Abstract
This article 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 dx; in particular, we allow for both stationary and nonstationary 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. Supplementary materials for this article are available online.
Highlights
To better model and forecast the volatility of economic and ...nancial time series, empirical researchers and practitioners often include exogenous regressors in the speci...cation of volatility dynamics
Our theoretical results for the non-stationary case rely on some of the results developed in Han (2014) and Han and Park (2013) who analyze the time series properties of GARCH-X models with long-memory regressors
We provide a more detailed analysis of the quasi-maximum likelihood estimator (QMLE) compared to HP2012
Summary
To better model and forecast the volatility of economic and ...nancial time series, empirical researchers and practitioners often include exogenous regressors in the speci...cation of volatility dynamics. In the case of non-stationary regressors, on the other hand, we speci...cally model xt as an I (dx) process with 1=2 < dx < 3=2 This allows for a wide range of persistence as captured by the long-memory parameter dx, including unit root processes (dx = 1). Our theoretical results for the non-stationary case rely on some of the results developed in Han (2014) and Han and Park (2013) who analyze the time series properties of GARCH-X models with long-memory regressors. Notations for various convergences such as !a.s., !p and !d frequently appear, where all limits are taken as n ! 1 except where otherwise indicated
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