Parametric estimation of the driving Lévy process of multivariate CARMA processes from discrete observations

Abstract

We consider the parametric estimation of the driving Lévy process of a multivariate continuous-time autoregressive moving average (MCARMA) process, which is observed on the discrete time grid $(0,h,2h,...)$. Beginning with a new state space representation, we develop a method to recover the driving Lévy process exactly from a continuous record of the observed MCARMA process. We use tools from numerical analysis and the theory of infinitely divisible distributions to extend this result to allow for the approximate recovery of unit increments of the driving Lévy process from discrete-time observations of the MCARMA process. We show that, if the sampling interval $h=h_N$ is chosen dependent on $N$, the length of the observation horizon, such that $N{h}_N$ converges to zero as $N$ tends to infinity, then any suitable generalized method of moments estimator based on this reconstructed sample of unit increments has the same asymptotic distribution as the one based on the true increments, and is, in particular, asymptotically normally distributed.

38 pages, four figures; to appear in Journal of Multivariate Analysis

References in corpus (3)

• Multivariate supOU processes
• Quasi maximum likelihood estimation for strongly mixing state space models and multivariate Lévy-driven CARMA processes
• Multivariate CARMA processes, continuous-time state space models and complete regularity of the innovations of the sampled processes

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