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Reconstructing the Kalman Filter for Stationary and Non Stationary Time Series

Author

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  • Ralph D. Snyder
  • Catherine S. Forbes

Abstract

A Kalman filter, suitable for application to a stationary or a non-stationary time series, is proposed. It works on time series with missing values. It can be used on seasonal time series where the associated state space model may not satisfy the traditional observability condition. A new concept called an 'extended normal random vector' is introduced and used throughout the paper to simplify the specification of the Kalman filter. It is an aggregate of means, variances, covariances and other information needed to define the state of a system at a given point in time. By working with this aggregate, the algorithm is specified without direct recourse to those relatively complex formulae for calculating associated means and variances, normally found in traditional expositions of the Kalman filter. A computer implementation of the algorithm is also described where the extended normal random vector is treated as an object; the operations of addition, subtraction and multiplication are overloaded to work on instances of this object; and a form of statistical conditioning is implemented as an operator.

Suggested Citation

  • Ralph D. Snyder & Catherine S. Forbes, 2002. "Reconstructing the Kalman Filter for Stationary and Non Stationary Time Series," Monash Econometrics and Business Statistics Working Papers 14/02, Monash University, Department of Econometrics and Business Statistics.
    • Handle: RePEc:msh:ebswps:2002-14
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    File URL: http://www.buseco.monash.edu.au/ebs/pubs/wpapers/2002/wp14-02.pdf
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    References listed on IDEAS

    as
    1. Everette S. Gardner, Jr. & Ed. Mckenzie, 1985. "Forecasting Trends in Time Series," Management Science, INFORMS, vol. 31(10), pages 1237-1246, October.
    2. Ralph D. Snyder & Grant R. Saligari, 1996. "Initialization Of The Kalman Filter With Partially Diffuse Initial Conditions," Journal of Time Series Analysis, Wiley Blackwell, vol. 17(4), pages 409-424, July.
    3. Harvey,Andrew C., 1991. "Forecasting, Structural Time Series Models and the Kalman Filter," Cambridge Books, Cambridge University Press, number 9780521405737.
    4. S. J. Koopman & J. Durbin, 2000. "Fast Filtering and Smoothing for Multivariate State Space Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 21(3), pages 281-296, May.
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    Cited by:

    1. Rob Hyndman & Muhammad Akram & Blyth Archibald, 2008. "The admissible parameter space for exponential smoothing models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(2), pages 407-426, June.

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    More about this item

    Keywords

    Time series analysis; forecasting; Kalman filter; State space models; Object-oriented programming.;
    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
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

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