Semiparametric ARCH Models: An Estimating Function Approach

We introduce the method of estimating functions to study the class of autoregressive conditional heteroscedasticity (ARCH) models. We derive the optimal estimating functions by combining linear and quadratic estimating functions. The resultant estimators are more efficient than the quasi-maximum likelihood estimator. If the assumption of conditional normality is imposed, the estimator obtained by using the theory of estimating functions is identical to that obtained by using the maximum likelihood method in finite samples. The relative efficiencies of the estimating function (EF) approach in comparison with the quasi-maximum likelihood estimator are developed. We illustrate the EF approach using a univariate GARCH(1,1) model with conditional normal, Student-t, and gamma distributions. The efficiency benefits of the EF approach relative to the quasi-maximum likelihood approach are substantial for the gamma distribution with large skewness. Simulation analysis shows that the finite-sample properties of the estimators from the EF approach are attractive. EF estimators tend to display less bias and root mean squared error than the quasi-maximum likelihood estimator. The efficiency gains are substantial for highly nonnormal distributions. An example demonstrates that implementation of the method is straightforward.