A solution to the positivity problem in the state-space approach to modeling vector-valued time series
Date of Original Version
In the state-space approach to modeling vector-valued time series, obtaining a stochastic process model from a given autocovariance model requires an algebraic Riccati equation to be solved for the state-vector covariance matrix. This Riccati equation fails to have a positive definite solution for all cases where the Fourier transform of the sequence defined by the covariance model is not positive semidefinite. Such cases occur frequently in empirical work. The procedure presented in this paper overcomes this problem by showing how to modify the covariance model in such a way that the Riccati equation has a positive definite solution. An attractive feature of this procedure is that the value of the zero-lag autocovariance matrix of the modified covariance model is constrained to equal the unconditional error sum of squares of the data. An empirical example illustrating the performance of the algorithm is provided. © 1993.
Journal of Economic Dynamics and Control
Vaccaro, Richard J., and Tomislav Vukina. "A solution to the positivity problem in the state-space approach to modeling vector-valued time series." Journal of Economic Dynamics and Control 17, 3 (1993): 401-421. doi:10.1016/0165-1889(93)90004-C.