State Space and ARMA Models: An Overview of the Equivalence

In this paper known results about the equivalence of state space and autoregressive moving-average models with exogenous inputs (ARMAX) (including vector auto-regressive or VAR models) are reviewed. While most of these results are not new, no single reference appears to bring together several important related points. In addition, the original references have used a variety of different conventions for time-shift and parameter signs, have usually not included exogenous inputs (or have not distinguished them from noise), and have frequently not distinguished algebraic equivalence transformations from the statistical estimation. Furthermore, an overview of the subject area reveals some important gaps in the theory. While the paper does not consider statistical estimation, it does suggest a subtle but important shift in methodology from the approach usually taken in econometrics. The traditional approach in econometrics has been to specify a representation based on economic theory, convert to a reduced form if necessary, impose identifiability constraints, and then estimate model parameters. If, on the one hand, forecasting is the final objective, this approach can work very well, though it is clear from the equivalence that it is not necessary to proceed in this order. On the other hand, if the eventual purpose is to use statistical hypothesis testing techniques to arrive at conclusions about the structure of the economy, then the equivalence can nullify many conclusions. In particular, it is often difficult or impossible to disentangle the statistical evidence from the implications imposed by the original economic theory used to specify the representation. The alternative methodology suggested by the equivalence examined in this paper is first to estimate the model using whatever representation is convenient for estimation, then to impose economic constraints on the representation. This approach will disentangle structural conclusions that are supported by statistical evidence from those which are implied by economic theory. Statistical tests are applicable only to results that are invariant across all equivalent representations. Economic theory must be used to reach conclusions that do not correspond to invariants of the equivalence class. This latter observation has some implications for continuing debates in the econometrics literature. Algorithms for model conversion and comparison are presented in an appendix.