Notes and Communications “COMPANY FORECAST ACCURACY

FOR EXPONENTIAL SMOOTHING MODELS

DECISION MAKING”: A COMMENT OF EARNINGS-PER-SHARE DATA FOR FINANCIAL

John P. Dickinson School of Financial Studies, University of Glasgow. GIBLE, Scotland, UK

ABSTRACT

The practice of abandoning all but the most accurate among a set of alternative f o m t - ing methods is shown to result in the loss of potentially useful information. The particular case of two forecasts is considered in detail. It is demonstrated practically that the inclusion of wen a relatively poor forecasting method can enhance a superior one significantly.

Subject Anar. Corpomte Finance Decision Pmcessrs, Forecasting, and Rlrk and Uncertaintj

INTRODUCTION

In a recent paper, Brandon, Jarrett, and Khumawala [ 2 ] compared the forecast accuracy of three models applied to earnings-per-share data. Using linear and qua- dratic loss functions, significant differences in performance were detected among the three methods. The implication was that the two inferior methods be rejected in favor of the superior.

Earlier work [ l ] [3] suggests that in circumstances where several alternative forecasts are available, those discarded will contain information not present in the adopted forecast. As an alternative to abandoning all but one method, a composite forecast is suggested. The composite is a linear function of the individual forecasts.

COMBINING TWO FORECASTS

In the relatively simple case where n =2, the error variance of the combined forecast-using weights w and ( 1 - w), respectively-is

uf= w 2 4 + (1 - w)%7; + 2 w ( l - W ) @ U I U 2 (1)

where e is the correlation coefficient between the errors of the two forecast methods. The error variance is a maximum or minimum when the result of differentiat-

ing uf with respect to w, d 4 / d w , is zero. Now

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Furthermore, the sign of the second derivative, (d2&dd (which is obtained by differentiating a second time), is positive for a minimum value of .f and negative for a maximum value Here

indicating a minimum value for 4 when (duf)/dw is zero. It now is convenient to consider the two cases u1 = a2 and u1 # u2 separately.

The Case of al = u2

From equation (2).

Since, from equation (3), ( d 2 4 ) / d w r 0 for all w, (4) implies that uf is a minimum when e=1 or when w = % . Taking these possibilities in turn, from (1).

In this case, no improvement in the reduction of forecast error is available through combining the two forecasts.

Also from (l) ,

W = ~-(uf) ,~, = y2(1+ e)u2. (4b)

Since e s 1 by definition, this implies that ( U ~ ) , ~ ~ I U ~ with equality only in the particular case of e=1 (which has been considered in (4a)). Thus, a composite forecast with error variance strictly less than that of either component is available except in the special case of e = 1. Intuitively, we might expect no improvement from combining two perfectly correlated forecasts of equal accuracy.

The Case of ul#u2

From (2). (d4)/dw is zero when

19881 Dickinson 235

Since, from (3), (d2a:)/dw2 is strictly positive, the corresponding value of 4 is a minimum. That is

Without loss of generality, suppose a2<al. Then

((9/q)- el2 221

at = I + (aflmin 1- e2

since the second term on the. right-hand side is nonnegative.

it provides improvement when 4 = a2/al.

Note that a composite forecast is never worse than either component, and that > I , which is always the case except when

APPLICATION

Consider the effect of combining the best forecast of [2] with one of the less accurate forecasts: 4=4.315 and 0;=.327.' If e=O, then af=.304; if g = . 5 , then a:= .264. It is clear, therefore, that the inclusion of poorly performing forecasts can enhance a superior forecast. [Received: November 5, 1986. Accepted: January 26,1987.1

REFERENCES

[ I ]

[2]

Bates, J. M., & Granger, C. W. J. The combination of forecasts. Opemlionul Reseurrh Quurlerly, 1%9, 20, 451. Brandon, C., Jarrett. J. E., & Khumawala, S B. Comparing forecast accuracy for exponential smoothing models of earnings per share data for financial decision making. Decision Sciences, 1986, 17. 186-192. Dickinson, J. P. Some statistical rcsults in the combination of forecasts. Operulionul Reseurch Quurterlx 1973, 24, 253-260.

John P. Dickinson is Professor of Accounting and Chairman of the lkpartment of Accounting and Finance at the University of Glasgow. He is a graduate of the universities of Cambridge and Leeds and a Fellow of the British Institute of Management and the Royal SOciFty of A r t s . Dr. Dickinson has teaching and management training experience in the United Kingdom and overseas in areas of manage- ment accounting, business finance, and quantitative analysis. He has consulted to various organiza- tions and advised several overseas educational institutions on the dmlopmmt of accounting and manage- ment education programs. Dr. Dickinson's publications include five books as wcll as papers in academic and professional journals.

[3]

-- 'Here the mean square error values of [2. p. 1901 for the simple exponential smoothing and Holt

and Wnters models have been designated 0: and 0:. respectively.