QML Inference for Volatility Models With Covariates

Abstract

The asymptotic distribution of the Gaussian quasi-maximum likelihood estimator (QMLE) is obtained for a wide class of asymmetric GARCH models with exogenous covariates. The true value of the parameter is not restricted to belong to the interior of the parameter space, which allows us to derive tests for the significance of the parameters. In particular, the relevance of the exogenous variables can be assessed. The results are obtained without assuming that the innovations are independent, which allows conditioning on different information sets. Monte Carlo experiments and applications to financial series illustrate the asymptotic results. In particular, an empirical study demonstrates that the realized volatility is an helpful covariate for predicting squared returns, but does not constitute an ideal proxy of the volatility. Keywords: APARCH model augmented with explanatory variables, Boundary of the parameter space, Consistency and asymptotic distribution of the Gaussian quasi-maximum likelihood estimator, GARCH-X models, Power-transformed and Threshold GARCH with exogenous covariates.

Two are better than one: Volatility forecasting using multiplicative component GARCH‐MIDAS models

“…While we focus on multiplicative GARCH models,Han and Park (2014) andHan (2015) analyze the properties of a GARCH-X specification with an explanatory variable that enters additively into the conditional variance equation. See alsoFrancq and Thieu (2019).…”

Asymptotics of Cholesky GARCH models and time-varying conditional betas

“…0 ) are strictly positive, for any sufficiently small neighbor- +i) 0 . By an extension of (5.20) in Hamadeh and Zakoïan (2011) and (42) in Francq and Thieu (2015), we then have E sup…”

Testing Garch-X Type Models

“…In relation to Assumption 2.1, observe that Han and Kristensen (2014, Lemma 1) state a su¢ cient condition for the existence of a stationary and ergodic solution to the GARCH-X model which includes the case of 0 ; 0 0. In line with Han and Kristensen (2014) and Francq and Thieu (2015), one can relax Assumption 2.2 and the underlying assumption of z t being IID(0; 1). Indeed, one could instead assume that z t is a martingale di¤erence sequence with respect to F t with constant conditional higher-order moments, see Han and Kristensen (2014, Assumptions 1(i) and 2(i)).…”

“…In terms of existing literature, Han and Kristensen (2014) (see also Han and Park, 2012) consider the asymptotic properties of the (quasi-)maximum likelihood estimator for the GARCH-X model under the assumption that the true parameter value lies in the interior of the parameter spaces, which in particular excludes testing for the presence of exogenous covariates. More recently, Francq and Thieu (2015) consider the asymptotic properties of the (quasi-)maximum likelihood estimator in GARCH-X type models where the true parameter value is a boundary point. However, the assumptions in Francq and Thieu (2015) rule out the possibility of nuisance parameters on the boundary as allowed here.…”

“…More recently, Francq and Thieu (2015) consider the asymptotic properties of the (quasi-)maximum likelihood estimator in GARCH-X type models where the true parameter value is a boundary point. However, the assumptions in Francq and Thieu (2015) rule out the possibility of nuisance parameters on the boundary as allowed here. In terms of pure (G)ARCH models (i.e.…”