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 equivalence of Ordinary and Generalized Least Squares estimates in Linear Regression models with trending data. This note considers the relative e ciency of OLS versus GLS in the linear regression model yt = x 0 t + ut (t = 1 ;2; : : : ); (1) where xt and are K 1 and where the unobservable disturbances ut are autocorrelated but independent of the regressors xt = ( xt1; : : : ; xtK)0. 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 1968, Kramer 1980, Kramer and Donninger 1986, Busse et al. 1994 among many others). Research supported by Deutsche Forschungsgmeinschaft through SFB 475 1 One strand of this literature, originating with Grenander (1954), is concerned with conditions on regressors and disturbances which guarantee that OLS is at least asymptotically e cient (Rosenblatt 1956, Chipman 1979, Kramer 1982, 1986, Phillips and Park 1988, Kramer and Hassler 1997). One su cient condition 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 disturbances, 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 (1) are assumed stationary AR(p), ut + 1ut 1 + : : :+ put p = "t; (2) where the "t's are i.i.d. (0; ), and where stationarity implies that all roots of the polynomial 1 + 1z + : : :+ pz are outside the unit circle. Ignoring observations 1; : : : ; p , which are asymptotically irrelevant, the GLS{ estimator ~ for is obtained by applying OLS to ~ yt = ~ x 0 t + "t; where (3) ~ xt = xt + 1xt 1 + : : :+ pxt p and (4) ~ yt = yt + 1yt 1 + : : :+ pyt p (t > p ) (5) i.e. ̂ = ( P ~ xt~ x0t) 1P ~ xŷt and ~ = 0 @ T X t=p+1 ~ xt~ x 0 t 1 A 1 T X t=p+1 ~ xt"t: (6)