Masking Weak Identification: Omitted Variables in Models Identified by Heteroskedasticity

This paper shows how spurious evidence of identification can arise when a relevant variable in the data-generating process (DGP) is omitted from a structural VAR identified by heteroskedasticity. In these models, identification rests on differences in the relative volatilities of structural shocks across regimes. We prove that an omitted variable whose relative volatility differs across regimes can artificially widen the spectral gap between the eigenvalues used as identifying information, making a weakly identified or unidentified system appear better identified than it actually is. This phenomenon can affect pre-tests of identification’s strength, which are performed on the observed, perturbed covariances. Notably, at exact degeneracy the perturbation lifts the gap to a strictly positive value whenever it is heteroskedastic relative to the other shocks. A Monte Carlo experiment shows that the distortion worsens with the sample size, and that a single omitted variable can render a wholly unidentified system empirically indistinguishable from a partially identified one. We conclude with an empirical illustration drawing on US labour market data.