The standard approach to combining n expert forecasts involves taking a weighted average. Granger and Ramanathan proposed introducing an intercept term and unnormalized weights. This paper deduces their proposal from Bayesian principles. We find that their formula is equivalent to taking a weighted average of the n expert forecasts plus the decision-maker's prior forecast.
KEY WORDS Combining forecasts Bayesian expert use Biased forecasts
Suppose we are interested in combining n expert forecasts, F,, . . . , F,, to obtain some estimate of an unknown quantity, Q. The most popular approach to the problem estimates Q by C wkFk, where wk is chosen so as to minimize E ( 1 WkFk -Q)2 with C wk = 1. Bates and Granger (1969) pioneered this approach to combining forecasts, with significant enhancements by Dickinson (1975) and Winkler (1981). Bordley (1982) showed that this formula was deducible from a Bayesian approach, assuming: (a) That the decision-maker has a diffuse prior about Qs value. In other words, before consulting the experts, the decision-maker thinks that any value of Q is equally likely.
(b) That the decision-maker considers the vector of expert forecast errors, F,, -Q, to be normally distributed with mean zero and variance-covariance matrix, I;. (This is a fairly standard assumption in the literature (Agnew (1986).)) (c) That the expert forecast errors are uncorrelated with the decision-maker's prior assessments of Q. Thus the distribution of forecast errors remains the same regardless of whether or not the actual Q is higher or lower than the decision-maker expected.
Thus the Bates and Granger formula is consistent with a Bayesian interpretation. Clemen and Winkler (1986), Makridakis and Winkler (1983) and Winkler and Makridakis (1983) have recently applied the Bates and Granger model to a number of empirical problems. The results have been somewhat mixed. Concurrently, Granger and Ramanathan (1 984) generalized the additive model and proposed estimating Q with the model w o + C,w,F,, where the weights are chosen to minimize E(w, + 1 wkFk -Q)* and need not sum up to one. This more general model naturally leads to lower mean squared error. Although it is not always better for prediction than the original Bates and Granger model (Clemen, 1986), it clearly