Abstract
We consider a mixed moving average (MMA) process $X$ driven by a Lévy basis and prove that it is weakly dependent with rates computable in terms of the moving average kernel and the characteristic quadruple of the Lévy basis. Using this property, we show conditions ensuring that sample mean and autocovariances of $X$ have a limiting normal distribution. We extend these results to stochastic volatility models and then investigate a Generalized Method of Moments estimator for the supOU process and the supOU stochastic volatility model after choosing a suitable distribution for the mean reversion parameter. For these estimators, we analyze the asymptotic behavior in detail.
Highlights
Levy-driven continuous-time moving average processes, i.e. processes (Xt)t∈R of the form Xt = R f (t − s)dLs with f a deterministic function and L a Levy process, are frequently used to model time series, especially, when dealing with data observed at high frequency
We consider a mixed moving average (MMA) process X driven by a Levy basis and prove that it is weakly dependent with rates computable in terms of the moving average kernel and the characteristic quadruple of the Levy basis
We show conditions ensuring that sample mean and autocovariances of X have a limiting normal distribution. We extend these results to stochastic volatility models and investigate a Generalized Method of Moments estimator for the supOU process and the supOU stochastic volatility model after choosing a suitable distribution for the mean reversion parameter
Summary
To develop a statistical estimation procedure for the MMA and MMA SV model in a semi-parametric framework, where the distribution of the random parameter A is specified in detail, and establishing its asymptotic properties For this end, it is of high importance to understand the dependence structure of the class of MMA processes. For the MMA stochastic volatility models, we show the θ-weak dependence of the return process and the distributional limit of its sample moments. For short memory supOU processes and more general MMA and MMA SV model we can give, exploiting the weak dependence properties, conditions under which functional central limit theorems hold in distribution as well as consider general moments.
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