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Local polynomial estimators of the volatility function in nonparametric autoregression. (English) Zbl 0904.62047
Summary: 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.

62G07 Density estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P20 Applications of statistics to economics
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