Small Sample Estimation Bias in GARCH Models with Any Number of Exogenous Variables in the Mean Equation
研究了GARCH模型均值方程中加入外生变量时，准最大似然估计的偏差如何变化，发现偏差随外生变量数量呈加性比例变化，并可用于模型误设检验。
In this article we show how bias approximations for the quasi maximum likelihood estimators of the parameters in Generalized Autoregressive Conditional Heteroskedastic (GARCH)(p, q) models change when any number of exogenous variables are included in the mean equation. The approximate biases are shown to vary in an additive and proportional way in relation to the number of exogenous variables, and they do not depend on the moments of the regressors under the correct specification of the model. This suggests a rule of thumb in testing for misspecification in GARCH models. We also extend the theoretical bias approximations given in Linton (1997) for the GARCH(1, 1). Because the expressions are not in closed form, we concentrate in detail, and for simplicity of interpretation, on the ARCH(1) model. At each stage, we check our theoretical results by simulation and generally, we find that the approximations are quite accurate for sample sizes of at least 50. We find that the biases are not trivial in some circumstances and we discuss how the bias approximations may be used, in practice, to reduce the bias. We also carry out simulations for the GARCH(1, 1) model and show that the biases change as predicted by the approximations when the mean equation is augmented. Finally, we illustrate the usefulness of our approach for U.S. monthly inflation rates.