Maximum Likelihood Estimation of the APARCH model with Skew Distribution for the Innovation Process

A method normally used in empirical financial studies to estimate the parameters of a general autoregressive conditional heteroskedasticity model is the quasi-maximum likelihood, which maximizes the likelihood function assuming conditional normality, also if it can be a false assumption. When it is possible to assume a nonnormal distribution of errors for this kind of models, it has been shown that there is a loss of efficiency of quasi-maximum likelihood estimators in finite samples with respect to maximum likelihood estimators. In this paper we study, with an empirical application to the daily returns of NASDAQ stock market index, the maximum likelihood estimates of the parameters of the asymmetric power ARCH model, a generalization of the general autoregressive conditional heteroskedasticity model, with skew distributions for the innovation process. The distributions considered are the Student-t, the exponential power and the generalized secant hyperbolic distributions, with reparametrization of the densities which adds inverse scale factors in positive and negative orthants in order to take the skewness into account. For comparison, we have analyzed the daily returns also with the quasi-maximum and the semiparametric maximum likelihood estimation procedures. We have used a quasi-Newton algorithm to optimize the average log-likelihood functions, in which analytical derivatives of the parameters have been obtained by MathStatica, a package of the computer algebra system Mathematica.