On the Identification of Time Varying Structures

The identifiability of reduced form econometric models with variable coefficients is investigated using the control theoretic concepts of uniform complete observability and uniform complete controllability. First, a variant of the state space representation of the traditional reduced form is introduced which transcribes the underlying non-stationary estimation problem into one particularly suited to a Kalman filtering solution. Using such a formulation, observability and controllability can be called upon to obtain a necessary and sufficient condition for identification of the specific parameterization. The results so obtained are completely analogous to those already established in the econometric literature, namely, that the parameters of the reduced form are always identified subject to the absence of multicollinearity(referred to as "persistent excitation" in the control literature). How-ever, now the multicollinearity condition is seen to depend on the structure of the parameter variations as well as the statistical nature of the explanatory variables. The verification of identifiability thus reduces to a check for uniform complete observability which can always be affected in econometric applications. Some consistency results are also presented which derive from the above approach.