Analytic derivatives and the computation of GARCH estimates
证明在单变量GARCH模型中，使用解析一阶和二阶导数进行最大似然估计是可行的，且能显著提升计算效率；同时比较了多种梯度算法，并提出一种结合信息矩阵与精确Hessian矩阵的全局高效计算算法。
In the context of univariate GARCH models we show how analytic first and second derivatives of the log-likelihood can be successfully employed for estimation purposes. Maximum likelihood GARCH estimation usually relies on the numerical approximation to the log-likelihood derivatives, on the grounds that an exact analytic differentiation is much too burdensome. We argue that this is not the case and that the computational benefit of using the analytic derivatives (first and second) may be substantial. Furthermore, we make a comparison of various gradient algorithms that are used for the maximization of the GARCH Gaussian likelihood. We suggest the implementation of a globally efficient computation algorithm that is obtained by suitably combining the use of the estimated information matrix with that of the exact Hessian during the maximization process. As this would appear a straightforward extension, we then study the finite sample performance of the exact Hessian and its approximations (that is, the estimated information, outer products and misspecification robust matrices) in inference.