Inference about Non-Identi?ed SVARs

We propose a method for conducting inference on impulse responses in structural vector autoregressions (SVARs) when the impulse response is not point identi?ed because the number of equality restrictions one can credibly impose is not su?cient for point identi?cation and/or one imposes sign restrictions. We proceed in three steps. We ?rst de?ne the object of interest as the identi?ed set for a given impulse response at a given horizon and discuss how inference is simple when the identi?ed set is convex, as one can limit attention to the set’s upper and lower bounds. We then provide easily veri?able conditions on the type of equality and sign restrictions that guarantee convexity. These cover most cases of practical interest, with exceptions including sign restrictions on multiple shocks and equality restrictions that make the impulse response locally, but not globally, identi?ed. Second, we show how to conduct inference on the identi?ed set. We adopt a robust Bayes approach that considers the class of all possible priors for the non-identi?ed aspects of the model and delivers a class of associated posteriors. We summarize the posterior class by reporting the "posterior mean bounds", which can be interpreted as an estimator of the identi?ed set. We also consider a "robusti?ed credible region" which is a measure of the posterior uncertainty about the identi?ed set. The two intervals can be obtained using a computationally convenient numerical procedure. Third, we show that the posterior bounds converge asymptotically to the identi?ed set if the set is convex. If the identi?ed set is not convex, our posterior bounds can be interpreted as an estimator of the convex hull of the identi?ed set. Finally, a useful diagnostic tool delivered by our procedure is the posterior belief about the plausibility of the imposed identifying restrictions.