Estimation and Testing for Unit Root Processes with GARCH (1, 1) Errors: Theory and Monte Carlo Evidence

Abstract

Least squares (LS) and maximum likelihood (ML) estimation are considered for unit root processes with GARCH (1, 1) errors. The asymptotic distributions of LS and ML estimators are derived under the condition α + β < 1. The former has the usual unit root distribution and the latter is a functional of a bivariate Brownian motion, as in Ling and Li [Ling, S., Li, W. K. (1998). Limiting distributions of maximum likelihood estimators for unstable autoregressive moving‐average time series with GARCH errors. Ann. Statist.26:84–125]. Several unit root tests based on LS estimators, ML estimators, and mixing LS and ML estimators, are constructed. Simulation results show that tests based on mixing LS and ML estimators perform better than Dickey–Fuller tests which are based on LS estimators, and that tests based on the ML estimators perform better than the mixed estimators.