An anonymous commenter on my post agonizing over the problems that expectations pose for dealing with endogeneity in macroeconomic empirical work shared "Causal Effects of Monetary Shocks: Semiparametric Conditional Independence Tests with a Multinomial Propensity Score" by Joshua Angrist and Guido Kuersteiner (forthcoming in RESTAT). Two things ought to jump out at you from that: "propensity score" and "monetary shocks" don't often appear together, and "Joshua Angrist" and "monetary shocks" don't often appear together (Angrist and "propensity score" would be less surprising).
The authors are struggling with identifying an empirical model of the impact of monetary policy and they end up doing it a lot like an impact analysis that a labor economist might perform. They categorize a series of monetary responses so that they can predict these responses with a multinomial logit model, but then they generate propensity scores from the predicted probabilities in that multinomial logit. A propensity score is basically a probability, based on observable characteristics, that an observation receives a treatment. Propensity score matching techniques rely on taking an observation that did receive treatment and comparing it to an observation that did not receive treatment but that has the same propensity score. In effect this process takes all the data in your control group and generates an artificial control group that is identical to your treatment group in terms of its observable characteristics. If selection intro treatment is principally on observable variables (or variables that those observables proxy), then you are effectively getting a randomized trial from your non-random data.
If there is substantial selection on unobservable characteristics, these methods aren't so good.
So this paper assigns propensity scores to monetary policy decisions because observable characteristics of the economy at the time of a policy decision ought to be the principle determinant of monetary policy.
Complications arise in doing this in a time-series setting so much of the contribution of this paper is discussing how you actually do that.
The appendix makes me feel stupid. If you suspect this might be the case for you, I suggest for the sake of your self-confidence you skip it. But do read their findings. They come up with similar conclusions to earlier work by Romer, and have bigger monetary policy effects than the VAR literature. They suggest that this is likely because of the endogeneity bias, which seems sensible to me.