Parameter estimation

The Kalman filter is has long been a standard tool for control engineers. However, the initial introduction of the filter to an audience of economists, while emphasizing the relevance of the filter in modeling economic systems, also pointed out the need to assume known covariance matrices for the various noise processes in the model. In an article in the Annals of Economic and Social Measurement, Athens (1972), after pointing out that the noise parameter in the system equation could represent “input uncertainties and deterministic modeling errors”, continues by saying: “Thus the covariance [of the error term in the system equation that is] selected by the designer should incorporate his judgment on the importance of the higher order terms in the validity of the linearized model. Thus, the ‘more nonlinear’ the system dynamics, the ‘larger’ the [covariance] should be. The white noise ... in the observation equation plays a similar role. Not only should it reflect the inherent uncertainty of the measurements due to sensor inaccuracies, but it should also be used to model the implications of neglecting [higher order terms] to obtain a linear equation.” (Athens (1972), p. 472, emphasis added)