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.
26 November 2014
Working Paper (CWP45/14)