A Reduced Form Representation for State Space Models

Estimating structural state space models with maximum likelihood is often infeasible. If the model can be expressed as a reduced form vector-autoregression (VAR) in the observable data, then two step techniques such as minimum chi-square estimation can reliably recover structural parameter estimates. However, macroeconomists cannot always rely on the existence of a VAR reduced form – as is often the case when estimating dynamic stochastic general equilibrium models. This paper introduces a reduced form representation for general state space models. I use a prediction based criterion to normalize the state vector and generate the reduced form. In particular, I force the state vector to be the best linear recursive predictor of the data. As a byproduct, this representation is intrinsically related to orthonormalized partial least squares regression. Also, a pair of quadratic parameter restrictions characterizes the representation, which enables constrained maximum likelihood estimation. This reduced form disentangles statistical inference from structural identification, greatly clarifying and simplifying the analysis of a broad class of dynamic structural econometric models. Using these results, economists can use a reduced form approach to structural estimation without needing a VAR representation.