Linear Combination of Forecasts -- A CommentAuthor(s): Simon FrenchSource: The Journal of the Operational Research Society, Vol. 32, No. 10 (Oct., 1981), pp. 937-938Published by: Palgrave Macmillan Journals on behalf of the Operational Research SocietyStable URL: http://www.jstor.org/stable/2581241 .

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Letters and Viewpoints

In applying the simplified computational scheme, we first compute T^ = Tf2 =0.05

years, T^i = T\2 = 0.1 years, T\^ = 0.11547 years, Tf21 = 0.06124 years, Tfn =

0.1225 years, Tf21 = 0.0707 years and, Tmin = 0.05 years. Other results and the two iterations are given in Table 1.

Table 1. Results of the simplified solution

REFERENCE 1 M. G. Korgaonker (1979) Integrated production inventory policies for multistage multiproduct batch production systems. J. Opl Res Soc. 30, 355-362.

Diosd, Ifjusdg U.8/a.2 Hany A. Makroum

2049?Hungary

LINEAR COMBINATION OF FORECAST&-A COMMENT

Readers of JORS will know that Bunn and I disagree quite sharply about the method?

ology of combining distinct forecasts into a composite forecast.1,2 Even though I am still

unconvinced by Bunn's arguments, I had decided not to continue the discussion further; debates about methodology can decay quickly into entrenched restatements of the

opposing positions. However, in a recent paper3 Bunn again compares his outranking

approach to combination of forecasts with the Bayesian or, as he terms it, veridical

approach. I have no intention of commenting upon the methodological and philosophi? cal points in his discussion; but he does report the results of a simulation study in which

he shows that his method performs signifieantly better than the Bayesian. Although it is

quite possible to remain unconvinced by a battery of methodological arguments, it is an

act of extreme bigotry to ignore the results of a valid simulation study. Indeed, I assure

Bunn that I will adopt his forecasting method and stop championing the Bayesian?if he

can convince me that his simulation study is valid. However, from my reading of his

paper I believe that it contains a significant flaw.

It will be helpful, I think, to return to Bates and Granger's original suggestion for the

combination of forecasts.4 Suppose that we are concerned with a time series

Yu Y2,.., 7t,... For the present assume that we are considering the series from time

zero when none of the Ts have been observed. We have available two forecasting

methods/S1} and/<2); i.e.f\i){Y1, Y2,..,Yt) (i = 1,2) are forecasts of Yt+1. The forecast

errors at time (t + 1) are

e\%i = Yt+,-f?{YuY2,..,Yt). (i=l,2) (1)

Now for t ^ 1, but considering the knowledge available at time zero, there are at least

two unobserved random variables, viz. Yu Y2,.., Yt+ u involved in the definition of e{t+\

and e{2+\. Thus it is conceptually possible for the two forecast errors e{^i and e{2+\ to be

mutually independent. It is this assumption of mutual independence of forecast errors

that Bates and Granger make in deducing their formula for the weighted combination of

forecasts. (See Bunn3 equations (1) and (2).)

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Journal ofthe Operational Research Society Vol. 32, No. 10

However, now consider the forecasts of Yt+1 when we are at time t, i.e. when we have observed Y^ = yu Y2 = y2,..,Yt = yt. The one step ahead forecast errors are:

etlli = Yt+1-f^{yuy29..9yt)

e$2=Yt+1-f2\yuy2,..,yt). (2)

Note that there is only one unobserved random variable in their definition, namely Yt+1. Thus efti and e{2^ cannot be independent conditional on the information available at time t. In fact, ej!^ and e^ are totally dependent random variables. (To see this, suppose that e(ti\ is known, then e{2+\ = eH\ + /|1)(}>i,}>2,.. ,}>,) -f\2)(yi,y2,-.,yt) ? also known exactly.) In other words the random variables, which may be independent conditional on the information at time zero, must be fully dependent conditional on the information at time t.

This point is vital to Bunn's simulation study. He describes this in two sentences

(Bunn,3 p. 218).

The forecaster has two predictors available which need not be modelled explicitly since the combinations are only a function of their forecast errors. He assumes these one-step ahead forecast errors to be independently normally distributed with unknown mean and variances.

If by this Bunn means that he has generated efti and e^^ as independent conditional on the information at time t, then, as we have seen above, he has generated an impossible stream of data. He would have effectively told the methods that Yt+1 was realised both as the quantity (ej+i +/?1)()>i,J>2> ? ?>)>?)) and als? as the quantity (e{2+\ +f{t2)(yuy2,..,yn)\ quantities that with probability 1 cannot be equal. Such con?

tradictory information would naturally confuse the Bayesian forecasting system and lead to its poor performance. Indeed, I would be worried about any forecasting method that did not become confused by such impossible data.

On the other hand Bunn's assumption of independence may mean that sequential errors, i.e. e\ll2 and e\%u are generated independently. However, I cannot conceive of how this may be done without specifying the predictors explicitly; for then he would be

generating the Y's and calculating the forecast errors from (2).

Perhaps Bunn would explain his simulation study in a little more detail than given in his paper. I can assure him that if he can convince me that Bayesian forecasting performs very poorly on a sensible stream of data, then I shall cease to advocate it.

Department of Decision Theory Simon French

University of Manchester

REFERENCES

1S. French and D. W. Bunn (1980) Outranking probabilities and the synthesis of forecasts. J. Opl Res. Soc. 31, 545-551.

2R. I. Phelps, S. French and D. W. Bunn (1981) Forecast Combination. J. Opl Res. Soc. 32, 68-70. 3D. W. Bunn (1981) Two methodologies for the linear combination of forecasts. J. Opl Res. Soc. 32, 213-222. 4M. Bates and C. W. J. Granger (1969) The combination of forecasts. Opl Res. Q. 20, 451-468.

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