It is well known that a linear combination of forecasts can outperform individual forecasts. The common practice, however, is to obtain a weighted average of forecasts, with the weights adding up to unity. This paper considers three alternative approaches to obtaining linear combinations. It is shown that the best method is to add a constant term and not to constrain the weights to add to unity. These methods are tested with data on forecasts of quarterly hog prices, both within and out of sample. It is demonstrated that the optimum method proposed here is superior to the common practice of letting the weights add up to one.
KEY WORDS Combining ARMA models Econometrics
It is clear that there are many ways to forecast time series. For a given information set forecasts may be optimal or suboptimal, use a linear formulation or some other specific functional form, use time-invariant or time-varying coefficients, be rich in dynamic specifications or use naive, unsophisticated ones. There are usually also a variety of information sets to be considered. Choices of forecast types can be made on cost considerations, be based on specific economic theories or depend on the abilities of the analyst. If there are available two sets of one-step forecasts from two competing theories, functional forms or information sets, then it has been known for some time that a linear combination of the two forecasts may outperform both of them. In this paper we extend previous discussions about how combinations should be formed, the properties of the combined forecasts and how the results may be interpreted. However, the discussion here is only of linear combinations.
Consider initially the case where there are two unbiased one-step forecastsf,,,, g,,l of x , + made at time n, so that E[(x,+ -f,,l)] = 0 and similarly for g,,l. The combinations considered by Bates and Granger (1969), Nelson (1972), Dickinson (1975) and many other writers in the field are of the form cn,1 = &,,I + (1 -a)gn,, (1) and it was shown in the above references that typically c , , ~ is a superior forecast, in terms of the mean squared error of the forecast error, than either component,for g . A combination of type (1) will be called the 'constrained form'. It does have the advantage that iffand g are unbiased then