Two Gaussians, Too Many: A bootstrap-based approach to assess identifiability in non-Gaussian structural vector autoregressions

Normality pre-tests on reduced-form innovations cannot detect whether two or more underlying structural shocks are Gaussian, and hence cannot detect a failure of identification in non-Gaussian SVARs. We propose a bootstrap-based approach that reveals such a failure. Under the null of valid identification and certain regularity conditions, the conditional bootstrap distribution of the maximum likelihood estimator of the impact matrix is asymptotically normal, so the diagnostic reduces to a normality test on the bootstrap replications. The impact-matrix estimator remains consistent and asymptotically normal when one shock is Gaussian and its shape parameter lies on the boundary of the parameter space, so the diagnostic is valid across the entire null hypothesis. When the number of bootstrap replications and the sample size diverge jointly at an appropriate rate, the test statistic, conditional on the data, is asymptotically pivotal. This implies that conditioning on this diagnostic does not induce pre-testing bias asymptotically. In simulations with Normal-inverse Gaussian shocks, the multivariate diagnostic is close to its nominal size under the null, its power against an additional Gaussian shock increases with the sample size, and its univariate version localizes the failure in large samples. Based on the estimates of a SVAR in the macroeconomic and financial uncertainty literature, we demonstrate its potential as a practical, robust tool for assessing non-Gaussian identification.