A theory is essentially a restriction on a reduced form. It can be imposed directly (hard restrictions) or used as as a prior mean in a more flexible Bayesian analysis (soft restrictions). The soft restriction approach -- "theory as a shrinkage direction" -- is appealing: coax parameter configurations toward a prior mean suggested by theory, but also respect the likelihood, and govern the mix by prior precision.
(1) Important macro-econometric DSGE work, dating at least to the classic Ingram and Whiteman (1994) paper, finds that using theory as a VAR shrinkage direction is helpful for forecasting.
(2) But that's not what most Bayesian DSGE work now does. Instead it imposes hard theory restrictions on a VAR, conditioning completely on an assumed DSGE model, using Bayesian methods simply to coax the assumed model's parameters toward "reasonable" values.
It's not at all clear that approach (2) should dominate approach (1) for prediction, and indeed research like Del Negro and Schorfheide (2004) and Del Negro and Schorfheide (2007) indicates that it doesn't.
I like (1) and I think it needs renewed attention.
[A related issue is whether "theory priors" will supplant others, like the "Minnesota prior." I'll save that for a later post.]