A note on estimation of [formula omitted]-stable CARMA processes sampled at low frequencies

Abstract

In this paper, we investigate estimators for α-stable CARMA processes sampled equidistantly. Simulation studies suggest that the Whittle estimator and the estimator presented in Garcia et al. (2011) are consistent estimators for the parameters of stable CARMA processes. For CARMA processes with finite second moments it is well-known that the Whittle estimator is consistent and asymptotically normally distributed. Therefore, in the light-tailed setting the properties of the Whittle estimator for CARMA processes are similar to those of the Whittle estimator for ARMA processes. However, in the present paper we prove that, in general, the Whittle estimator for α-stable CARMA processes sampled at low frequencies is not consistent and highlight why simulation studies suggest something else. Thus, in contrast to the light-tailed setting the properties of the Whittle estimator for heavy-tailed ARMA processes cannot be transferred to heavy-tailed CARMA processes. We elaborate as well that the estimator presented in Garcia et al. (2011) faces the same problems. However, the Whittle estimator for stable CAR(1) processes is consistent.