Likelihood Functions for State Space Models with Diffuse Initial Conditions

Marc K. Francke (), Siem Jan Koopman and Aart de Vos ()
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Marc K. Francke: Vrije Universiteit Amsterdam
Aart de Vos: Vrije Universiteit Amsterdam

No 08-040/4, Tinbergen Institute Discussion Papers from Tinbergen Institute

Abstract: This discussion paper led to an article in the Journal of Time Series Analysis (2010). Vol. 31, pages 407-414.

State space models with nonstationary processes and fixed regression effects require a state vector with diffuse initial conditions. Different likelihood functions can be adopted for the estimation of parameters in time series models with diffuse initial conditions. In this paper we consider profile, diffuse and marginal likelihood functions. The marginal likelihood is defined as the likelihood function of a transformation of the data vector. The transformation is not unique. The diffuse likelihood is a marginal likelihood for a specific data transformation that may depend on parameters. Therefore, the diffuse likelihood can not be used generally for parameter estimation. Our newly proposed marginal likelihood function is based on an orthonormal transformation that does not depend on parameters. Likelihood functions for state space models are evaluated using the Kalman filter. The diffuse Kalman filter is specifically designed for computing the diffuse likelihood function. We show that a modification of the diffuse Kalman filter is needed for the evaluation of our proposed marginal likelihood function. Diffuse and marginal likelihood functions have better small sample properties compared to the profile likelihood function for the estimation of parameters in linear time series models. The results in our paper confirm the earlier findings and show that the diffuse likelihood function is not appropriate for a range of state space model specifications.

Keywords: Diffuse likelihood; Kalman filter; Marginal likelihood; Multivariate time series models; Profile likelihood (search for similar items in EconPapers)
JEL-codes: C13 C22 C32 (search for similar items in EconPapers)
Date: 2008-04-14
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Related works:
Journal Article: Likelihood functions for state space models with diffuse initial conditions (2010)
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