Abstract
We estimate the parameter of a stationary time series process by minimizing the integrated weighted mean squared error between the empirical and simulated characteristic function, when the true characteristic functions cannot be explicitly computed. Motivated by Indirect Inference, we use a Monte Carlo approximation of the characteristic function based on i.i.d. simulated blocks. As a classical variance reduction technique, we propose the use of control variates for reducing the variance of this Monte Carlo approximation. These two approximations yield two new estimators that are applicable to a large class of time series processes. We show consistency and asymptotic normality of the parameter estimators under strong mixing, moment conditions, and smoothness of the simulated blocks with respect to its parameter. In a simulation study we show the good performance of these new simulation based estimators, and the superiority of the control variates based estimator for Poisson driven time series of counts.
Highlights
Let (Xj)j∈Z be a stationary time series, whose distribution depends on θ ∈ Θ ⊂ Rq for some q ∈ N
Feuerverger (1990) proposed an estimator based on matching the empirical characteristic function computed from blocks of the observed time series and the true chf
We find that the simulation based parameter estimator is asymptotically normal with asymptotic covariance matrix equal to the one of the oracle estimator as derived in Knight and Yu (2002)
Summary
Let (Xj)j∈Z be a stationary time series, whose distribution depends on θ ∈ Θ ⊂ Rq for some q ∈ N. The practical choice of this set depends on the problem at hand and the asymptotic results derived in Feuerverger (1990) do not offer practical guidance for choosing these points To overcome this limitation Yu (1998) and Knight and Yu (2002) considered a integrated weighted squared distance between the empirical and the true chfs. The oracle estimator does not apply, since the chf of a Poisson-AR process cannot be computed in closed form For this model and different parameter sets, both the simulation based and the control variates based estimators perform satisfactorily, and the control variates based estimator improves the performance of the simulation based estimator considerably.
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