Local likelihood for non-parametric ARCH(1) models

We propose a non-parametric local likelihood estimator for the log-transformed autoregressive conditional heteroscedastic (ARCH) (1) model. Our non-parametric estimator is constructed within the likelihood framework for non-Gaussian observations: it is different from standard kernel regression smoothing, where the innovations are assumed to be normally distributed. We derive consistency and asymptotic normality for our estimators and show, by a simulation experiment and some real-data examples, that the local likelihood estimator has better predictive potential than classical local regression. A possible extension of the estimation procedure to more general multiplicative ARCH(p) models with p > 1 predictor variables is also described.