英文文献：Asymptotic Theory for the QMLE in GARCH-X Models with Stationary and Non-Stationary Covariates-含平稳和非平稳协变量GARCH-X模型QMLE的渐近理论
英文文献作者：Heejoon Han,Dennis Kristensen
英文文献摘要：
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 dx; in particular, we allow for both stationary and non-stationary covariates. We show that the QMLE'?s of the regression coefficients entering the volatility equation are consistent and normally distributed in large samples independently of the degree of persistence. This implies that standard inferential tools, such as t-statistics, do not have to be adjusted to the level of persistence. On the other hand, the intercept in the volatility equation is not identifi?ed when the covariate is non-stationary which is akin to the results of Jensen and Rahbek (2004, Econometric Theory 20) who develop similar results for the pure GARCH model with explosive volatility.

摘要研究了加了一个解释变量GARCH- x模型的GARCH模型的高斯拟极大似然估计的渐近性质。附加的协变量允许显示由其长内存参数dx捕获的任何程度的持久性;特别地，我们考虑到平稳和非平稳协变量。我们证明QMLE'?进入波动率方程的回归系数的s是一致的，并且在大样本中呈正态分布，与持久性的程度无关。这意味着标准推理工具(如t-statistics)不必根据持久性级别进行调整。另一方面，波动率方程中的截距不是identifi?这与Jensen和Rahbek(2004，计量经济学理论20)的结果相似，他们为具有爆炸性波动的纯GARCH模型开发了类似的结果。