Single and multiple error state-space models for signal extraction

Abstract

We compare the results obtained by applying the same signal-extraction procedures to two observationally equivalent state-space forms. The first model has different errors affecting the states and the observations, while the second has a single perturbation term which coincides with the one-step-ahead forecast error. The signals extracted from both forms are very similar but their variances are drastically different, because the states for the single-source error representation collapse to exact values while those coming from the multiple-error model remain uncertain. The implications of this result are discussed both with theoretical arguments and practical examples. We find that single error representations have advantages to compute the likelihood or to adjust for seasonality, while multiple error models are better suited to extract a trend indicator. Building on this analysis, it is natural to adopt a ‘best of both worlds’ approach, which applies each representation to the task in which it has comparative advantage.

Keywords:

• state-space model
• signal extraction
• seasonal adjustment
• Kalman filter
• trend

MSC Classification:

• 60G35
• 93E11
• 93E14
• 91B84

Acknowledgements

We benefitted from the ideas and suggestions by Antonio Garcia-Ferrer, Marcos Bujosa, Esther Ruiz and an anonymous referee. Their contribution is gratefully acknowledged although any errors are our sole responsibility. This work was finished while Sonia Sotoca and Miguel Jerez were visiting UC Davis. Special thanks are due to the Department of Economics for providing ideal working conditions.

Funding

This work was supported by the Programa Becas Complutenses Del Amo, from Universidad Complutense de Madrid, UC Davis and Ministerio de Economia y Competitividad [grant number ECO2011-23972].