## The Annals of Statistics

### Inference in nonstationary asymmetric GARCH models

#### Abstract

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

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

Dates
First available in Project Euclid: 23 October 2013

https://projecteuclid.org/euclid.aos/1382547510

Digital Object Identifier
doi:10.1214/13-AOS1132

Mathematical Reviews number (MathSciNet)
MR3127855

Zentralblatt MATH identifier
1277.62210

#### Citation

Francq, Christian; Zakoïan, Jean-Michel. Inference in nonstationary asymmetric GARCH models. Ann. Statist. 41 (2013), no. 4, 1970--1998. doi:10.1214/13-AOS1132. https://projecteuclid.org/euclid.aos/1382547510

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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.