Forecasting Non-Stationary Economic Time Series

The purpose of this paper is to show how certain results obtained by Peter Whittle [Whittle, P. 1963. Prediction and Regulation by Linear Least Square Methods. English Universities Press, London.] may be used to derive the results on adaptive forecasting obtained by Wage and Nerlove [Nerlove, M., S. Wage. 1964. On the optimality of adaptive forecasting. Management Sci. 10(January) 207-24.] and to carry forward the general program of that paper; namely, to study the prediction of those types of non-stationary and non-deterministic series 1 which can be reduced to stationary series by a finite linear transformation. For this class of non-stationary series it may be shown in general that the optimal forecast for any future period may be expressed as a linear combination of past values of the series, and past one-step prediction errors; they are, in this sense, "adaptive." Unfortunately, this class contains series which, when transformed, have infinite moving average representations. For these series the adaptive representation of the optimal forescats will contain an infinite number of terms, and thus, in terms of the original motivation of adaptive forecasting, be less than helpful. Such considerations do, however, suggest what the limits of applicability of the adaptive representation are, and give a criterion for choosing between the autoregressive and adaptive representations in the case of any particular series. The analysis is illustrated by forecasts of unemployment of males, aged 14-19, one, four, and fifteen months ahead.