Joint Confidence Sets for Structural Impulse Responses

44 Pages Posted: 2 Jun 2014

See all articles by Atsushi Inoue

Atsushi Inoue

Southern Methodist University

Lutz Kilian

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

Date Written: March 2014


Many users of structural VAR models are primarily interested in learning about the shape of structural impulse response functions. This 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 structural impulse response 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 demonstrate by simulation the coverage accuracy of these sets in finite samples under realistic conditions. We make the case for representing these 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 provides economically relevant additional information about the shape of response functions that is lost when reducing the joint confidence set to two-dimensional bands.

Keywords: Bootstrap, Confidence regions, Degenerate limiting distribution, Impulse response shapes, Joint inference, Shotgun plots

JEL Classification: C32, C52, C53

Suggested Citation

Inoue, Atsushi and Kilian, Lutz, Joint Confidence Sets for Structural Impulse Responses (March 2014). CEPR Discussion Paper No. DP9892. Available at SSRN:

Atsushi Inoue (Contact Author)

Southern Methodist University ( email )

Dallas, TX 75275
United States

Lutz Kilian

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)

United Kingdom

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