Analytic derivatives of asymmetric Garch models

ABSTRACT The model parameters of a Garch process can be estimated by using numerical optimization to maximize the log-likelihood. However, widely used optimization techniques (such as Newton methods) require information concerning the derivatives of the log-likelihood. This paper presents computer algorithms that compute the analytic derivatives of asymmetric regression-Garch(p,q) processes, with series shocks from either a Gaussian distribution or a Student’s t-distribution. Initial estimates and pre-observed values for the Garch model parameters are also discussed. Monte Carlo simulation results are presented which compare the results obtained using both analytic and numeric derivatives. It is found that the numeric derivatives are faster but provide less accurate parameter estimates for short Garch sequences.