Asymptotic Equivalence of Ordinary Least Squares and Generalized Least Squares in Regressions with Integrated Regressors

Abstract

Abstract It is shown that in the multiple regression model y 1 = x 1′β + u t, where u t, is a stationary autoregressive process and x 1 is an integrated m-vector process, the asymptotic distributions of the ordinary least squares (OLS) and generalized least squares (GLS) estimators of β are identical. This generalizes a result obtained by Krämer (1986) for two-variate regression and extends fixed regressor theory developed by Grenander and Rosenblatt (1957). Our approach uses a multivariate invariance principle and yields explicit representations of the asymptotic distributions in terms of functionals of vector Brownian motion. We also provide some useful asymptotic results for hypothesis tests of the model. Thus if x 1 is generated by a vector (autoregressive integrated moving average) ARIMA(r, l, s) model and u t is generated by an independent (autoregressive) AR(p) process, then and have the same limiting distribution (where and are the OLS and GLS estimators, respectively). This distribution is nonnormal and most conveniently represented in terms of a vector Brownian motion (B 1(r), B 2(r)′) as the functional , where B 2 is an m-vector Brownian motion independent of B 1. Furthermore, if X(T × m) is a matrix of T observations of x 1 then and have the same limiting normal distribution as T ↑ ∞. But the variance of this normal distribution is given by σ12 = 2πfu(0) [where f u(λ) is the spectral density of u t] and not σ2 = var(u t). Traditionally constructed asymptotic tests of significance are invalid in the present context; however, these tests may be made robust by simply replacing the usual estimators of σ2 with consistent estimates of σ12 .