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+