Joint Confidence Sets for Structural Impulse Responses

40 Pages Posted: 12 Mar 2016

See all articles by Atsushi Inoue

Atsushi Inoue

Vanderbilt University - College of Arts and Science - Department of Economics

Lutz Kilian

University of Michigan at Ann Arbor - Department of Economics; Centre for Economic Policy Research (CEPR)

Date Written: February 8, 2016

Abstract

Many questions of economic interest in structural VAR analysis involve estimates of multiple impulse response functions. Other questions relate to the shape of a given impulse response function. Answering these questions requires joint inference about sets of structural impulse responses, allowing for dependencies across time as well as across response functions. Such joint inference is complicated by the fact that the joint distribution of the structural impulse response estimators becomes degenerate when the number of structural impulse responses of interest exceeds the number of model parameters, as is often the case in applied work. This degeneracy may be overcome by transforming the estimator appropriately. We show that the joint Wald test is invariant to this transformation and converges to a nonstandard distribution, which can be approximated by the bootstrap, allowing the construction of asymptotically valid joint confidence sets for any subset of structural impulse responses, regardless of whether the joint distribution of the structural impulse responses is degenerate or not. We propose to represent the joint confidence sets in the form of “shotgun plots” rather than joint confidence bands for impulse response functions. Several empirical examples demonstrate that this approach not only conveys the same information as confidence bands about the statistical significance of response functions, but may be used to provide economically relevant additional information about the shape of and comovement across response functions that is lost when reducing the joint confidence set to two-dimensional bands.

Keywords: joint inference, shotgun plots, confidence bands, impulse response shapes, bootstrap, degenerate limiting distribution

JEL Classification: C320, C520, C530

Suggested Citation

Inoue, Atsushi and Kilian, Lutz, Joint Confidence Sets for Structural Impulse Responses (February 8, 2016). CESifo Working Paper Series No. 5746. Available at SSRN: https://ssrn.com/abstract=2745829

Atsushi Inoue

Vanderbilt University - College of Arts and Science - Department of Economics ( email )

Box 1819 Station B
Nashville, TN 37235
United States

Lutz Kilian (Contact Author)

University of Michigan at Ann Arbor - Department of Economics ( email )

611 Tappan Street
Ann Arbor, MI 48109-1220
United States
734-764-2320 (Phone)
734-764-2769 (Fax)

Centre for Economic Policy Research (CEPR)

London
United Kingdom

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