How do invariant transformations affect the calibration and optimization of the Kalman filtering algorithm used in the estimation of continuous-time affine term structure models?

Abstract

In studying affine term structure models (ATSM), researchers have made significant theoretical advances in simplifying the burden of the Kalman filtering algorithm (KF) procedure and its optimization process by incorporating mathematical relationships implied by invariant transformations. Through the usage of National Science Foundation granted supercomputing resources, we assess the effects of invariant transformations on the calibration and optimization of the KF when used in the estimation of three, four, and five factor ATSM. From our analysis, we find that restrictions imposed on ATSM by these transformations do affect the estimation risk in the optimization process of the KF, albeit in different ways. Of all components connected to the optimization process, the prediction error of the measurement of the bond yield constructed by the KF and its affiliated covariance matrix demonstrate the least variability. The conditional covariance of the state vector at the prediction and update step demonstrates the largest variability. Estimation of the Kalman gain and conditional mean of the state vector at both the prediction and update step demonstrate a moderate degree of estimation risk. We interpret from our research that some considerations contributing to variation in the effect of such restrictions include the increase in the number of constraints that come with the greater number of factors and the type of invariant transformation undertaken. Our findings support theoretical predictions made by economic theory. Directions for further research are provided.