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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.

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  • 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
    DOI: 10.1111/j.1467-9892.2010.00673.x
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    4. 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.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. repec:jss:jstsof:41:i07 is not listed on IDEAS
    10. 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.
    11. 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.
    12. 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.
    13. Webel, Karsten & Smyk, Anna, 2023. "Towards seasonal adjustment of infra-monthly time series with JDemetra+," Discussion Papers 24/2023, Deutsche Bundesbank.
    14. 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.
    15. Proietti, Tommaso & Pedregal, Diego J., 2023. "Seasonality in High Frequency Time Series," Econometrics and Statistics, Elsevier, vol. 27(C), pages 62-82.
    16. 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.
    17. 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.

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    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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