Identification Of Covariance Structures

The issue of identification of covariance structures, which arises in a number of different contexts, has been so far linked to conditions on the true parameters to be estimated. In this paper, this limitation is removed. As done by Johansen (1995, Journal of Econometrics 69, 112–132) in the con- text of linear models, the present paper provides necessary and sufficient conditions for the identification of a covariance structure that depends only on the constraints and can therefore be checked independently of estimated parameters. A structure condition is developed, which only depends on the structure of the constraints. It is shown that this condition, if coupled with the familiar order condition, provides a sufficient condition for identification. In practice, because the structure condition holds if and only if a certain matrix, constructed from the constraint matrices, is invertible, automatic software checking for identification is feasible even for large-scale systems. Most of the paper focuses on structural vector autoregressions, but extensions to other statistical models are also briefly discussed.