Estimation of the volatility function : non-parametric and semiparametric approaches
We investigate two problems in modelling time series data that exhibit conditional heteroscedasticity. The first part deals with the local maximum likelihood estimation of volatility functions which are in the form of conditional variance functions. The existing estimation procedures yield plausible results. Yet, they often fail to take into account special features of the data at the cost of reduced accuracy of prediction. More precisely, many of the parametric and nonparametric conditional variance models ignore the fact that the error distribution departs significantly from gaussian distribution. We propose a novel nonparametric estimation procedure that replaces popular local least squares method with local maximum likelihood estimation. Intuitively, using information from the error distribution improves the estimators and therefore increases the accuracy in prediction. This conclusion is proved theoretically and illustrated by numerical examples. In addition, we show that the proposed estimator adapts asymptotically to the error distribution as well as to the mean regression function. Applications with real data examples demonstrate the potential use of the adaptive maximum likelihood estimator in financial risk management. The second part deals with the variable selection for a particular class of semipara-metric models known as the partial linear models. The existing selection methods are computationally demanding. The proposed selection procedure is computationally more efficient. In particular, if P and Q are the number of linear and nonparametric candidate regressors, respectively, then the proposed procedure reduces the order of the number of variable subsets to be investigated from 2 Q+P to 2Q + 2 P. At the same time, it maintains all the good properties of existing methods, such as consistency. The latter is proven theoretically and confirmed numerically by simulated examples. The results are presented for the mean regression function while the generalization to the conditional variance function is discussed separately.