Quasi maximum likelihood estimation for strongly mixing state space models and multivariate L\'evy-driven CARMA processes

TL;DR

This paper develops and analyzes quasi maximum likelihood estimation methods for non-Gaussian state space models and multivariate Le9vy-driven MCARMA processes, establishing their consistency and asymptotic normality.

Contribution

It extends QML estimation theory to non-Gaussian and continuous-time models, providing new asymptotic results and practical applicability for econometric data.

Findings

QML estimators are consistent and asymptotically normal under standard conditions.

The methods are applicable for any sampling frequency in continuous-time models.

Simulation and real data demonstrate the effectiveness of the proposed approach.

Abstract

We consider quasi maximum likelihood (QML) estimation for general non-Gaussian discrete-ime linear state space models and equidistantly observed multivariate L\'evy-driven continuoustime autoregressive moving average (MCARMA) processes. In the discrete-time setting, we prove strong consistency and asymptotic normality of the QML estimator under standard moment assumptions and a strong-mixing condition on the output process of the state space model. In the second part of the paper, we investigate probabilistic and analytical properties of equidistantly sampled continuous-time state space models and apply our results from the discrete-time setting to derive the asymptotic properties of the QML estimator of discretely recorded MCARMA processes. Under natural identifiability conditions, the estimators are again consistent and asymptotically normally distributed for any sampling frequency. We also demonstrate the practical applicability of our method through a simulation study and a data example from econometrics.