Econometrics is about confronting economic models with the data. In doing so it is crucial to choose a statistical model that not only contains the economic model as a submodel, but also contains the data generating process. When this is the case, the statistical model can be analyzed by likelihood methods. When this is not the case, but likelihood methods are applied nonetheless, the result is incorrect inference. In his paper we illustrate the problem of possible incorrect inference with a recent application of a DSGE model to US data (Ireland, 2004). Specifically, this paper discusses two broad methodological questions.

   ·  How should a statistical model be chosen to achieve valid inference for the economic model?
   ·  Given a correctly chosen statistical model, can we rely on the asymptotic results found in the statistical literature for the analysis of the data at hand?

Using some simple examples, the paper first discusses some unfortunate consequences of applying Gaussian maximum likelihood when the chosen statistical model does not properly describe the data. It also demonstrates that even when the correct statistical model is chosen, asymptotic results derived for stationary processes cannot be used to conduct inference on the steady state value for a highly persistent stationary process.