Efficient likelihood estimation in state space models

The Annals of Statistics · 2006 · Volume 34 · Issue 4

Abstract

Motivated by studying asymptotic properties of the maximum likelihood estimator (MLE) in stochastic volatility (SV) models, in this paper we investigate likelihood estimation in state space models. We first prove, under some regularity conditions, there is a consistent sequence of roots of the likelihood equation that is asymptotically normal with the inverse of the Fisher information as its variance. With an extra assumption that the likelihood equation has a unique root for each n, then there is a consistent sequence of estimators of the unknown parameters. If, in addition, the supremum of the log likelihood function is integrable, the MLE exists and is strongly consistent. Edgeworth expansion of the approximate solution of likelihood equation is also established. Several examples, including Markov switching models, ARMA models, (G)ARCH models and stochastic volatility (SV) models, are given for illustration.

Keywords

(G)ARCH models, 62E25, 62F12, 62M09, ARMA models, asymptotic expansion, asymptotic normality, consistency, efficiency, incomplete data, iterated random functions, Markov switching models, maximum likelihood, stochastic volatility models

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