COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1665R

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

Donald W.K. Andrews and Patrick Guggenberger

June 2008
Revised March 2010

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) approaching infinity and (ii) n(1 – rho_n) approaching h in [0,infinity) as n approaching 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 Number: C22