Optimal bandwidth criteria for nonparametric trend estimation under stochastic volatility error processes

Karim Benhenni (1) , Didier A. Girard (1) , Sana Louhichi (1)

1 IPS - Inférence Processus Stochastiques

This paper is concerned with the optimal selection of the smoothing parameter $h$ in kernel estimation of a trend in nonparametric regression models with (dependent) stochastic volatility errors $\epsilon_i= \sigma_i Z_i$, $i=1,\cdots, n$, where $(\sigma_i)_i$ is referred as the volatility sequences and $(Z_i)_i$ a sequence of i.i.d random variables. We consider three types of volatility sequences; the log-normal volatility, the Gamma volatility and the log-linear volatility with Bernoulli innovations. In fact, based on three criteria for deriving optimal smoothing parameters, namely the average squared error, the mean average squared error and an adjusted Mallows-type criterion to the dependent case, we show that these three minimizers are first-order equivalent in probability. Moreover, we derive the normal asymptotic distribution of the difference between the minimizer of the average squared error and the minimizer based on the Mallows-type criterion. A Monte-Carlo simulation is conducted for a log-normal stochastic volatility model.

Mots clés: Nonparametric trend estimation; Average squared error; Cross validation; Generalized cross validation; Kernel nonparametric models; Mallows criterion; Mean average squared error; Martingale difference sequences; Bandwidth selection; Kernel smoothing; Stochastic volatility processes

Domaines: Statistiques [math.ST]

Karim Benhenni, Didier A. Girard, Sana Louhichi. Optimal bandwidth criteria for nonparametric trend estimation under stochastic volatility error processes. 2020. ⟨hal-02889802v1⟩