Numerical optimization methods for financial time series GARCH(p, q) model, a comparative approach

Maximum likelihood estimation (MLE) is often used in econometric and other statistical models despite its computational considerations and because of its strong theoretical appeal. The non-linear optimization discipline provides feasible alternative methods for calculating MLE's, especially when the special structure may be exploited, for example in probabilistic choice models. This paper examines the estimation of the financial time series model parameters named GARCH(p, q) using four numerical optimization methods and gives numerical comparisons of these methods. Among the issues considered in this paper are the theoretical background of MLE. Also, methods of approximating the Hessian are presented. These include (DFP and BFGS) and statistical approximations (BHHH).