Asymptotic Theory for the QMLE in GARCH-X Models With Stationary and Nonstationary Covariates

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may May 2013Abstract This paper investigates the asymptotic properties of the Gaussian quasi-maximum-likelihood estimators (QMLE's) of the GARCH model augmented by including an additional explanatory variable -the so-called GARCH-X model. The additional covariate is allowed to exhibit any degree of persistence as captured by its long-memory parameter d x ; in particular, we allow for both stationary and non-stationary covariates. We show that the QMLE's of the parameters entering the volatility equation are consistent and mixed-normally distributed in large samples. The convergence rates and limiting distributions of the QMLE's depend on whether the regressor is stationary or not. However, standard inferential tools for the parameters are robust to the level of persistence of the regressor with t-statistics following standard Normal distributions in large sample irrespective of whether the regressor is stationary or not.

Estimation in nonlinear regression with Harris recurrent Markov chains

“…When the covariates are integrated, Han and Park (2012) study the maximum likelihood estimation of the parameters in the ARCH and GARCH models. A recent paper by Han and Kristensen (2014) further considers the quasi maximum likelihood estimation in the GARCH-X models with stationary and nonstationary covariates. We next consider the general Harris recurrent Markov process {X t } and use a robust estimation method for model (4.1).…”

QML Inference for Volatility Models With Covariates

“…Actually, even if practitioners often add exogenous variables to volatility models, the probabilistic properties and the statistical inference of ARCH models with exogenous variables have not been yet extensively studied. Notable exceptions are the papers of Han (2013), Han and Kristensen (2014) and Han andPark (2012, 2014), which studied the inference of the GARCH(1,1) model augmented by an additional covariate which can be persistent. A common assumption to all the references previously given in this section, is that the true value of the parameter belongs to the interior of the parameter space.…”

“…For the GARCH-type models, it is usual to assume the stronger assumption that (η t ) is iid (0, 1). Note, however, that Escanciano (2009) and Han and Kristensen (2014) employed A2. The advantage of using A2 is that (2) becomes a semistrong model, that can be satisfied for different σ-fields F t , corresponding for example to different sequences of exogenous variables (x t ).…”

“…When there is no covariate (r = 0), we retrieve the results obtained by Francq (2004) in the GARCH case (δ = 2 and α 0i+ = α 0i− ) and when (η t ) is iid, by Escanciano (2009) in the GARCH case when (η t ) is a conditionally homoscedastic martingale difference, and by HZ in the general APARCH case. In the presence of covariates, (15) allows to retrieve some of the results obtained by Han and Kristensen (2014) for the GARCH-X(1,1) model, under slightly different assumptions.…”

Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models

“…The theoretical properties of the QMLE in the GARCH model need to be developed further, especially in statistical applications, to include situations where these sorts of moment conditions are not satis ed. Han and Kristensen (2014) [15] applied the asymptotic properties of Gaussian QMLE to the GARCH model with an additional explanatory variable, and showed that the QMLE of the parameters for the volatility equation is consistent and mixed-normally distributed in large samples.…”