Estimation, Testing, and Finite Sample Properties of Quasi-Maximum Likelihood Estimators in GARCH-M Models
研究了GARCH-M模型中准最大似然估计的三个新结果，包括模型性质变化、一种更简单的GARCH效应检验方法，以及有限样本下估计量的偏差和方差近似，对从事金融时间序列和波动率建模的研究者有参考价值。
We provide three new results concerning quasi-maximum likelihood (QML) estimators in generalized autoregressive conditional heteroskedastic in mean (GARCH-M) models. We first show that, depending on the functional form that we impose in the mean equation, the properties of the model may change and the conditional variance parameter space may be restricted, in contrast to the theory of traditional GARCH processes. Second, we also present a new test for GARCH effects in the GARCH-M context which is simpler to implement than alternative procedures such as in Beg et al. (2001). We propose a new way of dealing with parameters that are not identified by creating composites of parameters that are identified. Third, the finite sample properties of QML estimators are explored in a restricted ARCH-M model and bias and variance approximations are found which show that the larger the volatility of the process the better the variance parameters are estimated. The invariance properties that Lumsdaine (1995) proved for the traditional GARCH are shown not to hold in the GARCH-M. For those researchers who choose not to rely on the first order asymptotic approximation of our proposed test statistic, we also show how our bias expressions can be used to bias correct the QML estimates with a view to improving the finite sample performance of the test. Finally, we show how our new proposed test works in practice in an empirical economic application.