Computing Observation Weights for Signal Extraction and Filtering

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.