Priors about observables in vector autoregressions

Standard practice in Bayesian VARs is to formulate priors on the autoregressive parameters, but economists and policy makers actually have priors about the behavior of observable variables. We show how to translate the prior on observables into a prior on parameters using strict probability theory principles, a posterior can then be formed with standard procedures. We state the inverse problem to be solved and we propose a numerical algorithm that works well in practical situations. We prove equivalence to a fixed point formulation and a convergence theorem for the algorithm. We use this framework in two well known applications in the VAR literature, we show how priors on observables can address some weaknesses of standard priors, serving as a cross check and an alternative formulation.