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Strict Stationarity Testing and Estimation of Explosive and Stationary Generalized Autoregressive Conditional Heteroscedasticity Models

Christian Francq
Jean-Michel Zakoïan

Abstract

This paper studies the asymptotic properties of the quasi-maximum likelihood estimator of (generalized autoregressive conditional heteroscedasticity) 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 nonstationary situations, but is consistently estimated with the same estimator in both cases. Tests of strict stationarity and nonstationarity are proposed. The tests developed for the classical GARCH(1, 1) model are able to detect nonstationarity in more general GARCH models. A numerical illustration based on stock indices and individual stock returns is proposed.