Double Robustness of Local Projections and Some Unpleasant VARithmetic

TL;DR

The paper analyzes impulse-response inference under local misspecification in a local-to-SVAR framework, showing that conventional local projection (LP) confidence intervals retain correct coverage even when misspecification is large, due to a double-robust property. In contrast, conventional VAR confidence intervals are fragile to even small misspecifications unless the lag length is so large that VAR becomes asymptotically equivalent to LP, which makes intervals wide. The authors derive analytical results on worst-case bias and coverage under bounded misspecification, demonstrate a robust LP performance in theory and in empirical simulations calibrated to real data, and provide practical guidance on lag-length selection and bias-aware inference. The findings suggest that, for reliable coverage across plausible DGPs, LP should be preferred for impulse-response inference, with lag augmentation and careful control for predictive lags; VAR-based inference remains appealing mainly for point estimation or when lags are excessively long. The work offers a rigorous framework to assess robustness of LP and VAR procedures and highlights the limitations of standard inference in applied macroeconomics, with implications for practitioners and future methodological developments.

Abstract

We consider impulse response inference in a locally misspecified 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 result follows from a "double robustness" property analogous to that of popular partially linear regression estimators. By contrast, the conventional VAR confidence interval with short-to-moderate lag length can severely undercover for misspecification that is small, difficult to detect statistically, and cannot be ruled out based on economic theory. The VAR confidence interval has robust coverage if, and only if, the lag length is so large that the interval is as wide as the LP interval.

Figures (6)

• Figure 4.1: Worst-case asymptotic coverage probability of the conventional 90% VAR confidence interval. Horizontal axis: relative asymptotic standard deviation of VAR vs. LP. Different lines: different bounds $M$ on $\|\alpha(L)\|$. Shaded area: empirical 10th--90th percentile range of relative standard errors based on Ramey2016, see the online replication materials for details. The solid horizontal line marks the nominal coverage probability $1-a=90\%$.
• Figure 4.2: Worst-case asymptotic probability of the joint event that the conventional VAR confidence interval fails to cover the true impulse response and yet the Hausman test fails to reject misspecification. Horizontal axis: relative asymptotic standard deviation of VAR vs. LP. The dotted horizontal line marks the nominal significance level $a=10\%$.
• Figure 4.3: Relative length of bias-aware VAR confidence interval vs. conventional LP interval. Significance level $a=10\%$. Horizontal axis: relative asymptotic standard deviation of VAR vs. LP. Different lines: different bounds $M$ on $\|\alpha(L)\|$. The solid horizontal line marks the value 1.
• Figure 5.1: Coverage probability (left) and median length (right) for VAR (red) and LP (blue) nominal 90% confidence intervals computed via the delta method or bootstrap (the latter are indicated with subscript "b" in the legends). Lag length: fixed at $p = 12$ in the top panel, and selected using AIC in the bottom panel.
• Figure A.1: Least favorable $\alpha^\dagger(L;h)$ for horizons $h \in \{ 1, 5, 10 \}$ for local-to-AR(1) models with different persistence parameters $\rho$ (left, middle, and right panel).

Theorems & Definitions (26)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Corollary 3.1
• proof
• Proposition 3.3
• proof
• Corollary 3.2
• proof