Double Robustness of Local Projections and Some Unpleasant VARithmetic

We consider impulse response inference in a locally misspecified stationary vector autoregression (VAR) model. The conventional local projection (LP) confidence interval has correct coverage even when the misspecification is so large that it can be detected with probability approaching 1. This follows from a "double robustness" property analogous to that of modern estimators for partially linear regressions. In contrast, VAR confidence intervals dramatically undercover even for misspecification so small that it is difficult to detect statistically and cannot be ruled out based on economic theory. This is because of a "no free lunch" result for VARs: the worst-case bias and coverage distortion are small if, and only if, the variance is close to that of LP. While VAR coverage can be restored by using a bias-aware critical value or a large lag length, the resulting confidence interval tends to be at least as wide as the LP interval.