Working Paper: CEPR ID: DP9892
Authors: Atsushi Inoue; Lutz Kilian
Abstract: 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 additionalinformation 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 Codes: C32; C52; C53
Edges that are evidenced by causal inference methods are in orange, and the rest are in light blue.
| Cause | Effect |
|---|---|
| joint Wald test statistic (C29) | asymptotically valid joint confidence sets (C14) |
| bootstrap method (Y20) | valid approximation of nonstandard distribution of Wald test statistic (C46) |
| joint confidence sets (C71) | more informative than traditional pointwise confidence intervals (C46) |
| shotgun plots (Y70) | convey additional information about shape of impulse response functions (C22) |