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Likelihood functions for state space models with diffuse initial conditions

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  • Marc K. Francke
  • Siem Jan Koopman
  • Aart F. de Vos

Abstract

State space models with non-stationary processes and/or 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 article, we consider profile, diffuse and marginal likelihood functions. The marginal likelihood function 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 data transformation that may depend on parameters. Therefore, the diffuse likelihood cannot be used generally for parameter estimation. The marginal likelihood function is based on an orthonormal data transformation that does not depend on parameters. Here we develop a marginal likelihood function for state space models that can be evaluated by the Kalman filter. The so-called diffuse Kalman filter is designed for computing the diffuse likelihood function. We show that a minor modification of the diffuse Kalman filter is needed for the evaluation of our marginal likelihood function. Diffuse and marginal likelihood functions have better small sample properties compared with the profile likelihood function for the estimation of parameters in linear time series models. The results in our article confirm the earlier findings and show that the diffuse likelihood function is not appropriate for a range of state space model specifications. Copyright 2010 Blackwell Publishing Ltd

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File URL: http://www.blackwell-synergy.com/doi/abs/10.1111/j.1467-9892.2010.00673.x
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Bibliographic Info

Article provided by Wiley Blackwell in its journal Journal of Time Series Analysis.

Volume (Year): 31 (2010)
Issue (Month): 6 (November)
Pages: 407-414

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Handle: RePEc:bla:jtsera:v:31:y:2010:i:6:p:407-414

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References

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  1. Rahman, Shahidur & King, Maxwell L., 1997. "Marginal-likelihood score-based tests of regression disturbances in the presence of nuisance parameters," Journal of Econometrics, Elsevier, vol. 82(1), pages 81-106.
  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.
  3. 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.
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Cited by:
  1. José Casals & Sonia Sotoca & Miguel Jerez, 2012. "Minimally Conditioned Likelihood for a Nonstationary State Space Model," Documentos del Instituto Complutense de Análisis Económico 2012-04, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales.
  2. Tommaso, Proietti & Alessandra, Luati, 2012. "Maximum likelihood estimation of time series models: the Kalman filter and beyond," MPRA Paper 39600, University Library of Munich, Germany.
  3. Søren Johansen & Marco Riani & Anthony C. Atkinson, 2012. "The Selection of ARIMA Models with or without Regressors," CREATES Research Papers 2012-46, School of Economics and Management, University of Aarhus.

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