Joint Inference and Counterfactual Experimentation for Impulse Response Functions By Local Projections

Abstract

This paper provides three measures of the uncertainty associated to an impulse response path: (1) conditional confidence bands which isolate the uncertainty of individual response coefficients given the temporal path experienced up to that point; (2) response percentile bounds which provide bounds on the universe of permissible paths at a given probability level; and (3) Wald tests of joint significance and joint cumulative significance. These results rely on general assumptions for the joint distribution of the system's impulse responses. Given this distribution, the paper then shows how to construct counterfactual experiments formally; provides a test on the likelihood of observing the counterfactual; and derives the distribution of the system's responses conditional on the counterfactual. The paper then derives the asymptotic joint distribution of structural impulse responses identified by either short- or long-run recursive assumptions and estimated by local projections (Jorda, 2005). An application to a two country system implements all of these new methods.