF ESTIMATED COEFF IC IE NTS-RUBBER

BO 81 B 2 B3 B4 B5 B6

ACCFLER ATOR T -0.068730 0. 041030 0. 026393 0.0 0.166944 -0.114804 0 .256874 AB -0.000760 0.023179 0.0^3015 0.0 0.255441 -0.064222 -0.010838 C 0.0 0.029790 0.039799 0.0 0.259010-0.151215 0.009810 NC -0.000760 -0.001610 0.003226 0.0 -0.003569 0.086992 -0.020648 SB -0.001972 -0.027158 -0.059413 0.0 -0.071603 -0.02147? 0.017628 C 0.0 -0.018105 -0.069413 0.0 -0.117755 0.105153 0.0 NC -0.001972 -0.009053 0.0 0.0 0.046152 -0.126625 0.017628 SE 0.002683 0.002008 0.0 0.0 0.490178 -0.233872-0.033974 E -0.061610 0.044060 0.0 0.0 0.840960 -0.434370 0.229690

EXPECTED PPDFITS T -0.112870 0.009150 0.012372 -0.007910 0.334662-0.242130 0.290292 AB -0.001422 0.028055 0.005679 0.012649 0.017762 -0.160415 0.024150 C 0.0 0.036600 0.003395 0.014953 0.207613 -0.135850 0.009959 NC -0.001422 -0.008545 0.002284 -0.002304 -0.189851 -0.024564 0.014191 SB 0.002013 -0.035360 -0.018051 -0.004739 -0.047828 0.047906 -0.007385 __C 0.0 -0.034255 -0.018051 -0.004739 -0.148862 0.097954 0.0 NC 0.002013-0.001105 0.0 0.0 0.101034 -0.05004B -0.007385 SE -0.000653 0.010616 0.0 0.0 0.460954 -0.151951 -0.041896 E -0.085070 0.012460 0.0 0.0 0.765550 -0.506590 0.265160

LIQUIDITY T _ -0.092570 0.267268 0.533582 0.478162 0.026958 -0.164244 0.281690 AB 0.000788 0.476590 0.493630 0.342338 -0.075409 0.199689 -0.009891 C 0.0 0.413526 0.341194 0.331707 -0.146502 -0.282362 0.002785 NC 0.000788 0.063064 0.152435 0.010631 0.071093 0.482051 -0.012666 SB -0.001505 -0.317840 -0.274650 -0.252244 0.007731 -0.035445 0.007658 C 0.0 -0.338577 -0.419689 -0.246079 0.029087 -0.035445 0.0 ro _NC_ _ ■_ -0.001505 0.020737 0. 145039 -0.006164 -0.021356 0.0 _____ 0.007658 - N SE 0.000674 0.289352 0.139268 -0.168696 0.439340 0.0 -0.058047 o E -0.061320 0.7153 70 0.891P30 0.399560 0.398620 0.0 0.221420 COMPONENTS OF ESTIMATED COEFFICIFNTS-RUBBER

BO Rl R2 B3 B4 B5 B6

NEOCLASSICAL I T -0.126500 0.001940 0.001836 -0.000532 0.163326 -0.157392 0.310222 AB 0.003478 0.001496 0.003085 -0.000349 0.553306 -0.492312 -0.005758 C 0.0 0.001906 0,002728 -0.000960 0.653304-0.629568 -0.004417 NC 0.003478 -0.000409 0.000357 0.000111 -0.099993 0.137256 -0.001341 SB -0.000196-0.001270-0.004971 0.001381 -0.507761 0.409476 0.001323 C 0.0 -0.001661 -0.004971 0.001381 -0.599015 0.548354 0.0 NC -0.000196 0.000391 0.0 0.0 0.091255 -0.139378 0.001323 SE -0.003348 0.000193 0.0 0.0 0.401378 -0.17078? -0.030037 E -0.091470 0.002360 0.0 0.0 0.610250 -0.411010 0.275750

NEOCLASSICAL I I T -0.133310 0.004288 0.003222 0.002300 0.286360 -0.353904 0.311648 AB 0.002434 0. 006S15 0.003893 0.001264 0.158223 -0.252607 0,000024 C 0.0 0.006570 0.005549 0.001133 0.144846-0.194142-0.004084 NC 0.002434 0.000245 -0.001656 0.000126 0.013377 -0.058465 0.004108 SB 0.001129 -0.005856 -0.007115 -0.003564 -0.143217 0.227130 -0.001732 _C _ 0.0 -0.005482 -0.007115 -0.003564 -0.116482 0.139023 0.0 . ... NC 0.00LI29 -0.000374 0.0 0.0 -0.026734 0.088157 -0.001732 SE -0.003629 -0 . 000947 0.0 0.0 0.385784 -0.115319 -0.035231 E -0.092320 0.004300 0.0 0.0 0.687650-0.494650 0.274710 COMPONENTS 0“ ESTIMATE COEFFICIENTS -STEEL

BO Rl 82 B3 B4 B5 B6

ACCELERATOR T 0.054700 0.049713 0.057555 -0.015263 0.535920 -0.568138 0.145663 AB -0.029161 0.056165 0.155448 0.105705 0.136919 0.212968 -0.000344 C 0.0 0.024315 0.057359 -0.041713 0.163370 0.090437 0.003324 NC -0.029161 0.031349 0. 098 0a9 0.147413 0.023550 0.122531 -0.004167 SB 0.004108 0.009669 -0.096951 -0.089445 -0.169951 0.355170 -0.015210 C 0.0 -0.008955 -0.123403 -0.039445 -0.170421 0.355170 0.0 NC 0.004108 0.018524 0.026552 0.0 0.000470 0.0 -0.015210 SE 0.024868 0.023043 0.019233 0.0 0.040521 0.0 0.012600 E -0.175290 0.138590 0.134390 0.0 0.594410 0.0 0.142210

EXPECTED PROFITS T -0.000330 0.053239 0.008103 -0.019603 0.599927 -0.654338 0.154610 AB -0.013162-0.000353 0.033854 -0.056580 0.073559 0.095809-0.011606 C 0.0 -0.001327 0.040517 -0.033223 -0.013101 0.040208 -0.002637 NC -0.013162 0.001474 -0.001663 -0.023357 0.086660 0.045601 -0.008969 SB 0.012091 -0.008050 -0.046957 0.055494 0.067781 -0.053813 -0.013358 C „ 0.0 -0.005056 -0.046957 0.064446 0.0 0.0 0.0 NC 0.012091 -0.002994 0.0 -0.008962 0.067781 -0.053813 -0.013358 SE 0.000973 0.013434 0.0 -0.029636 0.134633 -0.032877 0.014324 E -0.097820 0.058270 0.0 -0.049340 0.975900 -0.655220 0.143970

LIQUIDITY T _ 0.365550-0.294253-0.109238 0.146153 0.392940-0.672257 0.129520 AB -0.056915 -0.124391 -0.274420 -0.000706 -0.019678 0.080863 0.006599 C 0.0 -0.212486 -0.42B901 -0.326674 -0.066310 -0.072015 -0.008037 NC -0.056915 0.088095 0.154491 0.325967 0.046632 0.152877 0.014636 SB -0.000617 0.164193 0.050493 -0.145452 0.025815-0.025116 0.001620 C 0.0 0.137552 0.058274 -0.145452 0.0 0.0 0.0 NC -0.000617 0.026641 -0.007781 0.0 0.025815 -0.025116 0.001620 SE 0.057493 -0.227729 -0.141274 0.0 -0.037646 -0.030399 0.003291 E 0.027260 -0.482180 -0.473440 0.0 0.361430 -0.696910 0.141030 COMPONENTS OF ESTIMATED C DEFF IC I ENT S-STEEL

BO B1 B2 83 84 B5 B6

NEOCLASSICAL I T 0.168800 0.009630 0.00P930 -0.002892 0.504600 -0.542866 0.135193 AB -0.020915 0.000800 0.010032 -0.030124 0.059213 0.104837 -0.014403 C 0.0 0.000798 0.014650 -0.014458 -0.021073 0.084702 -0.001824 NC -0.020915 0.000001 -0.004618 -0.015666 0.080286 0.020135-0.012579 SB -0.00582° -0.001137 -0.012962 0.030008 0.075028 -0.02737? 0.003858 C 0.0 -0.000516 -0.012962 0.031473 0.0 0.0 0.0 NC -0.005329 -0.000621 0.0 -0.001470 0.075028 -0.027373 0.003858 SE 0.026596 0.001317 0.0 -0.004932 0.013869 0.065824 0.020801 € -0.120710 0.010610 0.0 -0.007940 0.652710 -0.399580 0.145450

_ . NEOCLASSICAL I I T 0.215570 0.011742 0.005613-0.004227 0.450595-0.536398 0.129967 AB -0.026408 0.002795 0.009695 -0.029411 0.077556 0.098147 -0.011559 C 0.0 0.003127 0.013022 -0.021133 0.018476 0.107828 -0.002136 NC -0.026408 -0.000332 -0.003326 -0.008278 0.059080 -0.009681 -0.009423 SB -0.006988-0.001634-0.015309 0.027836 0.069734 0.018539 0.004033 _ C 0.0 0.0 -0.015309 0.032271 -0.043045 0.0 0.0 NC -0.006988 -0.001634 0.0 -0.004435 0.112779 0.018539 0.004033 SE 0.033315 -0.000242 0.0 -0.002933 0.015455 -0.001708 0.010859 E -0.044430 0.012660 0.0 -0.009790 0.613340 -0.421420 0.133300

tvs WINS APPENDIX C :QMPrjNE NTS OF ESTIMATED COEFFICIENTS - AUTOS { P90P0RTI ON) BO B1 3 2 83 34 B5 B 6

ACCELERATOR T/E 1. 09 1. 05 1. 02 0.0 0.48 0.0 0.97 AB/E -o .o i -0 . 03 -0.01 0.0 0.48 0.0 0.04 SB/E 0.03 0. 03 -0 . 11 0.0 -0.09 0.0 -0.01 SE/E -0.02 - 0. 06 0. 09 0.0 0.14 0.0 -0 .0 0

EXPECTEO PROFITS T/E 1. 30 1.13 1.19 0.0 0.89 0.0 0.97 AB/E -0 .0 3 - 0. 27 -0 . 11 0.0 -0 .04 0.0 0.03 SB/E -0 .0 4 0. 02 -0.01 0.0 0.04 0.0 0.01 SE/E 0.07 0. 07 -0 . 07 0.0 0.11 0.0 0.00

LIQUIDITY T/E 1.06 0. 0 I . 14 0.0 0.80 0.0 1.02 AB/E -0 . 00 0. 0 -0.21 0.0 0.06 0.0 -0 .0 2 SB/E 0.00 0. 0 -0 .19 0.0 0.05 0.0 -0 .0 0 SE/E 0.00 0. 0 0.25 0.0 0.09 0.0 -0 .0 0

NE OS LA SSI C AL I T/E 1.44 1.64 0. 0 0.0 0.45 0.41 0.94 AB/E 0.07 -1.01 0.0 0.0 1.65 2.43 -0 .0 4 SB/E -0 . 19 0. 17 0.0 0.0 -1.20 -1.91 0.08 SE/E 0. 13 0. 19 0. 0 0.0 0.10 0.06 0.02

NEOCLASSICAL 11 T/E 0.57 0. 0 11.22 0.0 0.54 0.0 1.15 AB/E 0.12 0. 0 14.25 0.0 1.42 0.0 -0 .0 4 SB/E 0.15 0.0 -24.66 0.0 -1.08 0.0 -0 .1 2 SE/E -0 .2 6 0.0 0. 19 0.0 0.13 0.0 0.01 COMPONENTS OF ESTIMATED COEFF I C I ENT S -CHEMICALS CPROPORTI ON) BO 31 B2 B3 B4 B5 B6

ACCELERATOR T/E -5 .6 4 1. 69 1.88 0.0 0.24 0.58 1.05 AB/E 0.61 -0 .0 9 -0.42 0.0 -0.23 0.63 0.07 SB/E 0.31 -0.41 -0 .2 7 0.0 0.36 -0 .4 3 0.04 SE/E -0.9 2 -0 .2 0 -0.19 0.0 0.60 0.22 -0 .1 6

EXPECTED PROFITS T/E 1.69 1.23 1.12 0.0 0.24 0.43 1.23 AB/E -0 .2 3 0. 67 -0 . 17 0.0 -0.09 0.82 -0 .0 6 SB/E 0.13 -1 .3 4 -0.3 7 0.0 0.11 -0.4 9 0.07 SE/E 0. 10 0.43 0.42 0.0 0.74 0.24 -0 .2 3

LIQUIDITY T/E 4.55 0. 0 1.30 0.0 0.30 0.51 1.17 AB/E -0 .2 2 0. 0 -0.48 0.0 -0 .2 2 0.36 0.07 SB/E -0 .1 0 0. 0 -0 .1 0 0.0 0.17 -0 .0 8 0.02 SE/E 0.32 0.0 0.29 0.0 0.75 0.22 -0 .2 6

NEOCLASSICAL I T/E 6.94 0. 0 0.58 0.0 0.64 0.70 1.12 AB/E -1 .6 7 0. 0 0.65 0.0 0.07 0.42 -0 .1 0 SB/E 1.13 0.0 -0 . 83 0.0 -0 .2 6 -0 .0 6 0.15 SE/E 0.54 0.0 0.60 0.0 0.55 -0 .0 6 -0 .1 7

NEOCLASSICAL 11 T/E 3.04 1. 83 0.37 0.0 0.53 0.94 1.29 AB/E -0 .2 6 - 0. 09 0.58 0.0 0.19 -0 .0 2 -0 .1 3 ro SB/E 0.01 -0 .6 9 -0.15 0.0 -0 .2 3 . -0 .1 5 -0 .0 1 ro SE/E 0.26 -0 . 04 0.21 0.0 0.52 0.23 -0 .1 5 components OF estimated COEFFICIENTS -Electrical I PROPORTION) BO B1 B2 B3 B4 B5 B6

ACCELERATOR T/E 1.04 1.03 1.08 0.0 0.90 0.52 0.99 AB/E 0. 06 -0 . 01 0. 06 0.0 0.02 0.46 0.03 S9/E -0 .1 3 -0 . 01 -0.07 0.0 0.00 -0.18 - -0.02 SE/E 0.07 -0 . 01 -0.0 7 0.0 0.03 0.21 0.01

EXPECTED PROFITS T/E - 0 . 86 1. 05 0.41 0.0 0.89 1.02 0.90 AB/E 0. 14 1.15 0.45 0.0 0.08 0.06 0.02 SB/E 0.24 -1 .7 7 -0.2 5 0.0 0.04 0.02 0.04 SE/E -0 .3 8 0.5 3 0.39 0.0 -0.00 -0.09 0.04

LIQUIDITY T/E 2.29 0. 0 0.0 0.0 0.90 0.79 0.97 AB/E -0 .0 3 0. 0 0. 0 0.0 0.05 0.12 -0 .0 2 SB/E -0 .0 7 0.0 -0.42 0.0 0.03 0.04 0.02 SE/E 0.10 0.0 1.42 0.0 0.02 0.05 0.03

NEOCLASSICAL I

T/E -1 .5 8 0.67 0.0 0.33 0.81 0.61 Q . 00 AB/E -0 .0 2 0. 69 0.0 0.34 -0 .0 1 0.01 -0 .0 4 SB/E 0.73 -0 .3 7 0. 0 -0.48 0.08 0.26 0.13 SE/E -0 .7 1 0.01 0.0 0.76 0.11 0.11 0.07

NEOCLASSICAL 11 T/E 0. 75 0.56 0.0 0.0 0.80 0.50 0.97 AB/E -0 .1 0 0.56 0.0 0.0 -0.0 2 -0.0 1 -0 .0 2 SB/E -0 .1 2 -0 .2 9 0.0 0.0 0.14 . 0.40 _ 0.06 SE/E 0.22 0.17 0.0 0.0 0.08 0.11 -0 .0 1 COMPONENTS OF E ST I MATEO COEFF 1C IENTS-0IL (DRQPORTI ON) BO 81 R? 83 84 B5 B6

ACCELERATOR T/E 1.24 0.54 0.73 0.59 0.56 0.0 1.42 AB/E 0.00 0. 01 0.92 0.39 0.89 0.0 -0.02 SB/E 0.07 -0.15 -0.87 -0.67 -0.94 0.0 -0.01 - SE/E -0.07 0.60 0.22 0.19 0.49 0.0 -0.39

EXPECTED PROFITS T/E 0.65 0.0 0.0 1.07 0.39 0.19 1.14 AB/E 0.02 0.0 0.0 0.25 0.47 0.39 -0.08 SB/E 0.05 0.0 0.0 -1.15 -0.18 -0.07 -0.00 SE/E -0.07 0.0 0.0 0.83 0.31 0.49 -0.06

LIQUIDITY T/E 1.08 0.75 0.46 0.0 0.29 0.0 1.34 AB/E -0.02 0.02 0.25 0.0 0.88 0.0 -0.16 SB/E 0.01 -0.10 -0.16 0.0 -0.45 0.0 . ..-0*04 SE/E 0.01 0.34 0.45 0.0 0.29 0.0 -0.13

NEOCLASSICAL I T/E 0.87 1.76 0.0 1.40 0.47 0.59 1.15 AB/E -0.02 -0.73 0.0 -0.15 0.53 0.31 -0.12 SB/E 0.03 -0.11 0.0 -0.20 -0.32 -0.16 0.01 SE/E -0.01 0.09 0.0 -0.05 0.31 0.26 -0.05

NEOCLASSICAL I I T/E 2.58 0.0 0.15 0.56 0.33 0.29 1.23 AB/E 0.07 0.0 1.16 0.30 0.48 0.36 -0.08 SB/E -0.21 0.0 -0.59 „ -0.il -0.12 _ 0.07 r0.06 £ SE/E 0.14 0.0 0.28 0.25 0.30 0.28 -0.09 00 COMPONENTS of E ST I MATEO. COEFF IC I ENTS -R jBBER (PROPORT I ON) BO 81 82 R3 B4 B5 H6

ACCELERATOR T/E 1. 12 0. 93 0. 0 0.0 0.20 0.26 1.12 AB/E O.Ol 0. 64 0. 0 0.0 0.30 0. 15 -0 .0 5 SR/E 0.03 -0.6? 0.0 0.0 -0.09 - 0.05 0.08 SE/E -0 . 04 0. 06 0.0 0.0 0.58 0.54 -0 .1 5

EXPECTED PROFITS T/E 1.33 0. 73 0.0 0.0 0.44 0.48 1.09 AB/E 0.02 2. 26 0.0 0.0 0.02 0.32 0.09 SB/E -0 .0 2 -2 . 84 0. 0 0.0 -0.06 -0 .0 9 -0 .0 3 SE/E 0.01 0.85 0.0 0.0 0.60 0.30 - 0 . 16

LIQUIDITY T/E 1.51 0.37 0.60 1.20 0.07 0.0 1.27 AB/E -0.01 0. 67 0.65 0.86 -0 . 19 0.0 -0 .0 4 SR/E 0.02 -0 . 44 -0. 31 -0.63 0.02 0.0 0.03 SE/E -0 .0 1 0.40 0. 16 -0.42 1.10 0.0 -0 .2 6

neoclassical I T/E 1.38 0. 82 0.0 0.0 0.27 0.38 1.13 AB/E -0 .0 4 0. 63 0.0 0.0 0.91 1.20 -0 .0 2 SB/E 0.00 -0 . 54 0.0 0.0 -0 .8 3 -1 .0 0 _ 0.00 SE/E 0.04 0. 08 0.0 0.0 0.66 0.42 -0 .1 1

NE OC LA SSI C AL 11 T/E 1.44 1.00 0.0 0.0 0.42 0.72 1.13 AB/E - 0 . 03 1.58 0.0 0.0 0.23 0.51 0.00 SB/E -0 .0 1 -1 . 36 0.0 0.0 -0.21 -0 .4 6 -o .o i SE/E 0.04 -0 .2 2 0. 0 0.0 0.56 0.23 -0 .1 3 COMPONE NTS OF ESTIMATED COEFFICIENTS -STEEL (PROPORTION) BO B 1 32 B3 B4 B5 B6

ACCELERATOR T/E -0.31 0.36 0.43 0.0 0.90 0.0 1.02 AB/E 0.17 0.41 1. 16 0.0 0.31 0.0 -0 .0 1 SB/E -0 .0 2 0. 07 -0.72 0.0 -0.29 0.0 -0 .1 1 SE/E - 0 . 14 0. 17 0. 14 0.0 0.07 0.0 0.09

EXPECTED PROFITS T/E 0.00 0. 91 0.0 0.38 0.68 1.00 1.07 AB/E 0. 13 -0.01 0.0 1.15 0.08 -0 .1 3 -0 .0 3 SB/E -0 .1 2 -0 . 14 0.0 -1 .1 2 0.08 0.08 -0 .0 9 SE/E -0 .0 1 0.23 0.0 0.60 0.15 0.05 0.10

LI QUIDITY T/E 13.41 0.61 0.23 0.0 1.09 0.96 0.92 AB/E -2 .0 9 0.26 0. 58 0.0 -0.05 -0 .1 2 0.05 SB/E -0 .0 2 -0 .3 4 -0.11 0.0 0.07 0.04 0.01 SE/E 2.11 0.47 0.30 0.0 -0.10 0 . 1 2 0.02

NEOCLASSICAL I T/E -1 .4 0 0. 91 0. 0 0.36 0.77 1.36 0.93 AB/E 0.17 0. 08 0.0 3.79 0.09 -0 .2 6 -0 .1 0 SB/E 0,05 - o . u 0.0 -3.78 0.11 0.07 .. 0.03 SE/E -0 .2 2 0.12 0.0 0.62 0.02 -0 .1 6 0.14

NEOCLASSICAL I I T/E -4 .8 5 0. 93 0.0 0.48 0.73 1.27 0.97 AB/E 0.59 0.22 0.0 3.35 0.13 -0 .2 3 -0 .0 9 SB/E 0.16 -0 .1 3 0.0 -3.17 O.U -0 .0 4 0.03 SE/E -0.7 5 -0 .0 2 0. 0 0.34 0.03 0 . 0 0 O.OB APPENDIX D COMPONENTS OF ESTI MAT ED COEFF ICI ENT S - AUTOS IPROPQRTION) BO B1 B2 B3 84 85 B6

ACCELERATOR AB/I E T) 0. 09 0.56 0.23 -1.03 0.91 0.0 1.45 S3/I E T) -0 .3 5 -0 . 56 4.52 2.03 -0.19 0.0 -0 .3 2 SE/( E T) 0.26 I. 00 -3.75 0.0 0.26 0.0 -0 .1 3

EXPECTED PROFITS AB/( E T) 0.12 1. 54 0.59 -0.63 -0.32 0.0 0.78 SB/I E T) 0.13 -0 .1 4 0.04 1.68 0.36 0.0 0.17 SE/IE T) -0 .2 5 -0 .4 0 0.37 0.0 0.96 0.0 0.05 _ - - — LIQUIDITY AB/t E T) 0.06 0.0 1.45 -0.89 0.28 0.0 0.65 SB/IE T) -0 .0 4 0.0 1.32 1.39 0.27 0.0 0.15 SE/IE T) -0 .0 3 0. 0 -1 .7 7 0.0 0.45 0.0 0.20

neoclassical 1 AB/I E T) -0 .1 6 1.57 -0.96 0.0 3.01 4.14 -0 .7 2 SB/l E T) 0.45 -0 .2 7 1.96 0.0 -2.19 -3 .2 5 1.43 SE/IE T) -0 .2 9 -0 .3 0 0. 0 0.0 0.18 O .U 0.29

NEOCLASSICAL 11 AB/(E T) 0.27 0. 87 -1 .3 9 0.0 3.07 -5 .1 3 0.29 SB/IE T> 0.34 0. 13 2.41 0.0 -2.3 5 6. 13 0.76 ro < jj SE/IE T) -0 .6 0 0. 0 -0.02 0.0 0.27 0.0 -0 .0 4 ro COMPONENTS OE ESTIMATED COEFFI C I ENTS -CHEMICALS {PROPORTION) BO B1 B2 B3 B4 B5 B6

ACCELERATOR AB/C E-T) 0.09 0.13 0.48 0.19 -0.27 1.51 -1 .3 3 SB/C E-T) 0.05 0.59 0.30 0.81 0.43 -1.03 - 0.86 SE/(E-T) ■0*14 0.23 0.21 0.0 0.79 0.52 3.19

EXPFCTED PROFITS AB/IE-T J 0. 33 -2 . 86 1.40 2. 19 - 0.12 1.42 0.26 SB/t E-T) 0.19 5.71 3.07 -1.19 0.15 -0 .8 4 -0 .2 9 SE/CE-T) 0.14 -1. 85 -3 .4 7 0.0 0.97 0.42 1.03

LI OUIOITY AB/CE-T) 0.06 2.63 1.62 -0.40 -0.32 0.72 -0 .3 9 SB/C E-T) 0.03 1.63 0.35 1.40 0.25 -0.17 - 0.12 SE/C E-T) •0.09 0.0 -0.97 0.0 1.07 0.44 1.51

NEOCLASSICAL I AB/( E-T) 0.28 0.19 1.56 0.0 0.18 1.39 0.82 SB/CE-T) 0.19 1.19 -1.93 0.0 -0.73 - 0 . 19 -1 .2 7 SE/CE-T) ■0.09 0.0 1.42 0.0 1.55 - 0.20 1.45

NEOCLASSICAL I I AB/CE-T) 0. 13 0.11 0.91 1.83 0.40 -0.36 0.45 SB/CF-T) 0.00 0.83 -0.24 -0.83 -0.50 -2.39 0.04 233 SE/CE-T) 0.13 0.05 0.33 0.0 1.10 3.75 0 .5 1 COMPONENTS OF ESTIMATED COEFFICIENTS -ELECTRICAL ( PROPOP.T I ON) BO fll B2 83 B4 B5 B6

ACCELERATOR A3 /I E-T) -1 .4 3 0. 30 -0 .7 8 0.0 0.17 0.95 1.88 S3/(E-T) 3.20 0. 45 0.91 0.0 0.01 -0 .3 8 -1 .7 5 SE / (E-T) -1 .7 9 0.26 0. 87 0.0 0.82 0.43 0.87

EXPECTED PROFITS AB/tE-T) 0.07 -25. 13 0.75 -1.09 0.70 -3 .7 5 0.22 S 3 /(E -T ) 0.13 3 8. 75 -0.41 2.09 0.34 -1 .3 3 0.40 SE/CE-T) -0 .2 0 -12.62 0.66 0.0 -0 .0 3 6.07 0.37 - .. ■ - LIQUIDITY AB/(E-T) 0.02 -0.71 0.0 0.0 0.50 0.57 -0 .5 0 SB/CE-T) 0. 05 1.71 -0.42 0.0 0.26 0.19 0.57 SE/tE-T) -0 .0 8 0. 0 1.42 0.0 0.23 0.24 0.92

NEOCLASSICAL I AB/C E-T) -0 .0 1 2. 08 -1.0 0 0.55 -0 .0 3 0.04 -0 .2 2 SB/*E-T) 0.28 -1.1 1 2. 00 -0.78 0.44 0.68 0.79 SE/CE-T) -0 .2 8 0. 03 0.0 1.23 0.59 0.29 0.43

NEOCLASSICAL II AB/C E-T) -0 .3 9 1.27 -0 . 96 -0.93 -0.08 -0.01 -0 .5 8 SB/(E-T) -0 .4 9 -0 . 66 1. 96 1.93 0.69 0.79 1.92 SE/IE-T) 0.88 0.38 0.0 0.0 0.39 0.22 -0 .3 4 COMPONENTS OF ESTIMATED COEFF ICI ENT S-OIL CPROPORTION) BO B1 B1 82 8383 84 B5 86

ACCELERATOR AB/C E-T) O.Ol 0.020. 02 3.373. 37 2.IB2.18 2.03 -0.90 0.04 SB/CE-T) 0.29 -0.330.33 -3 -3.20 .2 0 -1.6 -1.65 5 -2.15 1.90 0.03 SE/CE-T) 0.30 1. 1.31 31 0.83 0.83 0.47 0.47 1.12 0.0 0.93

EXPFCTED PROFITS AB/C E-T) 0.07 -0.130. 13 -3.4 -3.42 2 -3.44 -3.44 0.78 0.48 0.54 S8/CE-T) 0.13 1.131. 13 4.424.42 16.01 16.01 -0.29 -0.09 0.03 S E /lE -T ) 0.20 0.00.0 0.00.0 -11.57 0.52 0.61 0.43

LIQUIDITY AB/CE-T) 0.22 0. 0.07 07 0.46 0.46 0.40 0.40 1.24 -1.46 0.48 SB/CE-T) >0.07 -0.400. 40 -0 -0.29 .2 9 0.60 0.60 -0.64 2.46 0.13 SE/C E-T) >0.17 1.33 1.33 0.820. 82 0.00.0 0.40 0.0 0.39

NEOCLASSICAL 1 AB/CE-T) ■0.14 0. 0.97 97 6.33 6.33 0.39 0.39 1.02 0.75 0.78 SB/CE-T) 0.21 0. 0.15 15 -5.33 -5.33 0.50 0.50 -0.61 -0.39 -0.09 SE/CE-T) •0.06 -0.120.12 0.0 0.0 0.120.12 0.59 0.65 0.30

neoclassicalNEOCLASSICAL 1 I II AB/CE-T) ■0.05 0.39 1.361.36 0.680.68 0.72 0.51 _ 0.34 SB/CE-T) 0.13 0.61 -0 -0.70 . 70 -0.25 -0.25 -0.17 0.10 0.27 ^ SE/CE-T) ■0.09 0.0 0.33 0.33 0.57 0.57 0.45 0.39 0.39 $ Z OMPQNE NTS OF ESTIMATED COEFF I Cl ENTS-RUBBER (PROPORTION) BO B1 32 B3 34 R5 36

ACCELERATOR A3/I E ■T) -0.11 0.30 -1.63 0.0 0.33 0.20 0.40 SB/l E T) -0.28 -8. 96 2.63 0.0 -o .u 0.07 -0*65 SE/IE T) 0.38 0.66 0.0 0.0 0.73 0.73 1.25

EXPECTED PROFITS AB/IE T) -0.05 8.43 -0.46 1.60 0.04 0.61 -0.96 SB/I E T) 0.07 -1 0. 63 L • 46 -0.60 -0.11 -0.18 0.29 SE/IE T) -0 .0 2 3.21 0.0 0.0 1.07 0.57 1.67

LIQUIDITY AB/IE T) 0.03 1.06 1.33 -4.35 -0.20 1.22 0.16 SB/IE T) -0.05 -0.71 -0.77 3.21 0.02 -0.22 -0.13 SE/IE T) 0.02 0.65 0.39 2.15 1.18 0.0 0.96

NEOCLASSICAL I ...... AR/l E T) 0.10 3.56 -1.64 -1.50 1.24 1.94 0.17 SB/I E T) -0.0 1 -3 . 02 2.64 2.60 -1.14 -1.61 -0.04 SE/IE T) -0.10 0.46 0.0 0.0 0.90 0.67 0.87

NEOCLASSICAL I I AB/IE ■T) 0.06 567.92 . -1.21 -0.55 0.39_ 1.79 _ -0 .0 0 SB/IE •T) 0.03 -488.00 2.21 1.55 -0.36 -1.61 0.05 236 SE/C E •T) -0.09 -73.92 0.0 0.0 0.96 0.82 0.95

\ COMPONENTS OF ESTIMATED COEFFICIENTS -STEEL CPROPORTION) BO B1 B2 B3 B4 B5 B6

ACCELERATOR AS/IE-TI 0.13 0.63 2.02 6.50 3.25 0.37 0.24 S3/C E-T) •0.02 0.11 -1.26 -5.50 -2.96 0.63 4.40 SE/t F-T J ■0.11 0.26 0.24 0.0 0.70 0.0 -3.65

EXPECTED PROFITS AB/lE- 1 ) 0.14 -0.07 -4.80 1.84 0.27 -97.29 1.09 SB/( E-T) ■0.12 -1.60 5.80 -1.81 0.25 61.01 1.26 SE/C E-T) •0.01 2. 67 0.0 0.96 0.49 37.28 -1.35

LIQUIDITY AB/CE-T) 0.17 0.66 0.75 0.00 0.62 -3.28 0.57 S3/C E-T) 0.00 -0.87 -0.14 1.00 -0.82 1.02 0.14 SE/C E-T) -0.17 1.21 0.39 0.0 1.19 3.26 0.29

NEOCLASSICAL I AB / C E- T) 0.07 0.82 -3.42 5.97 0.40 0.73 -1.40 SB/CE-T) 0.02 -1.16 4.42 -5.94 0.51 -0.19 0.38 SE/(E-T) *0.09 1.34 0.0 0.98 0.09 0.46 2.03

NEOCLASSICAL I I AB/C E-T) 0.10 3.04 -1.73 6.45 0.48 0.85 -3.47 SB/C E-T) 0.03 -1.78 2.73 -6.10 0.43 0.16 1.21 rvj SE/CE-T) -0.13 -0.26 0.0 0.65 0.09 -0.01 3.26 ^ APPENDIX E Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical I theory.

Armco Steel

OLS -.00428 -.01735 .72151 -.84141 .18158 (.00535) (.15935) (.15424) (.02925)

D-TSLS -.00872 -.00467 .17517 (.00379) (.02710)

Bethlehem Steel

OLS -.11504 .00571 .40390 -.51953 .27358 (.00419) (.23911) (.24879) (.07526)

D-TSLS -.16163 .00745 .28834 (.00478) (.05450)

Chrysler

OLS .03644 .00518 .65951 -.33999 .26663 (.00298) (.23690) (.30090) (.09444)

D-TSLS .04968 .00292 .36677 (.00212) (.13641)

ro VOCO Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical I theory.

Continental Oil

OLS ■.04737 .01128 .29397 -.35826 .23906 (.00556) (.24944) (.27908) (.04133)

D-TSLS -.05417 .01121 .23769 (.00522) (.02446)

Dow Chemical

OLS .00861 .01233 .30853 -.45668 .19581 (.01666) (.24549) (.24088) (.06141)

D-TSLS .00291 .01077 .18517 (.01390) (.05783)

Dupont

OLS .04349 .00342 -.55163 .29151 (.00520) (.27540) (.04919)

D-TSLS -.09083 .00759 .22483 (.00521) (.02666) Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical I theory.

Eastman Kodak

OLS .04429 .00837 .41591 .30377 (.00285) (.25208) (.07189)

D-TSLS -.03869 .00708 .47993 (.00267 (.03859)

Firestone

OLS -.03090 .00401 -.00741 .34634 (.00320) (.00312) (.03671)

D-TSLS -.03294 .00515 -.00930 .35333 (.00400) (.00386) (.04047)

General Electric

OLS .02544 .00742 .00411 1.13785 -.44717 .13470 (.00353) (.00353) (.22945) (.27930) (.07150)

D-TSLS .04041 .00457 .00763 .39070 (.00290) (.00339) (.11905) Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical I theory.

General Motors

OLS .20558 .06170 .05099 .20808 (.01007) (.01007) (.02294)

D-TSLS .19578 .06295 .05168 .21000 (.01090) (.01089) (.02700)

General Tire

OLS .00045 .00666 .00288 .22421 (.00208) (.00253) (.02332)

D-TSLS .00076 .00748 .00320 .22174 (.00227) (.00260) (.02555)

Goodri ch

OLS .02552 .00542 .00349 .32295 (.00301) (.00306) (.03984)

D-TSLS -.02622 .00558 .00372 .32590 (.00338) (.00344) (.04624) Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical I theory.

Goodyear Tire and Rubber

OLS -.03284 .00304 -.00256 .30575 (.00238) (.00232) (.03412)

D-TSLS -.03090 .00314 -.00218 .30120 (.00268) (.00259) (.03910)

Gulf

OLS .04903 .03036 .02433 .77765 -.53087 .16374 (.01080) (.01201) (.24718) (.28677) (.02751)

D-TSLS ,05511 .02271 .01234 .23393 (.00701) (.00680) (.03513)

Inland Steel

OLS .00441 .00804 .66791 -.44924 .10743 (.00756) (.23251) (.24047) (.03348)

D-TSLS .01128 .00278 .12395 (.00587) (.04336) Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical I theory.

X3 X4 'i

Monsanto

OLS .01628 .00374 .66702 .05106 (.00225) (.25111) (.03361)

D-TSLS .02516 .00178 .10495 (.00177) (.05991)

National Steel

OLS .02232 .01792 .28420 -.40706 .09248 (.01167) (.19178) (.24334) (.03815)

D-TSLS .04017 .01197 .05426 (.01012) (.03943)

Republic Steel

OLS .00686 .01030 .39488 -.69760 .14444 (.00542) (.18649) (.18480) (.05459)

D-TSLS .01563 .00540 .12567 (.00404) (.05099) Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical I theory.

Shell

OLS .02363 .02655 .02488 .91545 -.49581 .15625 (.01137) (.01136) (.22276) (.23027) (.03533)

D-TSLS .01130 .02538 .00685 .32455 (.00677) (.00600) (.04644)

Standard Oil of California

OLS .03067 .01905 .11645 (.00852) (.01326)

D-TSLS .02769 .02040 .11739 (.00891) (.01552)

Standard Oil of Indiana

OLS -.18132 .03250 .01570 .01827 .21547 (.01009) (.00983) (.00902) (.03210)

D-TSLS -.18457 .03836 .01202 .01808 .21681 (.01162) (.01042) (.00995) (.03879) Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical I theory.

Standard Oil of New Jersey

OLS -.04238 .00819 -.00687 .00725 .87409 -.58600 .12356 (.00543) (.00547) (.00610) (.25015) (.29316) (.02485)

D-TSLS -.04277 .01010 .00134 .00044 .17615 (.00645) (.00712) (.00583) (.03584)

Texaco

OLS .01433 .02590 .01760 .27248 .16197 (.01364) (.01386) (.25235) (.02449)

D-TSLS -.02181 .02107 .01684 .22537 (.01407) (.01443) (.03270)

Union Carbide

OLS .00943 .00877 .48007 -.44931 .17150 (.00647) (.26990) (.30311) (.04928)

D-TSLS .00765 .00635 .18366 (.00637) (.05111)

ro -t*CT» Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical I theory.

Uni royal

OLS .03769 .00094 .81663 -.78696 .35186 (.00174) (.23790) (.35489) (.10368)

D-TSLS -.05656 -.00009 .46745 (.00137) (.06303)

U.S. Steel

OLS .26825 .01581 .01758 .55520 -.34237 .01165 (.00725) (.00724) (.19921) (.20232) (.06458)

D-TSLS .46509 .00831 .01277 -.07456 (.00662) (.00653) (.10476)

Mestinghouse Electric

OLS -.00523 -.00304 1.14659 -.42998 .16949 (.00250) (.27266) (.28787) (.06403)

D-TSLS .02057 .00080 .31101 (.00178) (.21965) Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical I theory.

Autos

OLS .16859 .02036 .73056 -.41217 .25137 (.01059) (.22386) (.29124) (.05124)

D-TSLS .25977 .01176 .34862 (.00712) (.08099)

Chemicals

OLS .00770 .00543 .58058 -.41645 .18107 (.00283) (.22040) (.21259) (.03011)

D-TSLS -.04225 .00365 .23292 (.00243) (.02288)

Electrical

OLS -.01283 .00555 -.00399 1.41056 -.71959 .18086 (.00288) (.00316) (.19203) (.20274) (.05545)

D-TSLS .07514 .00218 .00067 .31814 (.00217) (.00234) (.15012) Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical I theory.

Oil

OLS -.31229 .01251 .00943 .86332 -.41137 .14579 (.00624) (.00604) (.22587) (.25925) (.01635)

D-TSLS -.21368 .01256 .26135 (.00449) (.02508)

Rubber

OLS -.09147 .00236 .61025 -.41101 .27575 (.00179) (.24769) (.25265) (.04497)

D-TSLS -.09991 .00146 .35800 (.00154) (.02892)

Steel

OLS -.12071 .01061 -.00794 .65271 -.39958 .14547 (.00497) (.00496) (.20136) (.21050) (.03902)

D-TSLS -.02310 .00419 -.00711 .17450 (.00421) (.00402) (.06438) Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical II theory.

Armco Steel

OLS .00333 -.00972 -.02536 .77569 -.97949 .19307 (.00642) (.00658) (.15268) (.15531) (.02769) D-TSLS .00743 -.01165 -.00515 .17244 (.00396) (.00403) (.02207)

Bethlehem Steel

OLS .11119 .01106 .42518 -.53157 .26695 (.00644) (.23265) (.23649) (.07299)

D-TSLS .15868 .01359 .28484 (.00690) (.05307)

Chrysler

OLS .03840 .00320 .67543 -.32634 .26143 (.00195) (.24242) (.30307) (.09486)

D-TSLS .05289 .00202 .36924 (.00146) (.14082)

oir\a o Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical II theory.

Continental Oil

OLS -.05382 .03742 .23202 (.01141) (.02104)

D-TSLS -.05583 .03865 .23349 (.01251) (.02433)

Dow Chemical

OLS .00929 .06028 -.06462 .38015 -.54909 .19558 (.02959) (.03909) (.21679) (.19812) (.05856)

D-TSLS .00813 .03894 -.05279 .18041 (.02500) (.03192) (.00595)

Dupont

OLS -.05416 .01838 -.47990 .29446 (.01728) (.28230) (.04676)

D-TSLS -.10861 .03002 .23768 (.01456) (.02722)

PO CJl Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical II theory. .

*1

Eastman Kodak

OLS ■.03903 .00852 1.01424 -.90323 .29706 (.01066) (.30982) (.38560) (.08151)

D-TSLS -.04858 .00547 .45103 (.00509) (.04978)

Firestone

OLS -.03186 -.01317 .35535 (.00700) (.03908)

D-TSLS -.03038 -.01391 .35147 (.00815) (.04518)

General Electric

OLS .00050 .01333 .01154 1.02933 -.32891 .15267 (.00542) (.00682) (.24669) (.28519) (.06738)

D-TSLS .02260 .00816 .01670 .45866 (.00451) (.00580) (.11409)

tnro ro Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical II theory.

General Motors

OLS .10975 .07269 .08169 .22371 (.01633) (.01644) (.02505)

D-TSLS .15449 .06963 .08066 .21282 (.01764) (.01730) (.03036)

General Tire

OLS -.00072 .00994 .00392 .00405 .22232 (.00267) (.00335) (.00328) (.02225)

D-TSLS -.00011 .01003 .00364 .00382 .21891 (.00291) (.00352) (.00348) (.02512)

Goodrich

OLS -.02722 .01219 .00560 .30506 -.50907 .33074 (.00555) (.00481) (.27887) (.33658) (.07261)

D-TSLS -.03215 .01131 .00801 .30118 (.00480) (.00491) (.04192)

(S3 (71 CO Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors, Neoclassical II theory.

x, X2 x3 X4 x5 x6 x7

Goodyear Tire and Rubber

OLS -.03777 .01120 .01119 .29705 -.46936 .30985 (.00583) (.00558) (.26440) (.27316) (.06221)

D-TSLS -.04402 .01020 .00542 .29053 (.00479) (.00472) (.03290)

Gulf

OLS -.05730 .02983 .02710 .82995 -.52075 .16427 (.01335) (.01484) (.26641) (.30595) (.02933)

D-TSLS -.05586 .02137 .01541 .25118 (.00787) (.00771) (.03903)

Inland Steel

OLS .00533 .00763 .65854 -.43777 .10467 (.00962) (.23861) (.24381) (.03377)

D-TSLS .01157 .00353 .12273 (.00748) (.04331)

UlPO Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical II theory.

1

Monsanto

OLS .00318 .02638 .01657 .01658 .11744 (.00621) (.00562) (.01049) (.02060)

D-TSLS -.00109 .02599 .01640 .01648 .11540 (.00681) (.00610) (.01123) (.02340)

National Steel

OLS .02557 .02910 .01458 -.28608 .08450 (.01269) (.00985) (.22177) (.03679)

D-TSLS .03828 .03201 .01502 .04404 (.01166) (.01161) (.03314)

Republic Steel

OLS -.00101 .01136 .39704 -.69016 .13263 (.00611) (.18725) (.18546) (.05982)

D-TSLS -.01206 .00618 .12081 (.00473) (.05096)

ro tn ( j i Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassi c a l. 11 theory.

Shell

OLS .03321 .03340 .04140 1.04812 -.59047 .14290 (.01495) (.01835) (.22276) (.23825) (.03341) D-TSLS .01493 .02806 .00823 .33814 (.00733) (.00699) (.08995)

Standard Oil of California

OLS .03216 .01936 .11570 (.01138) (.01397)

D-TSLS .03066 .02079 .11596 (.01197) (.01645)

Standard Oil of Indiana

OLS .12579 .02715 .19456 (.01016) (.03165)

D-TSLS .10875 .02810 .18647 (.01136) (.03828)

ro inCTl Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical II theory.

Standard Oil of New Jersey

OLS .00991 .00955 -.01848 .01846 1.05416 -.76950 .11327 (.00702) (.00733) (.00899) (.22315) (.27019) (.02129)

D-TSLS -.01534 .01495 -.00296 .00168 .16878 (.00922) (.01124) (.00829) (.03212)

Texaco

OLS -.02571 .02019 .33230 .17218 (.01870) (.25524) (.02476)

D-TSLS -.01194 .01644 .24195 (.01639) (.03515)

Union Carbide

OLS .00406 .06769 .32038 -.29692 .15392 (.01565) (.18706) (.21781) (.03486)

D-TSLS -.01666 .07084 .18325 (.01639) (.03481)

ro cn " 4 Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical II theory.

Uni royal

OLS -.03574 .00383 .83219 -.79109 .33998 (.00547) (.23819) (.34824) (.10299)

D-TSLS -.05678 -.00002 .47379 (.00377) (.06578)

U.S. Steel

OLS .30020 .02102 .01910 .44712 -.29332 -.00202 (.00825) (.00889) (.21623) (.20614) (.06452)

D-TSLS .46020 .01530 .01894 -.06948 (.00794) (.00791) (.09539)

Westinghouse Electric

OLS -.00366 -.00537 1.10880 -.36863 .16651 (.00409) (.26490) (.29099) (.06369)

D-TSLS .02229 .00127 .31520 (.00272) (.22428)

uiro CO Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical II theory. .

*i h h h h x 6 x7

Autos

OLS .26165 .00364 .62713 .21050 (.00643) (.21066) (.04741)

D-TSLS .45739 .00239 .39085 (.00552) (.12476)

Chemicals

OLS -.03147 .01688 .01911 .65438 -.44481 .17077 (.01197) (.01253) (.24629) (.21231) (.03121)

D-TSLS -.06202 .01209 .02454 .23404 (.01202) (.01233) (.02379)

Electrical

OLS -.00419 .01183 1.33478 -.70030 .16453 (.00450) (.18056) (.20267) (.05394)

D-TSLS .04440 .00445 .35351 (.00307) (.12546)

PO CJ1 VO Ordinary least-squares and Durbin two-stage least-squares regression coefficients and their standard errors. Neoclassical II theory.

'1

Oil

OLS .07271 -.01731 .02232 1.26737 -.75645 .13165 (.00955) (.01070) (.22772) (.26575) (.01560)

D-TSLS -.00672 -.01891 -.00176 .27569 (.00596) (.00595) (.02775)

Rubber

OLS -.09232 .00430 .68785 -.49465 .27471 (.00396) (.24679) (.25580) (.04570)

D-TSLS -.10663 .00365 .36242 (.00268) (.02875)

Steel

OLS -.04443 .01266 -.00879 .61334 -.42142 .13330 (.00659) (.00734) (.20458) (.21547) (.03941)

D-TSLS .02016 .00577 -.00646 .15439 (.00574) (.00565) (.06136)

ro o APPENDIX F Components of estimated coefficients (D-TSLS) (proportion)

AUTOS

NEOCLASSICAL I T/E 0*94 2. 80 0.0 0.0 0.45 0.41 0.83 AB/E 0.04 1. 88 0. 0 0.0 1.76 2.42 -0 .1 3 SB/E - 0 . 13 0.29 0.0 0.0 -1.29 -1 .9 0 0.06 se/e 0.08 0.21 0.0 0.0 0.07 0.06 0.24

NEOCLASSICAL 11 T/E 0.45 0. 0 16. 87 0.0 0.54 0.0 0.74 AB/E 0.06 0. 0 2 0. 64 0.0 1.45 0.0 -0 .1 3 SB/E 0.08 0. 0 -3 6.26 0.0 -1.12 0.0 -0 .0 6 SE/E -0 . 14 0. 0 -0 .2 6 0.0 0.13 0.0 0.44

CHEMICALS

NEOCLASSICAL I T/E 2.22 0. 0 1.01 0.0 0.64 0.70 1.01 AB/E - 0.21 0. 0 1.90 0.0 0.08 0.35 -0 .1 5 SB/E 0.15 0. 0 -1 .9 4 0.0 -0 .2 2 -0 .0 6 . . 0.09 SE/E 0.06 0.0 0.04 0.0 0.49 0.01 0.05

NEOCLASSICAL I I T/E 2.69 2.25 0.42 0.0 0.52 1.00 I .00 AB/E - 0 . 17 0.11 0.52 0.0 0.13 - 0.11 - 0 . 1 6 SB/E 0.01 0.95 -0.16 0.0 -0.19 - 0.11 - 0 . 0 1 r\3 SE/E 0. 16 0.40 0.21 0.0 0.54 0.22 0 .1 7 & Components of estimated coefficients (D-TSLS) (proportion)

LLttTRICAL N6 CC LA SSI C AL I T/E 0,81 1.05 0.0 0.60 0.91 0.61 1 . 1 0

AB/E 0.00 0. 96 0.0 0.14 0.01 - 0.01 0.04 SB/E •0.12 -0.30 0.0 -0.15 0.12 0.37 0.07 SE/E 0.12 -0 . 70 0.0 0.44 0.06 0.03 - 0.22

NEOCLASSICAL I I T/E 1.01 1.06 0.0 0.0 0.80 0.50 1 .0 9 AB/E 0.02 0.62 0 . 0 0.0 0.00 -0 .0 4 0.07 SB/E 0. 13 -0.31 0.0 0.0 0.18 0.45 0.01 SE/E 0. 14 -0 .3 7 0 . 0 0.0 0.01 0.09 -0 .1 7

OIL

NEOCLASSICAL I T/E 1.49 1.70 0.0 0.0 0.47 0.59 0.84 AB/E 0.03 -0.69 0.0 0.0 0.57 0.42 -0 .0 7 SB/E 0.00 -0 . 15 0. 0 0.0 -0 .2 4 -0 .0 5 - 0.01 SE/E 0.03 0.14 0.0 0.0 0.20 0.03 0.24

NEOCLASSICAL I I T/E 30.08 0.0 0.02 -2.06 0.33 0.29 0.80 AB/E 1.08 0.0 0.41 0.12 0.51 0.41 -0 .0 4 S9/E -3 .1 6 0.0 -0.17 -0.71 - 0.11 0.09 -0 .0 4 Ciro SE/E 2.07 0.0 0.74 3.65 0.27 0.20 0.29 to Components of estimated coefficients (D-TSLS) (proportion)

RUBBER

NEOCLASSICAL I T/E 1*46 1.45 0. 0 0.0 0.27 0.39 0.93 AB/E -0 .0 6 1.03 0.0 . 0.0 0.90 0.90 -0 .0 6 SB/E 0.00 -0 . 83 0.0 0.0 -0.79 -0 .6 6 0.01 SE/E 0.06 -0 .6 0 0.0 0.0 0.62 0.38 0.12

NEOCLASSICAL I I T/E 1. 53 1. 11 0. 0 0.0 0.42 0.72 0.90 AB/E -0 .0 5 1.75 0. 0 0.0 0.24 0.47 -0 .0 5 SB/E -0.01 -1 .5 0 0.0 0.0 -0.2 2 -0 .3 9 0.00 SE/E 0. 06 -0 .3 6 0.0 0.0 0.56 0.20 0.15

STEEL

NEOCLASSICAL I T/E -14.31 1.43 0.0 o . u 0.77 1.36 0.66 AB/E 1. 85 0.26 0.0 3.71 O.U -0 .2 3 -0 .0 8 SB/E 0.18 -0 .1 7 0.0 -3 .5 0 0.08 0.05 .. 0.01 SE/E -2 .0 2 -0.51 0.0 0.67 0.03 - 0 . 18 0.40

NEOCLASSICAL I I T/E 16.46 1.70 0.0 0.13 0.73 1.27 0.73 AB/E -2 .0 5 0.31 0.0 4. 10 0.18 - 0 . 18 -0 .0 7 SB/E .... -0 .3 5 -0 .3 0 0.0 -3 .3 9 0.10 -0 .0 4 _ 0.03 SE/E 2.40 -0 . 72 0.0 0.16 -0.0 1 -0 .0 5 0.31 APPENDIX 6 Constrained regression coefficients and their standard errors, 1949-1968.

Model 1

Armco Steel

Neoclassical 1 .00466 .01381 .01178 .72151 -.13014 .09429 (.00850) (.00857) (.03289)

Neoclassical I I .00984 .01176 .01515 .77569 -.15042 .07657 (.01088) (.01129) (.03458)

Bethlehem Steel

Neoclassical I ■.05394 .00806 .40390 -.04078 .19564 (.00419) (.00561)

Neoclassical I I ■.04995 .01395 .42518 -.04519 .18781 (.00663) (.05492)

Chrysler

Neoclassical I .05216 .00422 .65951 -.10874 .21667 (.00255) (.06955)

Neoclassical II .05271 .00266 .67543 -.11405 .21561 (.00166) (.06996) 266 Constrained regression coefficients and th e ir standard errors, 1949-1968.

Model 1

Continental Oil

Neoclassical 1 .03987 .01001 .29397 -.02160 .21055 (.00521) (.02234)

Neoclassical I I .05382 .03742 .23202 (.01141) (.02104)

Dow Chemical

Neoclassical I .00699 .02361 .30853 .02380 .16579 (.01544) (.05813)

Neoclassical I I .01237 .05217 .38015 .03613 .13485 (.03340) (.06097)

Dupont

Neoclassical I .01658 .00740 .22970 (.00521) (.04156)

Neoclassical II .04145 .03217 .24908 (.01608) (.04046)

ro o >4 Constrained regression coefficients and their standard errors, 1949-1968.

Model 1

Eastman Kodak

Neoclassical I -.04429 .00837 .41591 .30377 (.00285) (.25208) (.07189)

Neoclassical I I -.01789 .00585 1.03757 -.26914 .20619 (.00481) (.02995)

Firestone Tire and Rubber

Neoclassical I -.02724 .00455 .00427 .32619 (.00414) (.00359) (.04009)

Neoclassical I I -.02859 .01045 .01087 .32282 (.00788) (.00734) (.03983)

General Electric

Neoclassical I .04347 .00780 .00474 1.13785 -.32368 .10303 (.00327) (.00278) (.04633)

Neoclassical II .00855 .01383 .01201 1.02933 -.26488 .13736 (.00482) (.00476) (.04291)

ro m oo Constrained regression coefficients and their standard errors, 1949-1968.

Model '1

General Motors

Neoclassical I .20558 .06170 .05099 .20808 (.01007) (.01007) (.02294)

Neoclassical I I .10975 .07269 .08169 .22371 (.01633) (.01644) (.02505)

General Tire and Rubber

Neoclassical 1 .00097 .00636 .22338 (.00208) (.02351)

Neoclassical I I -.00072 .00994 .00392 .00405 .22232 (.00267) (.00335) (.00328) (.02225)

Goodrich

Neoclassical I -.02552 .00542 .00349 .32295 (.00301) (.00306) (.03984)

Neoclassical II -.01934 .00758 .00517 .30506 -.02327 .27098 (.00455) (.00472) (.03965)

CT>ro ic Constrained regression coefficients and their standard errors, 1949-1968,

Model 1

Goodyear Tire and Rubber

Neoclassical I -.03147 .00464 .00258 .29769 (.00240) (.00237) (.03413)

Neoclassical I I -.02323 .01076 .00507 .01068 .29705 -.02206 .24295 (.00588) (.00615) (.00597) (.03494)

Gulf Oil

Neoclassical I -.05411 .02316 .77765 -.15118 .15420 (.01056) (.02533)

Neoclassical I I -.05439 .02259 .82995 -.17220 .15322 (.01287) (.02640)

Inland Steel

Neoclassical I .00601 .00573 .66791 -.11153 .08508 (.00747) (.02989)

Neoclassical II .00677 .00460 .65854 -.10842 .08358 (.00931) (.03009) Constrained regression coefficients and their standard errors, 1949-1968.

Model

Monsanto

Neoclassical I .01628 .00374 .66702 .05106 (.00225) (.25111) (.03361)

Neoclassical I I -.00318 .02638 .01657 .01658 .11744 (.00621) (.00562) (.01049) (.02060)

National Steel

Neoclassical I .01884 .02951 .28420 -.02019 .08107 (.00878) (.03844)

Neoclassical I I .02315 .03962 .01190 .07526 (.00993) (.00983) (.03683)

Republic Steel

Neoclassical I .00929 .00940 .39488 -.03898 .09236 (.00695) {.06403)

Neoclassical II .01454 .01083 .39704 -.03941 .08184 (.00778) (.06399) Constrained regression coefficients and their standard errors, 1949-1968.

Model 1

Shell Oil

Neoclassical I .01991 .02944 .91545 .20951 .15978 (.01225) (.02861)

Neoclassical I I .02482 .04778 1.04812 .27464 .14357 (.01541) (.02737)

Standard Oil of California

Neoclassical I .03067 .01905 .11645 (.00852) (.01326)

Neoclassical I I .03216 .01936 .11570 (.01138) (.01397)

Standard Oil of Indiana

Neoclassical I -.18132 .03250 .01570 .01827 .21547 (.01009) (.00983) (.00902) (.03210)

Neoclassical II -.12579 .02715 .19456 (.01016) (.03165) Constrained regression coefficients and th e ir standard errors, 1949-1968.

Model

Standard Oil of New Jersey

Neoclassical I -.15545 .00757 .87409 -.19101 .12859 (.00529) (.02218)

Neoclassical I I -.13815 .00830 1.05416 -.27781 .12127 (.00858) (.02351)

Texaco

Neoclassical I .00578 .02977 .01203 .02454 .15868 (.01405) (.01326) (.01346) (.02628)

Neoclassical I I -.01632 .02371 .18374 (.01888) (.02359)

Union Carbide

Neoclassical 1 .00139 .01055 .48007 -.05762 .16034 (.00619) (.04139)

Neoclassical II -.00132 .07149 .32038 -.02566 .14485 (.01531) (.03015) Constrained regression coefficients and their standard errors, .1949-1968.

Model '1

Uni royal

Neoclassical I -.00919 .00005 .81663 -.16672 .19428 (.00172) (.05062)

Neoclassical I I -.00762 .00166 .83219 -.17313 .18543 (.00551) (.05450)

U.S. Steel

Neoclassical I .26121 .01766 .01870 .55520 -.07706 .01011 (.00705) (.00689) (.06482)

Neoclassical I I .26449 .02232 .01766 -.00404 .44712 -.04998 .00824 (.00789) (.00900) (.00853) (.07034)

Westinghouse Electric

Neoclassical 1 -.00509 .00199 1 .14659 -.32867 .16041 (.00262) (.06027)

Neoclassical II -.00698 .00425 1.10880 -.30736 .16507 . (.00392) (.05908) Constrained regression coefficients and their standard errors, 1949-1968.

Model

Autos

Neoclassical I .22147 .01425 .73056 -.13343 .21943 (.00866) (.04079)

Neoclassical I I .26165 .00364 .62713 .21050 (.00643) (.21066) (.04741)

Chemicals

Neoclassical I .02628 .00283 .00768 .58058 -.08427 .14761 (.00259) (.00255) (.01712)

Neoclassical II -.00964 .01964 .02775 .65438 -.10705 .13747 (. 0 1 1 2 1 ) (.01128) (.01736)

Electrical

Neoclassical I .04881 .00728 1.41056 -.49742 .10638 (.00283) (.04337)

Neoclassical II .03138 .01278 1.33478 -.44541 .11878 (.0.0456) (.04215) Constrained regression coefficients and their standard errors, 1949-1968.

Model

Oil

Neoclassical I -.41804 .01223 .86332 -.18633 .14533 (.00587) (.01352)

Neoclassical 11 ■.24861 .01685 1.26737 -.40156 .12359 (.01026) (.01499)

Rubber

Neoclassical I -.06247 .00257 .61025 -.09310 .23426 (.00174) (.01989)

Neoclassical 11 -.05723 .00361 .68765 -.11822 .22556 (.00398) (.02059)

Steel

Neoclassical I .02432 .01408 .00571 .65271 -.10651 .10942 (.00546) (.00548) (.04015)

Neoclassical 11 .03537 .01650 .61334 -.09405 .10696 (.00687) (.04021) APPENDIX H COMPONENTS OF ESTIMATED COEPFIC IENTS -AUTOS (constrained) BO B1 B? 33 B4 65 B6

NEOCLASSICAL I T 0.257740 0.032960 0.025495 0.0 0.329755 -0.054370 0.212375 AB 0.011904 -0 . 020667 0.024790 0.0 1.204046 -0.930689 0.007684 C 0.0 -0.020636 0.026495 0.0 0.329755 -0.054370 -0.003041 NC 0.011904 -0.000031 -0.000715 0.0 0.374291 -0.876319 0.010725 SB 0.034722 0.003135 -0.050275 0.0 -0.861929 0.634656 0.020568 c 0.0 0.0 -0.050275 0.0 -1.201137 0.793978 0.0 NC -0.034722 0.003195 0.0 0.0 0.339208 -0.159322 0.020568 SE 0.022911 -0. 001227 0.0 0.0 0.058688 0.216972 -0.021198 E ~ 0.221470 0. 014250 0.0 0.0 0.730560 -0.133430 0.219430

NEOCLASSICAL II T 0.162460 0.037675 0.040845 0.0 0.337715 -0.057025 0.219660 AB 0.030503 -0.033260 0.052612 0.0 0.879700 -0.763304 0.007396 C 0.0 -0.033276 0.040S45 0.0 0.337715 -0.057025 0,0028 74 — n: 0. 03 05 03 0.00 0 016 0 . 0117 6 7 0 .0 0.541985 -0.706279 0.004521 SB 0.033697 -0.004415 -0.090736 0.0 -0.646338 0.820329 -0.020886 c 0.0 -0.004415 -0.094027 0.0 -0.968312 0.820329 0.0 NC 0.033697 0.0 0.003291 0.0 0.321974 0.0 -0.020386 SE 0.064020 0.0 0.000918 0.0 0.055053 0.0 0.004330 E 0.261650 0.0 0.003640 0.0 0.627130 0.0 0.210500 278 COMPONENTS OF Fr ST 1 MATEO COEFF IC I ENTS -CHEMICALS (constrained) BO Bl B2 B3 B4 95 B6

NEOCLASSICAL I T -0.036210 0.004532 0.006202 0.0 0.374305-0.016284 0.182132 AB 0.009534 0.000739 0. 006457 0.0 0.031725 0.019616 -0.016493 C ...... 0.0 0.0013 78 0.004285 0.0 0.063979 -0.019693 -0.025283 NC 0.009534 - 0.0006 39 0.002172 0 .0 -0.032253 0.039309 0.009790 SB 0.001857 -0.001149 -0.003431 0.0 -0.081706 -0.028038 -0.002691 Z 0.0 -0.001513 - 0.008722 0.0 -0.10203? -0.001298 0.0 NC 0.001857 0.000464 0.000241 0.0 0.020326 -0.026741 -0.002691 SE ____-0.011357 -0.001292 0.003501 0.0 0.256255 -0.059564 -0.015337 E 0.026280 0.002830 0.007590 0.0 0.580580-0.094270 0.147610'

NEOCLASSICAL IT T -0.051470 0.031178 0.009743 0.003316 0.347620 -0.055922 0.170482 AB 0.002836 0.005204 0.009*41 0.006949 0.077530 0.140113-0.000510 C 0.0 0.003662 0.011503 0.013264 0.053017 0.034503 -0.007065 NC 0.002836 0.001541 -0.001662 -0.006415 0.024513 0.105610 0.006555 SB 0.005218 -0.010665 -0.001809 -0.010165 -0.069821 -0.049662 -0.018918 C 0.0 -0.010993 -0.003851 -0.010165 -0.159869 -0.022044 0.0 NC 0.005218 0.00032B 0.002043 0.0 0.090047 -0.027618 -0.018918 SE -0.008054-0.006077 0.009969 0.0 0.299052-0.141579-0.013584 E -0.009640 0.019640 0.027750 0.0 0.654380 -0.107050 0.137470 ro to COMPONENTS OF ESTIMATED CQE FFICI ENT S-ELECTRICAL (constrained) 90 Bl 82 B3 B4 B5 B6

NEOCLASSICAL I T 0.03B380 0.004895 0.002370 0.0 1.142220 -0.326175 0.131720 A8 0.000412 0.000899 0.002593 0.0 -0.003812 -0.030268 -0.009088 C 0.0 0.000834 0.002370 0.0 -0.002621 0.001237 -0.010376 NC 0.000412 0.000065 0.000223 0.0 -0.001191 -0.031505 0.001287 SB -0.006403 0.000553 -0.004963 0.0 0.112751 -0.161948 0.015347 C 0.0 0.0 -0.004963 0.0 0.0 0.0 0.0 NC -0.006403 0.000553 0.0 0.0 0.112751 -0.161948 0.015347 SE_ _ 0.006021 0.000934 0.0 0.0 0.159401 0.020970 -0.031598 E 0.048310 0.007280 0.0 0.0 1.410560 -0.497420 0.106380

NEOCLASSICAL II T 0.001570 0.009040 0.006005 0.0 1.069065-0.286120 0.151215 AB -0.000095 0.00146? 0.004903 0.0 -0.005145 -0.041571 -0.001105 _ C 0.0 0.001470 0.006005 0.0 -0.022437 0.009653 -0.004960 "NC” -0.000095 -0.000016 -0.001202 0.0 0.017292 -0.051228 0.003356 SB 0.004728 0.000918-0.010803 0.0 0.1649 24-0.193774-0.001382 C 0.00*728 0.0 -0.010803 0.0 0.0 0.0 0.0 NC 0.004728 0.000818 0.0 0.0 0.164924-0.193774-0.001382 SE -0.004602 0.001460 0.0 0.0 0.105936 0.076054-0.029949 E 0.031380 0. 012780 0.0 0.0 1.334780 -0.445410 0.118780 __

ooro o COMPONENTS OF ESTIMATED COEFFICI ENTS-OIL (constrained) BO B1 B2 83 B4 35 86

NEOCLASSICAL I T - O,374390 0.021643 0.003961 0.006116 0.403709 -0.081900 0.163389 AB 0.005785 -0.007728 0.007614 0.014741 0.453129 -0.060516 -0.015295 C 0.0 -0.007504 0.009616 0.015223 0.400513-0.098678-0.013895 NC 0.005785 -0. 0002 2'-t -0. 002002 -0.000487 0.052616 0.038161 -0.001401 SB -0.002846 -0.003651 -0.011576 -0.020857 -0.172004 0.000500 -0.000068 C 0.0 0.0 -0.011576 -0.020857 -0.198923 0.039490 0.0 NC -0.002846 -0.003651 0.0 0.0 0.026919 -0.038990 -0.000068 SE -0.003304 0.001966 0.0 0.0 0.173486 -0.044413 -0.002695 ”E ~ -0.418040 0.012230 0.0 0.0 0.863320-0.186330 0.145330

NEOCLASSICAL I I T -0.331490 0.026616 0.0 0.0 0.418890 -0.103521 0.163440 AB -0.003247 -0.009943 0.0 0.0 0.592491 -0.150636 -0.011388 * C _ 0.0 -0.009988 0.0 0.0 0.568607 -0.140915 -0.013954 NC -0.003247 0.000045 0.0 0.0 " 0.023884 -0.009721 0.002566 SB 0.005107 -0.005109 0.0 0.0 -0.265460 0.033307 -0.005688 C 0.0 0.0 0.0 0.0 -0.283948 0.063364 0.0 NC 0.005107 -0.005109 0.0 0.0 0.018488 -0.030056 -0.005688 SE -0.002061 0.005286 0.0 0.0 0.521449 -0.180710 -0.022774 E -0.248610 0.016850 0.0 0.0 1.267370 -0.401560 0.123590

ro oo COMPONENTS OF ESTIMATED COE FFIC I ENT S -RUBBER (constrained) BO 31 B2 B3 B4 B5 B6

NEOCLASSICAL I T -0.092450 0.003110 0.002454 0.000303 0.163326-0.033344 0.272898 A3 -0.000103 0.002121 0.001372 0.001773 0.645691 -0.239238 0.011637 C" 0.0 0.001305 0.001145 0.000352 0.653304 -0.133376 0.014070 MC -0.000103 0.000316 0.000227 0.000921 -0.007613 -0.105362 -0.002433 SB 0.002834 -0.002317 -0.003826 -0.002581 -0.496484 0.229352 -0.022330 C 0.0 -0.001556 -0.003826 -0.002581 -0.676237 0.230270 0.0 NC 0.002834 -0.000661 0.0 0.0 0.179754 -0.000913 -0.022330 SE __ -0.002775 -0.000344 0.0 0.0 0.297717 -0.049970 -0.027946 E -0.062470 0.002570 0.0 ' 0.0 0.610250 -0.093100 0.234260

NEOCLASSICAL I I T -0.079500 0.006230 0.005483 0.004312 0.286860 -0.043692 0.248902 AB -0.000666 0.005785 0.001760 0.003985 0.200004 -0.094353 0.005928 C 0.0 0.004217 0.001395 0.002976 0.147640 -0.010777 0.008273 NC " -0.000666 0.001568 0.000364 0.001309 0.052365-0.083576 -0.002345 S3 0.001944 -0.005272 -0.007249 -0.008297 -0.061337 0.056522 -0.005082 C 0.0 -0.003775 -0.007248 -0.008297 -0.127417 0.031642 0.0 NC 0.001944 -0.001498 0.0 0.0 0.066080 0.024980 -0.005082 SE -0.001319 -0.003133 0.0 0.0 0.262123 -0.036697 -0.024188 E -0.057230 0. 003610 0.687650 -0.118220 0.225560 ro ro03 COMPONENTS OF F ST I MAT ED COEFF IC I ENTS-STEEL (constrained) BO 81 82 33 B4 85 B6

NEXLASSICAL I T 0.246070 0.014028 0.005030 0.0 0.504600 -0.069780 0.093092 AB -0.016438 -0,001790 0.003260 0.0 0.073053 0.004461 -0.005204 Z" 0. 0 - 0 . 001 7 02 0 . 0 1 2 6 8 7 0 .0 -0.016661 0.004359 -0.002279 NC -0.016438 -0.000039 -0.004427 0.0 0.0R9714 0.000102 -0.002926 SB -0.000749 -0.000970 -0.007664 0.0 -0.029392 0.024631 0.001369 C 0.0 0.0 -0.007664 0.0 0 . 0 0 . 0 0 . 0 NC - 0.000749 -0.000070 0.0 0 .0 -0.029392 0.024631 0.001369 SE 0.017170 0.001912 0.000034 0.0 0.104949 -0.065822 0.020164 E 0.024320 0.014080 0.005710 0.0 0.652710 -0.106510 0.109420

NEOCLASSICAL I I T 0.268840 0.017180 0.007452 -0.000673 0.450595 -0.065570 0.085550 AB -0.021275 -0.000529 0.004037 -0.006815 0.120995 -0.003673 0.003118 C _ 0.0 0.000080 0.009177 -0.003367 0.026189 0.007612 0.000066 NC -0.021275 -0.000603 -0.005141 -0.003449 0.094305 -0.011285 ~ 0.003052 SB -0.001194 -0.000729 -0.011433 0.007439 0.011532 -0.031653 0.001484 C 0.0 0.0 -0.011488 0.007489 -0.050716 0.006354 0.0 NC -0.001194 -0.000729 0.0 0.0 0.062249 -0.038007 0.001484 SE 0.022459 0.000573 0.0 0.0 0.030218 0.006845 0.016809 E _ , 0.035370 0.016500 0.0 0.0 0.613340-0.094050 0.106960., APPENDIX I Components of estimated coefficients (constrained) (proportion) AUTOS b4 Bo B1 NEOCLASSICAL . B3 B5 B6 T/E 1.16 2. 31 0. 0 0.0 0.45 0.41 0.97 AB/E 0. 05 -1.45 0.0 0.0 1.65 6.98 0.04 SB/E -0.16 0.22 0.0 0.0 -1. 18 -4.76 0.09 SE/E 0. 10 -0. 09 0.0 0.0 0.08 -1.63 -0.10

NEOCLASSICAL 11 T/E 0.62 0.0 11.22 0.0 0.54 0.0 1.04 AB/E 0. 12 0.0 14.45 0.0 1.40 0.0 0.04 SB/E 0.13 0.0 -24.93 0.0 -1.03 0.0 -0.10 SE/E -0. 24 0. 0 0.25 0.0 0.09 0.0 0.02

CHEMICALS

NEOCLASSICAL I T/E ■1.38 1.60 0.81 0.0 0.64 0.19 1.23 AB/E 0.36 0.26 0.84 0.0 0.05 ■0.23 0.11 SB/E 0.07 0.41 -1.10 0.0 ■0.14 0.33 0.02 SE/E ■0.43 0.46 0.46 0.0 0.44 0.71 ■ 0.10

NEOCLASSICAL II T/E 5.34 1.59 0.35 0.0 0.53 0.52 1.24 AB/E ■0.29 0.26 0.35 0.0 0.12 1.31 0.00 SB/E ■0.54 0.54 -0.07 0.0 ■0 . 11 0.46 •0.14 r\3 SE/E 0. 84 03 0.31 0.36 0.0 0.46 1.32 - 0.10 cn Components of estimated coefficients (constrained) (proportion)

ELECTRICAL

b4 B0 Bi NEOC L a2; s I c a l I B3 B5 H T/E 0. 79 0. 67 0.0 0.0 0.91 0.66 1.24 AB/E 0.01 0. 12 0.0 0.0 -0.00 0.06 -0 .0 9 SB/E -0 .1 3 0. 03 0.0 0.0 0.08 0.33 0.14 SE/E 0.12 0. 13 0.0 0.0 0. Li -0 .0 4 -0 .3 0

NEOCLASSICAL 11 T/E 0.05 0.71 0.0 0.0 0.80 0.64 1.27 AB/E -0 . 00 0. 11 0.0 0.0 -0.00 0.09 -0 .0 1 SB/E 0. 15 0. 06 0. 0 0.0 0.12 0.44 -0.0 1 SE/E -0 . 15 0. 11 0.0 0.0 0.03 -0 . 17 -0 .2 5

OIL

NEOCLASSICAL I T/E ~ 0.90 1.77 0.0 ' 0.0 0 .4 7 0.44 1.12 AB/E ■ 0.01 ■0.63 0.0 0.0 0 .5 2 0.32 ■0 . 1 1 SB/E 0.01 ■0.30 0.0 0.0 - 0.20 - 0.00 ■ 0.00 SE/E 0 . 0 1 0.16 0.0 0.0 0.20 0 .2 4 0.02

NEOCLASSICAL I I T/E 1 .3 3 1. 58 " 0. 0 0 .0 0 .3 3 0 .2 6 1 .32 AB/E 0.01 0.59 0.0 0.0 0 .4 7 0.3B 0.09 SB/E 0.02 0.30 0.0 0.0 ■0.2L -0 .0 8 0 .0 5 jss SE/E 0.01 0.31 0.0 0.0 0 .4 1 0 .4 5 ■0.18 “ Components of estimated coefficients fconstrained) [proportion) RUBBER DC

4* B5 B o B0 B1 NEOT hsSICAL I B3 T/E 1.43 1.21 0 . 0 0.0 0.27 0. 36 1.16 AR/E 0.00 0.83 0 . 0 0.0 1.05 2.57 0.05 SB/E -0 . 05 -0 .9 0 0 . 0 0.0 -0.91 -2 .4 6 - 0 . 10 SE/E 0.04 -0. 13 0 . 0 0.0 0.49 0.54 -0 .1 2

NE XL ASS I CAL 11 T/E 1. 39 1. 73 0 . 0 0 . 0 0.42 0. 37 I . 10 AB/E 0.01 1.60 0 . 0 0.0 0.29 0.80 0.03 S8/E -0 . 03 -1 .4 6 0. 0 0.0 -0.09 -0.4 8 -0 .0 2 SE/E 0. 0? -0 . 37 0 . 0 0.0 0.38 0.31 - 0 . i l

STEEL

NEOCLASSICAL I T/E 10.12 I.00 0.89 0.0 0.77 0.66 0.85 AB/E - 0.68 •0.13 1.45 0.0 0.11 •0.04 ■0.05 SB/E - 0 . 03 ■0.00 -1.3 4 0.0 -0.05 •0.23 0.01 SE/E 0. 71 0.14 0.01 0.0 0.16 0.62 0.18

NEOCLASSICAL I I T/E 7.60 1.04 0.0 0.0 0.73 0.70 0.80 AB/E -0 .6 0 0.03 0.0 0.0 0.20 0.04 0.03 SB/E -0 .0 3 •0.04 0.0 0.0 0.02 0.34 0.01 fs j CO SE/E 0.63 0.04 0.0 0.0 0.05 •0.07 0.16 -~l APPENDIX J 289 Components of the bias in the estimated aggregate residual variance of a rational distributed lag function by theory and industry.

Autos ^ . ‘ I■ Neoclassical I U'U/T 35Q55VoV

Expected Profits Neoclassical 11 U'U/T 23170.91544266 17958 .56267782 2U' V/T 1076.73349342 6 4 1 3 .76520786 2U'Z/T -887.31593536 5044 .61179527 2U'E/T -667.49174151 -450 .14805174 V'V/T 338.37096990 11901 .45406 029 2VZ/T -o.oomnooo -5 9 5 0 .32659154 ZV'E/T ’ -o.oooo :ooo -9 2 0 .1405863? Z'Z/T 162.777/26J3 25146 .17719736 2Z'E/T -0.00000000 - 0 .00090000 E'E/T 333.74587076 635 .14431903

29527.58587619 59829.1000230? s2

Liquidity

U'U/T 5 48*34.060 2532 6 2U'V/T -58.66332338 2U'Z/T -187.42907734 2U'E/T —1423.8 00 98554 V'V/T 121.53908399 2V'Z/T 0.00000000 2V‘ E/T ' -0.00000000 Z'Z/T 535.05295876 2Z'E/T -0.00000000 E'E/T 711.90049777

S2 54532.65940251 « 290 Components of the bias in the estimated aggregate residual variance of a rational distributed lag function by theory and industry.

Chemicals

Accelerator Neoclassical I U'U/T 4603 ..797513? L. — 6 5 .9 .5 ,4 5 000608 2U'V/T 345.60 176320 -451.30838123 2U'Z/T 9 BA* 9 .& 5 5 M U 2 - _ -4 8 6 .7 7 5 3 0 8 04 2U'E/T -2494.77906970 -2774 . ->?6*75049 V'V/T 3 7 0 .6 3 76262 9 7_4, 5296773 9 2 V'Z/T -385.31125972 -46.54352802 2 V'E/T . -397,3973582.9. .19925063 Z'Z/T 1400.60152705 828.967RB196" 2Z'E/T 0. 0000 OOP Q._ 0 . 0 0 0 0 0 0 0 0 E'E/T 1446.08346399 14.33.21300056 3902.32291723 5.081 . 10194 758

Expected Profits Neoclassical II U'U/T . 615Q.*?9182529- 5700.. 38732549 2U'V/T 320.78244262 -239.18143700 2 U'Z/T -250.91745973 ='t5Z*A15_7.Q£56 -2079.70311011 2 U'E/T -4847.96972094 V'V/T 328.0565197 7_ - 6 12.32562 86 3. -419. 326 36194 2 V'Z/T -144.36437412 2 V'E/T _-35fl,.96 366282 .. ——1.7 *.10 7.0377.8 Z'Z/T 600.99142531 348.02926025 2 Z*E/T _____ 0 . 0 0 3 > 0 0 0 0 ______0,-0030 0 0 0 0 E'E/T 2 6 0 3 .3o6441 83 1049.40507394 4401.46393725 *497.*13*3204

Liquidity u'u/r . 7433.. 8 ii! 2.6243 _ 2U'V/T -3 3 5 .I 3620911 2U' Z/T - 1708.340)7669 2U'E/T -4 4 6 8 .5 9 5 70546 V'V/T 3.71..547.9 3 20 3 2V'Z/T 27.76710439 2V'E/T . —22 7.6 99.3864 3 Z'Z/T 104 .2 " 1 t 145 2Z' E/T O A 'i E'E/T 2345.64704594

6689.21598164 291 Components of the bias in the estimated aggregate residual variance of a rational distributed lag function by theory and industry.

Electrical

Accelerator Neoclassical 1 U'U/T Tff6R"r?T75V/9^ ■~TVLT.“6 f 4 3 2U'V/T -22.76097944 _ -in .79479820 ZU'Z/T -8.13018077 -4 2.08 79 1394 2U'E/T -35.97745723 -300.32730451 V'V/T 1.28330366“ 0.40724208 ZV'Z/T -1.9 3467357 — 1.228294 74 2V'E/T -0.50396231 -0.79002337 Z'Z/T 9.16405267 • 165.52406600 2Z'E/T 0.00030000 * -0 .“00030000 E'E/T 18.240 70977 ___ 150.55866394 — * - * 567.69316072 ‘ T 4 6 5 .8 7604241

Expected Profits Neoclassical II U'U/T 1923.47197?^^9 ' T4ttfl.533T0'617 " 2U'V/T 72.658 #5162 -9.23725641 2U'Z/T -3 7.360 74 20 5 —9 4. 3100 5 90 4" ~ 2U'E/T -229.92.112333 2 .1 2 4 . 82957680 V'V/T ' 10.41? #3Rl3 - "0.50 3 709*3 5" “ 2V'Z/T -3.30332629 -3.06354699 2V*E/T — TTT5 6 6 73 84”l =77757 T545S7T- Z'Z/T 61.65239709 25^.33126470 2Z'E/T -0.00390000 - 0. 000) 0000' E'E/T 121.74 3 AH087 62.72356124

190 6. 80666 73 3 1 5 7 5 7 0 3 3 75654

Liquidity

U'U/T >160.033 7SOT7T 2U’V/T __ 156.4002 8616 2U'Z/T -53.45083484“ 2U'E/T -366.11053818 V'V/T 59.58848198 - 3 . 3 1 8 >9 0^1? 2 V'Z/T 2 V'E/T '1 2 • 8 1 0 L747 1 Z'Z/T 80.52243316 2Z'E/T - 0.00000000 E'E/T 176.65018173 " I 2 2222.63066676" Components of the bias in the estimated aggregate 292 residual variance of a rational distributed lag function by theory and industry.

Oil

ucctilerator Neoclassical I U'U/T ~3‘5*9T.6264Cfl7* ~ ■ 3 * * 3 1 . 1 19 * > 1 9 1 7 “ 3 6 4 . 3 2 1 6 4 7 4 6 2 U'V/T 2842.15119119 2U'Z/T * 579.63817142 ”*4592.74020085 2U'E/T -37339.36852442 -10553.20936B69 V'V/T “ 1149.70624687 3668;44671259 -715.52929099 2 V'Z/T -2287.84125140 2V'E/T •-1544".£7B493?75~ --58 5 T 67*22 677" Z'Z/T 59P4.06127433 3285.08753497 2Z'E/T “ 0.00030000 “ * - 0.00000000 E'E/T 1944?.023*0903 5569.44404773 ?452 4•31°7S210 ” 41256'. 7421*762 7'

Expected Profits Neoclassical II U'U/T “5*>r4T:457r4l5£J7~ “ T6713T7L97 5b2TT~ 2 U'V/T _ -922.82294639 1033.40357001 2U'Z/T 6074.0 5619229 ' ' 627.45427715 2U'E/T -19698.21836749 -15781.60 302435 V'V/T 1252.68212027 "3009.97630399 2 V'Z/T 443 .86 874 7J5 0 -368.58060559 2V'E/T 8.72597740" =Vi9'.227*0371 “ Z'Z/T 2034.73566281 4798.16028787 2Z'E/T - 0.00000000 -O'.00030000 E'E/T 9844.74644604 7950.41526403 53236.43077551' '36752.19613158

L iq u id ity U'U/T ~~445T9V2'8454'45'7"" 2U' V/T -5397.74891740 2U'Z/T -4943.35971773 ' 2 U'E/T -22152 .0404.1428 V'V/T 1205.?2537792 2V'Z/T 546.86954066 2 VE/T 3TT. 59603 709” Z'Z/T 4996.43930401 2Z‘E/T " 0.30000000 E'E/T 10890.22220100

S2 "3011 6 .4 *3 7 9 3 0 9 4 293 Components of the bias in the estimated aggregate residual variance of a rational distributed • lag function by theory and industry.

Rubber Accelerator Neoclassical I U'U/T • -760.09569571 847.72731568 „ 2U'V/T ***• 15601924 -1 1.84595576 2U'Z/T _ -9 3 .a 1 78561 9 - y ♦ ._13 9 4 c 71 0_ _ 2U' E/T -371.064 02860 ■>7 8 • 1 39M r? I V'V/T 50.00786963 _l 3.86808*73 2V'Z/T -9 .6 6 8 6 1606 -3 .0 4 2 7 6 7 9 2 2V' E/T .-2 3 .5 4 3 7 8 2 1 1 0.54373094 Z'Z/T 79.56808618 62.74321306 2Z ’ E/T _____0.00 0 00000 __ 0.00000000 E'E/T 19 7.3036403 5 137*. 79 791 51 3 S2 564. 726 71967 697.46751 367

Expected Profits Neoclassical II U'U/T 611.23173745 727.67475194 2U'V/T ‘ 21 3. 39764583 128.53301794 2U'Z/T - 7 0 . 5 4 36912 7 -81.74147317 2U' E/T -266.87154518 -244.3 7878975 V'V/T 107.15186527 49•346942 R1 2VZ/T -24.7134174 4 -21.48414072 2V' E/T ^ 95.42928690 . -4.72605938 Z'Z/T ' 43.32169901 46.40665514 2Z'E/T ___ 0.00030000 0 . 0 0 0 0 0 0 0 0 E'E/T' 18 1.00041654 124.55217456 *s2 69*. 8454233? 723.18267938

L iq u id ity U'U/T . 60 3.54 39 035 8 2U'V/T -46.29497712 2U'Z/T ___46 .9 57 3186 3 2U'E/T -254.29135959 V'V/T 4 4.666 491 2 .2 . 2V'Z/T -6 4 • 39866341 2V'E/T _ - 3 4 . 1023M 41 Z'Z/T 5 4 .2 3821083 2Z' E/T 0 .0 0 000000 E'E/T .144 • 1°68 4550 g2 4 0 4 .5 1 5 .4 3821 Components of the bias in the estimated aggregate residual variance of a rational distributed lag function by theory and industry.

Steel

Accelerator Neoclassical 1 U'U/T *T ^ 7 3 ".4 T 9 7"237^_ ~l 3013 .2 36 5472 2 aj'v/T 2355.92922211 3286.56710703 2U'Z/T 6*6.411 19750 -73.37690700 2U'E/T -767* 20ft1025 3 -2014.45509533 V'V/T " 575.28211329 484.99943259

2V'Z/T -490.106 75657 - zM 7^PZHIXUL 2V'E/T -62.39034099 -7 0 1 . 898^5063 Z'Z/T 3053.57934244 119 1 .7 3 1 7 1 9 3 5 2Z'E/T -0.00003000 0 . 0 0 0 0 0 0 0 0 ' E’E/T 412.2^372176 1361.15702300 16168.37332242 S2 203 2 2 . 25ft82071

Expected Profits Neoclassical 11 U'U/T “ T27#9T?^509156' 13215.4°147151 2U'V/T 2119.7334787? 3662.12338962 2U' Z/T —435.30337407 ' -1058.71409573 2U'E/T -4183.996 20115 -308.982°3028 V'V/T 273.11077606 ‘ "798.7344965? 2V'Z/T — 5 4.ft 5 736479 -433.19418728 2V'E/T ‘ ~ - 8 ? .3 3 9 7 7 5 3 4 —237.8 f 4 n r r 1 fr Z'Z/T 1 1 8 9 .3 240 10 5? 826.77326589 2Z'E/T -0.00030000 0.00000000 E'E/T 21 38.66TQJ5B?t 273.39855573

18 755.13562956 1673 7. 82 078479

Liquidity U'U/T 1~437T.TT 19018 ft" 2U'V/T 5253.23462280 2U' Z/T -289.2 370292 8 2U' E/T -1000.53743331 V'V/i 2702.75235227' 2V'Z/T llP 3 _ .J 0 36721 5 2V'E/T -256.0069 48 2 3 Z'Z/T __21l.55673564 2Z'E/T -0.000 30000 E'E/~ .. *78.27214077 S2 21314.84236017 BIBLIOGRAPHY 296

Adelman, Irma, and Lobo, Orlando. "Some Observations on Full Employ­ ment versus Full Capacity," American Economic Review, XLIV (June, 1956), 412-419.

Alchian, Armen A. "The Rate of Interest, Fisher's Rate of Return over Costs, and Keynes' Internal Rate of Return," in The Management of Corporate Capital. Edited by E. Solomon. Glencoe, 1959. 67-71.

Allen, R.G.D. Mathematical Economics. 2d ed. London: Macmillan and Co., Ltd., 1965.

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