Diffuse Restricted Kalman Filtering

Abstract

This article investigates how the use of an initial diffuse state vector affects the use of the Kalman smoother under linear restrictions. The contribution is twofold. First, using elementary linear and matrix algebra, it is established that it is still possible to obtain restricted smoothed state vectors in the “diffuse” period under quite general conditions. Second, using results from general conditional expectation theory, it is proven that the extension of the restricted Kalman smoother also preserves conditional statistical efficiency, given some meaningful information sets.

Keywords:

• Diffuse initialization
• General conditional expectation theory
• Linear restrictions
• Restricted Kalman filtering
• State space models
• Statistical efficiency

Mathematics Subject Classification:

• 62M10, 15A03

Acknowledgement

I am grateful to the support from FAPERJ (PDR scholarship). It is also important to mention that most of the improvements in this version of this article have been achieved from comments and requirements of an anonymous referee.