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Likelihood Functions for State Space Models with Diffuse Initial Conditions

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Author Info
Marc K. Francke () (Vrije Universiteit Amsterdam)
Siem Jan Koopman () (Vrije Universiteit Amsterdam)
Aart de Vos () (Vrije Universiteit Amsterdam)

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Abstract

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.

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Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 08-040/4.

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Date of creation: 14 Apr 2008
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Handle: RePEc:dgr:uvatin:20080040

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Related research
Keywords: Diffuse likelihood; Kalman filter; Marginal likelihood; Multivariate time series models; Profile likelihood;

Find related papers by JEL classification:
C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Estimation
C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions
C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions

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References listed on IDEAS
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  1. Kuo, Biing-Shen, 1999. "Asymptotics Of Ml Estimator For Regression Models With A Stochastic Trend Component," Econometric Theory, Cambridge University Press, vol. 15(01), pages 24-49, February. [Downloadable!]
  2. Francke, Marc K. & de Vos, Aart F., 2007. "Marginal likelihood and unit roots," Journal of Econometrics, Elsevier, vol. 137(2), pages 708-728, April. [Downloadable!] (restricted)
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