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

Abstract

Standard pre-tests of normality on reduced-form innovations are insufficient to detect two or more Gaussian shocks and hence, the failure of identification in non-Gaussian SVARs. We instead propose a bootstrap-based approach to evaluate the asymptotic validity of this condition by measuring the divergence between the conditional bootstrap distribution of a maximum likelihood estimator and its limiting distribution under valid identification. We show that, under valid identification and certain regularity conditions, the conditional bootstrap distribution of the impact matrix is asymptotically normal, so the diagnostic reduces to a test of normality of the bootstrap replications. The diagnostic remains valid in the single-Gaussian case, where the shape parameter of the Gaussian shock lies on the boundary, and the full-parameter information is singular; this establishes its validity across the entire null. Under the null of valid identification, the diagnostic induces no pre-testing bias as bootstrap replications and sample size diverge jointly at an appropriate rate. The joint divergence ensures that the test statistic, conditional on the data, is asymptotically pivotal, so conditioning on the diagnostic does not distort subsequent inference. Monte Carlo simulations with Normal-Inverse Gaussian shocks show that the diagnostic attains near-exact nominal size under valid identification and detects the failure due to multiple Gaussian shocks with power increasing in the sample size. Under weak identification with a near-Gaussian shock, conditioning on the bootstrap diagnostic, unlike on residual-based normality pre-tests, preserves the probability coverage of the estimates. Based on estimates of a SVAR model in the macroeconomic and financial uncertainty literature, we demonstrate its potential as a practical, robust tool for validating non-Gaussian identification without pre-testing bias.