Asymptotic Theory for the QMLE in GARCH-X Models With Stationary and Nonstationary Covariates
研究了GARCH-X模型中高斯拟极大似然估计量（QMLE）的渐近性质，允许协变量具有任意程度的持久性（包括平稳和非平稳），发现参数估计量一致且混合正态分布，而t统计量在样本足够大时服从标准正态分布，不受协变量持久性影响。
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 <i>d<sub>x</sub></i>; 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 <i>t</i>-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.