Asymptotics for LS, GLS, and Feasible GLS Statistics in an AR(1) Model with Conditional Heteroskedasticity

Abstract

This paper considers a first-order autoregressive model with conditionally heteroskedastic innovations. The asymptotic distributions of least squares (LS), infeasible generalized least squares (GLS), and feasible GLS estimators and t statistics are determined. The GLS procedures allow for misspecification of the form of the conditional heteroskedasticity and, hence, are referred to as quasi-GLS procedures. The asymptotic results are established for drifting sequences of the autoregressive parameter and the distribution of the time series of innovations. In particular, we consider the full range of cases in which the autoregressive parameter rho_{n} satisfies (i) n(1 - rho_{n}) approaches infinity and (ii) n(1 - rho_{n}) approaches h_{1} in [0,infinity) as n approaches infinity, where n is the sample size. Results of this type are needed to establish the uniform asymptotic properties of the LS and quasi-GLS statistics.

Keywords: Asymptotic distribution, Autoregression, Conditional heteroskedasticity, Generalized least squares, Least squares

JEL Classification: C22