Abstract :
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+
