Abstract
In this paper, we give an autoregressive model of order 1 type of characterization covering all multivariate strictly stationary processes indexed by the set of integers. Consequently, under square integrability, we derive continuous time algebraic Riccati equations for the parameter matrix of the characterization. This provides us with a natural way to define the corresponding estimator. In addition, we show that the estimator inherits consistency from autocovariances of the stationary process. Furthermore, the limiting distribution is given by a linear function of the limiting distribution of the autocovariances. We also present the corresponding existing results of the continuous time setting paralleling them to the discrete case treated in this paper.
Highlights
Stationary stochastic processes provide a significant instrument for modeling numerous temporal phenomena related to different fields of science
The method is based on continuous time algebraic matrix Riccati equations (CAREs) and it is applicable in estimation of essentially any square integrable multivariate stationary process
The characterization leads to covariance based symmetric CAREs for the related parameter matrix providing us with a novel estimation method of multivariate discrete stationary processes
Summary
Stationary stochastic processes provide a significant instrument for modeling numerous temporal phenomena related to different fields of science. The method is based on continuous time algebraic matrix Riccati equations (CAREs) and it is applicable in estimation of essentially any square integrable multivariate stationary process. The characterization leads to covariance based symmetric CAREs for the related parameter matrix providing us with a novel estimation method of multivariate discrete stationary processes. In outline, this results in the following correspondence between the continuous time case. The characterizations provide us with models of stationary processes yielding symmetric CAREs for the corresponding parameter matrices These equations can be applied in a similar manner in estimation in both cases. For the reader’s convenience, all technical proofs are postponed to section 3
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