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Analytic derivatives and the computation of GARCH estimates
Author(s) -
Fiorentini Gabriele,
Calzolari Giorgio,
Panattoni Lorenzo
Publication year - 1996
Publication title -
journal of applied econometrics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.878
H-Index - 99
eISSN - 1099-1255
pISSN - 0883-7252
DOI - 10.1002/(sici)1099-1255(199607)11:4<399::aid-jae401>3.0.co;2-r
Subject(s) - hessian matrix , autoregressive conditional heteroskedasticity , univariate , inference , mathematics , expectation–maximization algorithm , computation , context (archaeology) , maximization , extension (predicate logic) , mathematical optimization , computer science , econometrics , algorithm , statistics , maximum likelihood , multivariate statistics , volatility (finance) , artificial intelligence , paleontology , biology , programming language
Abstract 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.