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Likelihood functions for state space models with diffuse initial conditions

Author

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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

Suggested Citation

  • Marc K. Francke & Siem Jan Koopman & Aart F. de Vos, 2010. "Likelihood functions for state space models with diffuse initial conditions," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(6), pages 407-414, November.
  • Handle: RePEc:bla:jtsera:v:31:y:2010:i:6:p:407-414
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    References listed on IDEAS

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    1. Barr Rosenberg, 1973. "Random Coefficients Models: The Analysis of a Cross Section of Time Series by Stochastically Convergent Parameter Regression," NBER Chapters,in: Annals of Economic and Social Measurement, Volume 2, number 4, pages 399-428 National Bureau of Economic Research, Inc.
    2. Durbin, James & Koopman, Siem Jan, 2012. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, edition 2, number 9780199641178, June.
    3. Francke, Marc K. & de Vos, Aart F., 2007. "Marginal likelihood and unit roots," Journal of Econometrics, Elsevier, vol. 137(2), pages 708-728, April.
    4. 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.
    5. 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. Marczak, Martyna & Proietti, Tommaso & Grassi, Stefano, 2018. "A data-cleaning augmented Kalman filter for robust estimation of state space models," Econometrics and Statistics, Elsevier, vol. 5(C), pages 107-123.
    2. Søren Johansen & Marco Riani & Anthony C. Atkinson, 2012. "The Selection of ARIMA Models with or without Regressors," Discussion Papers 12-17, University of Copenhagen. Department of Economics.
    3. repec:spr:empeco:v:53:y:2017:i:4:d:10.1007_s00181-016-1179-0 is not listed on IDEAS
    4. Tommaso Proietti & Alessandra Luati, 2013. "Maximum likelihood estimation of time series models: the Kalman filter and beyond," Chapters,in: Handbook of Research Methods and Applications in Empirical Macroeconomics, chapter 15, pages 334-362 Edward Elgar Publishing.
    5. José Casals & Sonia Sotoca & Miguel Jerez, 2012. "Minimally Conditioned Likelihood for a Nonstationary State Space Model," Documentos de Trabajo del ICAE 2012-04, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.

    More about this item

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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