Inference about non-identified SVARs

We propose a method for conducting inference on impulse responses in structural vector autoregressions (SVARs) when the impulse response is not point identified because the number of equality restrictions one can credibly impose is not sufficient for point identification and/or one imposes sign restrictions. We proceed in three steps. We first define the object of interest as the identified set for a given impulse response at a given horizon and discuss how inference is simple when the identified set is convex, as one can limit attention to the set's upper and lower bounds. We then provide easily verifiable 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, identified. Second, we show how to conduct inference on the identified set. We adopt a robust Bayes approach that considers the class of all possible priors for the non-identified 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 identified set. We also consider a "robustified credible region" which is a measure of the posterior uncertainty about the identified set. The two intervals can be obtained using a computationally convenient numerical procedure. Third, we show that the posterior bounds converge asymptotically to the identified set if the set is convex. If the identified set is not convex, our posterior bounds can be interpreted as an estimator of the convex hull of the identified set. Finally, a useful diagnostic tool delivered by our procedure is the posterior belief about the plausibility of the imposed identifying restrictions.