The Annals of Statistics

Inference in nonstationary asymmetric GARCH models

Christian Francq and Jean-Michel Zakoïan

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This paper considers the statistical inference of the class of asymmetric power-transformed $\operatorname{GARCH}(1,1)$ models in presence of possible explosiveness. We study the explosive behavior of volatility when the strict stationarity condition is not met. This allows us to establish the asymptotic normality of the quasi-maximum likelihood estimator (QMLE) of the parameter, including the power but without the intercept, when strict stationarity does not hold. Two important issues can be tested in this framework: asymmetry and stationarity. The tests exploit the existence of a universal estimator of the asymptotic covariance matrix of the QMLE. By establishing the local asymptotic normality (LAN) property in this nonstationary framework, we can also study optimality issues.

Article information

Ann. Statist. Volume 41, Number 4 (2013), 1970-1998.

First available in Project Euclid: 23 October 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62F12: Asymptotic properties of estimators 62F05: Asymptotic properties of tests

GARCH models inconsistency of estimators local power of tests nonstationarity quasi maximum likelihood estimation


Francq, Christian; Zakoïan, Jean-Michel. Inference in nonstationary asymmetric GARCH models. Ann. Statist. 41 (2013), no. 4, 1970--1998. doi:10.1214/13-AOS1132.

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Supplemental materials

  • Supplementary material: Supplement to “Inference in nonstationary asymmetric GARCH models.”. The supplementary file contains an illustration concerning the optimality of the asymmetry test, a Monte Carlo study of finite sample performance, an application to real time series, an explicit expression for the matrix $\mathcal{I}$ in Theorem 3.1, the proofs of Theorems 3.2 and 6.1.