Using Seasonal and Cyclical Components in Least Squares Forecasting Models

Although many articles have been written concerning the improved accuracy of combined forecasts, sometimes the obvious is overlooked. By combining seasonal indices and cyclical factors with other explanatory variables, forecasting models acquire increased accuracy for out-of-sample predictions.. This paper encourages the use of least squares forecasting models with time series components. It also provides new directions for research in combining forecasts. This approach to forecasting is also compared to other popular forecasting methods. Surprisingly, the use of seasonal indices and cyclical factors in least squares equations does not frequent the literature. INTRODUCTION The decomposition method of separating time series data into the four components of trend, cyclical movement, seasonal variation, and irregular fluctuations is well known. Indeed, combining these components in a multiplicative manner is one of the oldest methods of forecasting (Barton, June 1941). However, considerable advantages are obtained by including seasonal indices and cyclical factors in a least squares forecasting equation: Ŷt = b0 + b1Xt + b2Sj + b3Ct (1) where Xt are for trend values, Ct are cyclical factors, and Sj are seasonal indices repeated each year. This approach becomes attractive when compared with other forecasting methods. Equation (2) describes the dummy variable approach to quarterly seasonal variation: Ŷt = b0 + b1Xt + b2D2 + b3D3 + b4D4 + b5Ct, (2) where Xt and Ct are defined in (1) above; Dj = 1 if quarter j, j = 2, 3, 4, 0 otherwise. Equations (1) and (2) have approximately the same accuracy. The dummy variable method of including seasonal variation is described in most econometric textbooks (Greene, 2000; also Ramanathan, 2002). Although (1) has the advantage of using a single index variable, its applications to forecasting does not frequent the Southwestern Economic Review 190 literature. When describing monthly seasonal variation, the dummy variable approach must employ 11 binary variables as compared to the one seasonal index variable in (1). This alone has considerable computational and methodological implications. Time series components in unrestricted least squares models are highly conducive to judgement modification operations thereby increasing the accuracy of out-of-sample forecasts. Hence, this approach extends the capabilities of combining forecasts using unrestricted least squares coefficients as weights (Granger and Ramanathan, 1984). An example is given which compares (1) with other forecast methods. CONCEPTS AND NOTATIONS Although there are exceptions, the accuracy obtained by using (1) over the traditional decomposition method conforms to intuition. Least squares estimates by (1) are more accurate than non-least squares estimates from