Local polynomial estimators of the volatility function in nonparametric autoregression

In this paper we consider a class of dynamic models in which both the conditional mean and the conditional variance (volatility) are unknown functions of the past. We first derive probabilistic conditions under which nonparametric estimation of these functions is possible. We then construct an estimator based on local polynomial fitting. We examine the rates of convergence of these estimators and give a result on their asymptotic normality. The local polynomial fitting of the volatility function is applied to different foreign exchange rate series. We find an asymmetric U-shaped 'smiling face' form of the volatility function.