Electronic Journal of Statistics

Quasi maximum likelihood estimation for strongly mixing state space models and multivariate Lévy-driven CARMA processes

Eckhard Schlemm and Robert Stelzer

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We consider quasi maximum likelihood (QML) estimation for general non-Gaussian discrete-time linear state space models and equidistantly observed multivariate Lévy-driven continuous-time 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.

Article information

Electron. J. Statist., Volume 6 (2012), 2185-2234.

First available in Project Euclid: 30 November 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F10: Point estimation 62F12: Asymptotic properties of estimators 62M09: Non-Markovian processes: estimation
Secondary: 60G51: Processes with independent increments; Lévy processes 60G10: Stationary processes

Asymptotic normality linear state space model multivariate CARMA process quasi maximum likelihood estimation strong consistency strong mixing


Schlemm, Eckhard; Stelzer, Robert. Quasi maximum likelihood estimation for strongly mixing state space models and multivariate Lévy-driven CARMA processes. Electron. J. Statist. 6 (2012), 2185--2234. doi:10.1214/12-EJS743. https://projecteuclid.org/euclid.ejs/1354284418

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