Optimality Of Gls For One-Step-Ahead Forecasting With Regarima And Related Models When The Regression Is Misspecified

We consider the modeling of a time series described by a linear regression component whose regressor sequence satisfies the generalized asymptotic sample second moment stationarity conditions of Grenander (1954, Annals of Mathematical Statistics 25, 252–272). The associated disturbance process is only assumed to have sample second moments that converge with increasing series length, perhaps after a differencing operation. The model's regression component, which can be stochastic, is taken to be underspecified, perhaps as a result of simplifications, approximations, or parsimony. Also, the autoregressive moving average (ARMA) or autoregressive integrated moving average (ARIMA) model used for the disturbances need not be correct. Both ordinary least squares (OLS) and generalized least squares (GLS) estimates of the mean function are considered. An optimality property of GLS relative to OLS is obtained for one-step-ahead forecasting. Asymptotic bias characteristics of the regression estimates are shown to distinguish the forecasting performance. The results provide theoretical support for a procedure used by Statistics Netherlands to impute the values of late reporters in some economic surveys.The author thanks two referees and the co-editor for comments and suggestions that led to substantial improvements in the exposition and also thanks John Aston and Tucker McElroy for helpful comments on an earlier draft. Any views expressed are the author's and not necessarily those of the U.S. Census Bureau.