Strict stationarity testing and estimation of explosive and stationary GARCH models

Christian Francq and Jean-Michel Zakoļan
Keywords: ARCH model, Inconsistency of estimators, Local asymptotic power of tests, Nonstationarity, Quasi Maximum Likelihood Estimation
Abstract: This paper studies the asymptotic properties of the quasi-maximum likelihood estimator of GARCH(1,1) models without strict stationarity constraints, and considers applications to testing problems. The estimator is unrestricted, in the sense that the value of the intercept, which cannot be consistently estimated in the explosive case, is not fixed. A specific behavior of the estimator of the GARCH coefficients is obtained at the boundary of the stationarity region but, except for the intercept, this estimator remains consistent and asymptotically normal in every situation. The asymptotic variance is different in the stationary and non stationary situations, but is consistently estimated, with the same estimator, in both cases. Tests of strict stationarity and non stationarity are proposed. The tests developed for the classical GARCH(1,1) model are able to detect non-stationarity in more general GARCH models. A numerical illustration based on stock indices and individual stock returns is proposed.
Published paper and Econometrica link to the Econometrica website.
MPRA MPRA working paper
Slides of a talk given at Slides Paris (CREST), April 5, 2011