A central debate in applied macroeconomics is whether statistical tools that use minimal identifying assumptions are useful for isolating promising models within a broad class. In this paper, I compare three statistical models - a vector autoregressive moving average (VARMA) model, an unrestricted state space model, and a restricted state space model - that are all consistent with the same prototype business cycle model. The business cycle model is a prototype in the sense that many models, with various frictions and shocks, are observationally equivalent to it. The statistical models I consider differ in the amount of a priori theory that is imposed, with VARMAs imposing minimal assumptions and restricted state space models imposing the maximal. The objective is to determine if it is possible to successfully uncover statistics of interest for business cycle theorists with sample sizes used in practice and only minimal identifying assumptions imposed. I find that the identifying assumptions of VARMAs and unrestricted state space models are too minimal: The range of estimates are so large as to be uninformative for most statistics that business cycle researchers need to distinguish alternative theories.
