Asymptotic equivalence of ordinary least squares and generalized least squares with trending regressors and stationary autoregressive disturbances

This note generalizes previous results on the asymptotic equiva lence of Ordinary and Generalized Least Squares estimates in Li near Regression models with trending data This note considers the relative e ciency of OLS versus GLS in the linear regression model yt x t ut t where xt and are K and where the unobservable disturbances ut are autocorrelated but independent of the regressors xt xt xtK It is well known that given the regressors OLS is in general no longer BLUE when disturbances are correlated but as GLS the BLUE is often only of academic interest due to lack of knowledge of the disturbance correlation structure there has been an enormous interest in statistics and econometrics in the relative e ciency of OLS Watson Kr amer Kr amer and Donninger Busse et al among many others Research supported by Deutsche Forschungsgmeinschaft through SFB One strand of this literature originating with Grenander is concerned with conditions on regressors and disturbances which guarantee that OLS is at least asymptotically e cient Rosenblatt Chipman Kr amer Phillips and Park Kr amer and Hassler One su cient condi tion for the asymptotic e ciency of OLS that emerges in this literature is that the regressors are in some sense trending for a precise de nition see below In conjunction with stationarity in particular stationary autoregressive distur bances this is then shown to imply that the respective limiting distributions of OLS and GLS are identical The present note extends and uni es this literature by suggesting a generic form of trend and by showing that it is this generic property of trending data which implies the asymptotic equivalence of OLS and GLS In what follows the disturbances ut from are assumed stationary AR p ut ut put p t where the t s are i i d and where stationarity implies that all roots of the polynomial z pz p are outside the unit circle Ignoring observations p which are asymptotically irrelevant the GLS estimator  for is obtained by applying OLS to yt x t t where xt xt xt pxt p and yt yt yt pyt p t p i e P xt x t P x yt and