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Likelihood Functions for State Space Models with Diffuse Initial Conditions

Author

Listed:
  • Marc K. Francke

    (Vrije Universiteit Amsterdam)

  • Siem Jan Koopman

    (Vrije Universiteit Amsterdam)

  • Aart de Vos

    (Vrije Universiteit Amsterdam)

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.

Suggested Citation

  • Marc K. Francke & Siem Jan Koopman & Aart de Vos, 2008. "Likelihood Functions for State Space Models with Diffuse Initial Conditions," Tinbergen Institute Discussion Papers 08-040/4, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20080040
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    Cited by:

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    2. Helske, Jouni, 2017. "KFAS: Exponential Family State Space Models in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 78(i10).
    3. 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.
    4. Yue Zhao & Difang Wan, 2018. "Institutional high frequency trading and price discovery: Evidence from an emerging commodity futures market," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 38(2), pages 243-270, February.
    5. Webel, Karsten & Smyk, Anna, 2023. "Towards seasonal adjustment of infra-monthly time series with JDemetra+," Discussion Papers 24/2023, Deutsche Bundesbank.
    6. Proietti, Tommaso & Pedregal, Diego J., 2023. "Seasonality in High Frequency Time Series," Econometrics and Statistics, Elsevier, vol. 27(C), pages 62-82.
    7. Nilsen, Øivind Anti & Raknerud, Arvid & Skjerpen, Terje, 2011. "Using the Helmert-transformation to reduce dimensionality in a mixed model: Application to a wage equation with worker and firm heterogeneity," Discussion Paper Series in Economics 11/2011, Norwegian School of Economics, Department of Economics, revised 04 Oct 2011.
    8. Tommaso Proietti & Alessandra Luati, 2013. "Maximum likelihood estimation of time series models: the Kalman filter and beyond," Chapters, in: Nigar Hashimzade & Michael A. Thornton (ed.), Handbook of Research Methods and Applications in Empirical Macroeconomics, chapter 15, pages 334-362, Edward Elgar Publishing.
    9. 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.
    10. Poncela, Pilar & Ruiz, Esther & Miranda, Karen, 2021. "Factor extraction using Kalman filter and smoothing: This is not just another survey," International Journal of Forecasting, Elsevier, vol. 37(4), pages 1399-1425.
    11. Ledenyov, Dimitri O. & Ledenyov, Viktor O., 2015. "Wave function method to forecast foreign currencies exchange rates at ultra high frequency electronic trading in foreign currencies exchange markets," MPRA Paper 67470, University Library of Munich, Germany.
    12. 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.
    13. repec:jss:jstsof:41:i07 is not listed on IDEAS
    14. Victor Bystrov, 2020. "Identification and Estimation of Initial Conditions in Non-Minimal State-Space Models," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 12(4), pages 413-429, December.
    15. A. Smyk & K. Webel, 2024. "Vers une désaisonnalisation des séries temporelles infra-mensuelles avec JDemetra+," Documents de Travail de l'Insee - INSEE Working Papers m2024-04, Institut National de la Statistique et des Etudes Economiques.
    16. Ledenyov, Dimitri O. & Ledenyov, Viktor O., 2013. "On the Stratonovich – Kalman - Bucy filtering algorithm application for accurate characterization of financial time series with use of state-space model by central banks," MPRA Paper 50235, University Library of Munich, Germany.
    17. Raïsa Basselier & David de Antonio Liedo & Jana Jonckheere & Geert Langenus, 2018. "Can inflation expectations in business or consumer surveys improve inflation forecasts?," Working Paper Research 348, National Bank of Belgium.
    18. Øivind A. Nilsen & Arvid Raknerud & Terje Skjerpen, 2017. "Estimation of a model for matched panel data with high-dimensional two-way unobserved heterogeneity," Empirical Economics, Springer, vol. 53(4), pages 1657-1680, December.

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    More about this item

    Keywords

    Diffuse likelihood; Kalman filter; Marginal likelihood; Multivariate time series models; Profile likelihood;
    All these keywords.

    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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