Volatility estimation in a nonlinear heteroscedastic functional regression model with martingale difference errors

Abstract

This paper studies the asymptotic properties of the conditional variance estimator in a nonlinear heteroscedastic functional regression model with martingale difference errors. A kernel-type estimator of the conditional variance is introduced when the response is a real-valued random variable and the covariate takes values in an infinite-dimensional space endowed with a semi-metric. Under stationarity and ergodicity assumptions, a uniform almost sure consistency rate is established, along with the asymptotic distribution of the estimator. To build confidence intervals for the conditional variance, two approaches are discussed. One is based on a normal approximation and the other relies on the empirical likelihood. Errors are assumed to form a martingale difference and may depend on the covariate. The results are derived under an ergodic assumption covering a large class of dependent processes, including those satisfying a strong mixing condition. Simulation studies show the performance of the proposed estimator and compare the two methods of constructing confidence intervals. An application to volatility prediction of the daily peak electricity demand in France, using the previous intraday load curve, is also provided.