Propagation of initial errors on the parameters for linear and Gaussian state space models

TL;DR

This paper analyzes how initial parameter estimation errors affect the Kalman filter's accuracy in linear and Gaussian state space models, extending the analysis to almost linear models with the Extended Kalman filter, with applications in econometrics.

Contribution

It provides explicit expressions for the propagation of parameter bias in Kalman filtering and extends the analysis to Extended Kalman filters for nearly linear models.

Findings

Bias in parameter estimates propagates to state estimates.

Explicit formulas for bias propagation in linear Gaussian models.

Extension of results to almost linear models with Extended Kalman filter.

Abstract

For linear and Gaussian state space models parametrized by $\theta_0 \in \Theta \subset \mathbb{R}^r, r \geq 1$ corresponding to the vector of parameters of the model, the Kalman filter gives exactly the solution for the optimal filtering under weak assumptions. This result supposes that $\theta_0$ is perfectly known. In most real applications, this assumption is not realistic since $\theta_0$ is unknown and has to be estimated. In this paper, we analysis the Kalman filter for a biased estimator of $\theta_0$. We show the propagation of this bias on the estimation of the hidden state. We give an expression of this propagation for linear and Gaussian state space models and we extend this result for almost linear models estimated by the Extended Kalman filter. An illustration is given for the autoregressive process with measurement noises widely studied in econometrics to model economic and financial data.