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Keywords:

  • Asymptotic normality;
  • GARCH;
  • nonstationary processes;
  • quasi-maximum likelihood estimation;
  • strong consistency;
  • time-varying models
  • C22;
  • C13

This article considers GARCH(1,1) models in which the time-varying coefficients are functions of the realizations of an exogenous stochastic process. Time series generated by this model are in general nonstationary. Necessary and sufficient conditions are given for the existence of nonexplosive solutions, and for the existence of moments of these solutions. The asymptotic properties of the quasi-maximum likelihood estimator are derived under mild assumptions.