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 context 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.I thank all the participants at the meeting held in Pavia on June 11, 2004, in honor of Carlo Giannini for their comments; it goes without saying that Carlo himself provided not only acute observations on the day but also the main inspiration for this piece of work. Sadly, Carlo passed away on September 11, 2004, and this paper is dedicated to his memory. Pär Österholm spotted several mistakes in an earlier version and helped me clarify some implementation details. Thanks are also due to Gianni Amisano, Bruce Hansen, Giulio Palomba, Paolo Paruolo, and two anonymous referees. The usual disclaimer obviously applies.