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Biometrika 2007 94(4):827-839; doi:10.1093/biomet/asm074
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© 2007 Biometrika Trust

Articles

Monte Carlo Estimation for Nonlinear Non-Gaussian State Space Models

Borus Jungbacker and Siem Jan Koopman

Department of Econometrics, VU University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands bjungbacker{at}feweb.vu.nl s.j.koopman{at}feweb.vu.nl

Received for publication 1 December 2005. Revision received 1 May 2007.
  Abstract

We develop a proposal or importance density for state space models with a nonlinear non-Gaussian observation vector y ~ p(y¦{theta}) and an unobserved linear Gaussian signal vector {theta} ~ p({theta}). The proposal density is obtained from the Laplace approximation of the smoothing density p({theta}¦y). We present efficient algorithms to calculate the mode of p({theta}¦y) and to sample from the proposal density. The samples can be used for importance sampling and Markov chain Monte Carlo methods. The new results allow the application of these methods to state space models where the observation density p(y¦{theta}) is not log-concave. Additional results are presented that lead to computationally efficient implementations. We illustrate the methods for the stochastic volatility model with leverage.

Key Words: Importance sampling • Kalman filtering • Markov chain Monte Carlo • Newton–Raphson • Posterior mode • Simulation smoothing • Stochastic volatility model


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