Two main classes of models are used in the analysis of financial time series: the ARCH-type models and the stochastic volatility models. The first class of models was introduced by R.F. Engel [Econometrica 50, 987-1007 (1982; Zbl 0491.62099)] and had a widespread diffusion, with an evolution towards more complicated formulations allowing for more realistic hypotheses. The models of the second class have a different structure because they assume that the variance of the conditional distribution of the observations depends on a latent variable that may adequately represent the flow of information arriving into financial markets. The first direct maximum likelihood approach dates back to 1988 to M. Fridman and L. Harris [J. Bus. Econ. Stat. 16, 284-291 (1998)]. They described a quadrature method which allows one to compute the likelihood of the model with a required precision. In practice the method makes use of numerical derivatives.
The aim of this paper is to extend the approach of Fridman and Harris through the computation of the first and second analytical derivatives of the approximate likelihood. This strategy helps squeezing the computational time in the estimation of the parameters since the Newton-Raphson algorithm may be used to maximize the approximate likelihood. Moreover, these derivatives approximate the corresponding derivatives of the exact likelihood. On the basis of the second derivative it is possible to compute the standard error of the estimator and confidence intervals for the parameters may be constructed. The reliability of the procedure is established by a simulation study involving processes with parameters already selected by other authors in order to facilitate the comparison of the results with other estimation methods.