1

MODELLING RESEARCH GROUP

GIANNINI FOUNDATION Ow AGRICULTURAL ECONOMICS LIB

.-R 93 1987

DEPARTMENT OF ECONOMICS,1 !UNIVERSITY OF SOUTHERN CALIFORNIA UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90089-0152

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MODELLING RESEARCH GROUP

BIASED PREDICTORS, RATIONALITY AND

THE EVALUATION OF FORECASTS

ARNOLD ZELLNER*

MRG WORKING PAPER #M8612

GIANNINI FOUNDATION 00 AGRICULTURAL ECONOMICS LIB se"93 1987

DEPARTMENT OF ECONOMICS,! UNIVERSITY OF SOUTHERN CALIFORNIA UNIVERSITY PARK LOS ANGELES, CALIFORNIA

90089-0152 BIASED PREDICTORS, RATIONALITY AND THE EVALUATION OF FORECASTS

ARNOLD ZELLNER*

MRG WORKING PAPER #M8612

March 1986

ABSTRACT

Optimal, rational forecasts are often biased and thus the empirical finding that actual forecasts are biased is not necessarily evidence of irrational behavior. Using an asymmetric loss function and a regression forecasting problem,these points are explicitly demonstrated.

••-• *H.G.B. Alexander Research Foundation, Graduate School of Business, University of , visiting the University of Southern California. ,t

Biased Predictors, Rationality and the Evaluation of Forecasts

by * Arnold Zellner

Abstract

and thus the empir- Optimal, rational forecasts are often biased is not necessarily evidence ical finding that actual forecasts are biased loss function and a regres- of irrational behavior. Using an asymmetric demonstrated. sion forecasting problem, these points are explicitly

and estimators It has long been recognized that biased predictors estimators in terms of expected may be better than unbiased predictors and --see e.g. Stein (1955), loss or risk relative to various loss functions Chs. 3,4), and Zellner (1963, Efron and Morris (1973), Judge et al. (1985, based on proper infor- 1985). Indeed, Bayesian estimators and predictors, known to be admissible and to mative priors, are generally biased but are these well known facts, it is minimize average or Bayes risk. In view of to evaluate forecasts concen- surprising that many studies that attempt and interpret a depar- trate attention on whether forecasts are unbiased from rationality and/or ture from unbiasedness as evidence of a departure

* Foundation and Research financed in part by the National Science Fund, Graduate School of by income from the H.G.B. Alexander Endowment address is: Graduate School Business, University of Chicago. The author's Chicago, Illinois 60637. of Business, U. of Chicago, 1101 East 58th Street, 2

For a review of a large portion of the rational expectations hypothesis. who, at my suggestion, mentioned this literature, see Zarnowitz (1985) Grossman (1975) recognized that optimal forecasts may be biased. Earlier, does not imply that the antici- that the rational expectations hypothesis its mathematical expectation in all pated value of a variable is equal to circumstances. an explicit regression The purpose of this note is to provide relative to an asymmetric example in which a biased predictor is optimal for usual "tests of ration- loss function and to indicate the implications to estimate the shape parameter of an ality." Also, it will be shown how its degree of asymmetry. asymmetric loss function that determines vector y be a standard nor- Let the model for the nxl observation mal multiple regression model,

matrix of rank k, 8 is a kxl vector where X is a given nonstochastic nxk an nx1 vector of disturbance terms that of regression parameters and u is a normal distribution with zero mean are assumed independently drawn from 2 value of the dependent variable, and variance a . Further, let a future y be given by f yf + uf (2)

and u is a future error term drawn inde- where x is a given kxl vector f N(0,02) distribution. Our problem pendently of the errors in (1) from a in (1) and the vector xf. From is to forecast yi given the data, the model that the least squares predictor results in the literature, it is known

(LSP), 3

ALS ,A yf = -f_x - 0 (3)

f 1-1 f under where A = (X'X) X'y is a minimum variance, unbiased predictor of y *LS . is y is optimal rela- the normality assumption introduced above. That f 5f a tive to a squared error loss function, Ls = c(5f-yf)2, with c > 0 and that point predictor, subject to the side condition that 5f be unbiased, be costly in is E(5f-yd = 0. The restriction that 5f be unbiased can ryd + terms of say, mean squared error (MSE). Note that MSE = var(Y 112 [E(yryoj = variance plus squared bias. As is well known, a predictor LSP in with a small variance and a small bias can have lower MSE than the predictor (3)--see, e.g. Zellner (1963) for examples. Also a Bayesian will in based on a proper prior distribution for the regression parameters error general be biased but has lower average risk relative to a squared squared loss function than the LSP. Thus even with use of a symmetric error loss function, various good or optimal predictors are biased.

Instead of a symmetric squared error loss function, consider use

by Varian ••• of the following asymmetric LINEX loss function, introduced

(1975),

L(A) = aA - 1] a * 0 (4) b >0

the use of the point where A = yryf, the forecast error associated with L, and when forecast Yf. It is seen that L(0) = 0, the minimal value of linearly when a > 0, loss rises almost exponentially for A > 0 and almost than with A < 0. Thus with a > 0, over-forecasting results in higher loss O. When lal is equal under-forecasting and the reverse is true for a < 2 2 as can be small in value, L(A) = a A /2, a squared error loss function, 14

2 aA . inserting this expression in seen by expanding e = 1 + aA + aA /2 and

(14), yr, p(yfID), where D Given that we have a predictive pdf for it can be used to compute denotes given sample and prior information,

expected loss as follows, air'f, -ay f'" EL(A) = b[e Ee - auf-Eyf) - 1].

to Si'f, as in Zellner On minimizing this expression with respect

(1985), the result is -ayf (5) S;* = (-1/a)logjEe ].

with a diffuse prior pdf, the In the case of the regression model (1) ALS 2 y = x'A and variance v = predictive pdf given a is normal with mean f f- p.72). Thus, from (5), + x'(X'Xj-ix la2 --see e.g. Zellner (1971, El -f -f ALS • -* y - av/2, (6) Yf = f

Varian (1975). This predictor a special application of (5) presented in it has uniformly lower risk than is biased but as shown in Zellner (1985) A LS relative to the LINEX y and thus the latter is inadmissible the LSP f A v in (6) is replaced by v = loss function in (4). This is also true if

x'(X 1 X)-ix Is2 where s2 = (y-4Ny-X.4)/(n-k). [1 + -f -f ' A v = v to forecast yf. Now suppose that a forecaster uses (6) with

is e, given by Then his forecast error f

-* = ,,LsA _ av/2A (7) ef = yf 'f 'f "f

and his mean error or bias is

]a2/2. (8) = -av/2 = -a[l + -rx'(X'X)' i x 5

depends on the value of the LINEX Thus the algebraic sign of the bias greater the asymmetry of the loss parameter a. The larger lal, the in (8). However, the forecaster is function and the greater is the bias expected loss by using the rational in the sense that he is minimizing biased predictor 5; in (6). e_.= v If we have a sequence of forecast errors, t1 - fi- Yfi' 1'2' with yfi = x' B + u and pfti av i/2, s,liT ...,m, where --fi fi - A 2, and s2 = (y-XA)'(y-Xe)/(n-k), then (X'X)-/X'y, = [1 + ,s 1(X'X)-1,1f1ls

of the mean forecast error e = e /m is the expectation f i=1 fi Ee = -av/2 (9) f

sign of the 1 [1 + x' OPX)-/x I/m. From (9), the algebraic where v = 02 -fi -fi i=1 sign of the LINEX parameter a. "bias," will be determined by the algebraic of a Also, from (9), the following is an estimator

A A -2e /v (10) a = f A A -LS LS Since e = e + a/2, where e = where -1.1- is -%"; with a2 replaced by s2. f f m -LS r LS -LS 2 and Ee = 0, EA = a, that is .2, e./m, ef and s are independent, f 1=1 calculation Var(A) = estimator in (10) is unbiased. By a direct the m - 17i lima2c2, x = x ./m and c = ,(X- 1 X)- tv-2), where v = n-k, ...f 1 i-fi 4v[1 + ...f -fi .1=1 m -1 1 [1 + xi,i(X'X) xfi]/m. i=1 Table 3, p. 301) that his It will be noted from Zarnowitz (1985, from zero. As he explains, "Table 3 mean forecast errors tend to differ inflation....In contrast, shows that almost all forecasters underestimated overestimated...." (pp. 299-300). real growth [Of GNP] was predominately errors are (a) peculiar to the period Whether his non-zero mean forecast ••••

use studied, (b) evidence of irrationality and/or (c) due to forecasters' from just of optimal biased forecasts cannot, of course, be ascertained (c) study of past forecast errors' properties. However, under assumption (10) that the and an assumed use of a LINEX loss function, it appears from for parameter a may be positive for inflation forecasting and negative further forecasting real growth of GNP. To make such inferences secure, research is needed to determine the forms of forecasters' loss functions and how they use them, explicitly or implicitly, to generate forecasts. 7

References

Efron, B. and C.F. Morris (1973), "Stein's Estimation Rule and Its Com-

petitors--An Empirical Bayes Approach," Journal of the American Sta-

tistical Association, 68, 117-130.

Grossman, S. (1975), "Rational Expectations and the Econometric Modelling

of Markets Subject to Uncertainty: A Bayesian Approach," Journal of

Econometrics, 1, 255-272.

Judge, G. et al. (1985), The Theory and Practice of Econometrics, 2nd ed.,

New York: John Wiley & Sons, Inc.

Stein, C. (1955), "Inadmissibility of the Usual Estimator for the Mean of

a Multivariate Normal Distribution," Proceedings of the Third Berkeley

Symposium on Mathematical and Probability, Vol. I, Berkeley:

University of California Press, 197-206.

Varian, H.R. (1975), "A Bayesian Approach to Real Estate Assessment," in

S.E. Fienberg and A. Zellner (eds.), Studies in Bayesian Econometrics

Leonard J. Savage, Amsterdam: North-Holland and Statistics in Honor of •••

Publishing Co., 195-208.

Zarnowitz, V. (1985), "Rational Expectations and Macroeconomic Forecasts,"

Journal of Business and Economic Statistics, 3, 293-310.

Zellner, A. (1963), "Decision Rules for Economic Forecasting," Econometrica,

31, 111-130.

(1971), An Introduction to Bayesian Inference in Econometrics,

New York: John Wiley & Sons, Inc.

(1985), "Bayesian Estimation and Prediction Using Asymmetric Loss

Functions," Technical Report, H.G.B. Alexander Research Foundation,

Graduate School of Business, University of Chicago, to appear in the

Journal of the American Statistical Association. REPORTS IN THIS SERIES

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