Are DSGE approximating models invariant to shifts in policy?

Bayesian and maximum-likelihood estimates of structural parameters in DSGE approximating models are invariant to shifts in policy only when the model is correctly specified. Since DSGE models are approximations, strict invariance is unattainable. Despite that, DSGE models might be useful for policy analysis provided that structural parameters are approximately invariant. We discuss this issue in the context of an example involving a pair of new Keynesian DSGE models in which the central bank's policy model includes a misspecified Phillips curve. In our example, the approximating model is a useful guide for policy despite the failure of strict invariance.