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Computing Observation Weights for Signal Extraction and Filtering

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  • A. C. Harvey
  • Siem Jan Koopman

    (Free University of Amsterdam)

Abstract

We present algorithms for computing the weights implicitly assigned to observations when estimating unobserved components using a model in state space form. The algorithms are for both filtering and signal extraction. In linear time-invariant models such weights can sometimes be obtained analytically from the Wiener-Kolmogorov formulae. Our method is much more general, being applicable to any model with a linear state space form, including models with deterministic components and time-varying state matrices. It applies to multivariate models and it can be used when there are data irregularities, such as missing observations. The algorithms can be useful for a variety of purposes in econometrics and statistics: (i) the weights for signal extraction can be regarded as equivalent kernel functions and hence the weight pattern can be compared with the kernels typically used in nonparametric trend estimation; (ii) the weight algorithm for filtering implicitly computes the coefficients of the vector error-correction model (VECM) representation of any linear time series model; (iii) as a by-product the mean square errors associated with estimators may be obtained; (iv) the algorithm can be incorporated within a Markov chain Monte Carlo (MCMC) method enabling computation of weights assigned to observations when computing the posterior mean of unobserved components within a Bayesian treatment. A wide range of illustrations show how the algorithms may provide important insights in empirical analysis. The algorithms are provided and implemented for the software package SsfPack 2.3 , that is a set of filtering, smoothing and simulation algorithms for models in state space form (see www.ssfpack.com). Some details of implementation and example programs are given in the appendix of the paper.

Suggested Citation

  • A. C. Harvey & Siem Jan Koopman, 2000. "Computing Observation Weights for Signal Extraction and Filtering," Econometric Society World Congress 2000 Contributed Papers 0888, Econometric Society.
    • Handle: RePEc:ecm:wc2000:0888
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    References listed on IDEAS

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    1. Siem Jan Koopman & Neil Shephard & Jurgen A. Doornik, 1999. "Statistical algorithms for models in state space using SsfPack 2.2," Econometrics Journal, Royal Economic Society, vol. 2(1), pages 107-160.
    2. Balke, Nathan S, 1993. "Detecting Level Shifts in Time Series," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(1), pages 81-92, January.
    3. Andrew Harvey & Siem Jan Koopman, 2000. "Signal extraction and the formulation of unobserved components models," Econometrics Journal, Royal Economic Society, vol. 3(1), pages 84-107.
    4. Durbin, James & Koopman, Siem Jan, 2012. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, edition 2, number 9780199641178, December.
    5. Shephard, N. & Pitt, M.K., 1995. "Likelihood Analysis of Non-Gaussian Parameter-Driven Models," Economics Papers 108, Economics Group, Nuffield College, University of Oxford.
    6. Craig F. Ansley & Robert Kohn, 1990. "Filtering And Smoothing In State Space Models With Partially Diffuse Initial Conditions," Journal of Time Series Analysis, Wiley Blackwell, vol. 11(4), pages 275-293, July.
    7. Andrew Harvey & Chia‐Hui Chung, 2000. "Estimating the underlying change in unemployment in the UK," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 163(3), pages 303-309.
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