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Computing observation weights for signal extraction and filtering

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  • Koopman, Siem Jan
  • Harvey, Andrew

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.
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  • Koopman, Siem Jan & Harvey, Andrew, 2003. "Computing observation weights for signal extraction and filtering," Journal of Economic Dynamics and Control, Elsevier, vol. 27(7), pages 1317-1333, May.
    • Handle: RePEc:eee:dyncon:v:27:y:2003:i:7:p:1317-1333
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    1. 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.
    2. 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.
    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. 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.
    6. Neil Shephard & Michael K Pitt, 1995. "Likelihood analysis of non-Gaussian parameter driven models," Economics Papers 15 & 108., Economics Group, Nuffield College, University of Oxford.
    7. 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.
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