LOCAL ASYMPTOTIC NORMALITY OF GENERAL CONDITIONALLY HETEROSKEDASTIC AND SCORE-DRIVEN TIME-SERIES MODELS

Christian Francq () and Jean-Michel Zakoian
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Christian Francq: CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - GENES - Groupe des Écoles Nationales d'Économie et Statistique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - GENES - Groupe des Écoles Nationales d'Économie et Statistique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique, IP Paris - Institut Polytechnique de Paris

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Abstract: The paper establishes the local asymptotic normality property for general conditionally heteroskedastic time series models of multiplicative form, $\epsilon _t=\sigma _t(\boldsymbol {\theta }_0)\eta _t$ , where the volatility $\sigma _t(\boldsymbol {\theta }_0)$ is a parametric function of $\{\epsilon _{s}, s< t\}$ , and $(\eta _t)$ is a sequence of i.i.d. random variables with common density $f_{\boldsymbol {\theta }_0}$ . In contrast with earlier results, the finite dimensional parameter $\boldsymbol {\theta }_0$ enters in both the volatility and the density specifications. To deal with nondifferentiable functions, we introduce a conditional notion of the familiar quadratic mean differentiability condition which takes into account parameter variation in both the volatility and the errors density. Our results are illustrated on two particular models: the APARCH with asymmetric Student- t distribution, and the Beta- t -GARCH model, and are extended to handle a conditional mean.

Date: 2022-03-21
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Published in Econometric Theory, 2022, 39 (5), pp.1067-1092. ⟨10.1017/S0266466622000093⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-05417197

DOI: 10.1017/S0266466622000093

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