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.
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Notes
Please see Ang and Piazzesi (2003), Duffee (2011), Duffee and Stanton (2012), Kim and Orphanides (2012), Filipovic and Willems (2018), or Juneja (2020a) for studies on the optimization of ATSM that have reported facing computational challenges in regards to fitting the yield curve. Please see Hamilton and Wu (2012), Adrian et al. (2013), Joslin et al. (2013a), or Diez de Los Rios (2015) for studies devoted to making advances in the optimization algorithms used in their estimation that do not directly involve the invariant transformation.
Please see Matos et al. (2000) for a study that employs invariant transformations to derive a set of formulae that can be used to derive exact solutions to field equations used in the study of gravitational fields in Physics. Their motivation was to reduce the computational burden of estimation. Please see Kaji and Ochiai (2016) for a study of affine transformations for use among computer graphics researchers that lead to computational tractability with a reduced computational burden. Please see Chiarella et al. (2016) for an application of invariant transformations within the context of continuous-time Wishart-dynamic term structure models.
In our discussion and presentation of the state space representation and the related implementation of the Kalman filtering algorithm, for convenience, we retain much of the same notation as Juneja (2020b).
We are grateful to the National Science Foundation for providing us with a grant to use a supercomputer through their Extreme Science and Engineering Discovery Environment (XSEDE) platform to carry out these experiments.
The implications of the results obtained from the within simulation trial variation computations (i.e., variation in the calibration of the filter at the iteration at which convergence was achieved) are identical to those obtained from across the simulation trial variation reported in this section and so we do not report the results from the within simulation trial variation computations. They are available upon request of the author.
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Juneja, J.A. 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?. Comput Manag Sci 18, 73–97 (2021). https://doi.org/10.1007/s10287-020-00380-7
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DOI: https://doi.org/10.1007/s10287-020-00380-7