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Local Projection Inference is Simpler and More Robust Than You Think

José Luis Montiel Olea, Mikkel Plagborg-Møller

TL;DR

The paper tackles impulse response inference in macroeconomic models estimated with local projections, showing that lag-augmented LP provides uniform, robust confidence intervals across highly persistent data and long horizons. It proves that standard, heteroskedasticity-robust standard errors suffice for LP when lag augmentation is used, eliminating the need for HAC corrections. The authors develop a general VAR($p$) theory establishing uniform asymptotic normality of lag-augmented LP estimators for impulse responses, and provide a practical bootstrap procedure to improve finite-sample performance. Compared with traditional AR methods and AR-based bootstraps, lag-augmented LP offers a simpler yet highly robust inference method with favorable coverage properties in a wide range of persistence and horizon settings. The results have direct applied relevance, guiding macroeconomists toward a practically implementable, uniformly valid inference approach for impulse responses.

Abstract

Applied macroeconomists often compute confidence intervals for impulse responses using local projections, i.e., direct linear regressions of future outcomes on current covariates. This paper proves that local projection inference robustly handles two issues that commonly arise in applications: highly persistent data and the estimation of impulse responses at long horizons. We consider local projections that control for lags of the variables in the regression. We show that lag-augmented local projections with normal critical values are asymptotically valid uniformly over (i) both stationary and non-stationary data, and also over (ii) a wide range of response horizons. Moreover, lag augmentation obviates the need to correct standard errors for serial correlation in the regression residuals. Hence, local projection inference is arguably both simpler than previously thought and more robust than standard autoregressive inference, whose validity is known to depend sensitively on the persistence of the data and on the length of the horizon.

Local Projection Inference is Simpler and More Robust Than You Think

TL;DR

The paper tackles impulse response inference in macroeconomic models estimated with local projections, showing that lag-augmented LP provides uniform, robust confidence intervals across highly persistent data and long horizons. It proves that standard, heteroskedasticity-robust standard errors suffice for LP when lag augmentation is used, eliminating the need for HAC corrections. The authors develop a general VAR() theory establishing uniform asymptotic normality of lag-augmented LP estimators for impulse responses, and provide a practical bootstrap procedure to improve finite-sample performance. Compared with traditional AR methods and AR-based bootstraps, lag-augmented LP offers a simpler yet highly robust inference method with favorable coverage properties in a wide range of persistence and horizon settings. The results have direct applied relevance, guiding macroeconomists toward a practically implementable, uniformly valid inference approach for impulse responses.

Abstract

Applied macroeconomists often compute confidence intervals for impulse responses using local projections, i.e., direct linear regressions of future outcomes on current covariates. This paper proves that local projection inference robustly handles two issues that commonly arise in applications: highly persistent data and the estimation of impulse responses at long horizons. We consider local projections that control for lags of the variables in the regression. We show that lag-augmented local projections with normal critical values are asymptotically valid uniformly over (i) both stationary and non-stationary data, and also over (ii) a wide range of response horizons. Moreover, lag augmentation obviates the need to correct standard errors for serial correlation in the regression residuals. Hence, local projection inference is arguably both simpler than previously thought and more robust than standard autoregressive inference, whose validity is known to depend sensitively on the persistence of the data and on the length of the horizon.

Paper Structure

This paper contains 43 sections, 8 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Literature.
  3. Outline.
  4. Overview of the Results
  5. Lag-Augmented Local Projection
  6. Model.
  7. Local Projections With and Without Lag Augmentation.
  8. Standard Errors.
  9. Lag-Augmented Local Projection Inference.
  10. Illustrative Simulation Study
  11. Comparison With Other Inference Procedures
  12. Textbook Autoregressive Inference.
  13. Lag-Augmented AR Inference.
  14. AR Grid Bootstrap and Projection.
  15. Other Local Projection Approaches.
  16. ...and 28 more sections

Key Result

Proposition 1

Let asn:u_mdsasn:var_u_regasn:var_XpX hold. Let $C>0$ and $\epsilon \in (0,1)$.

Figures (1)

  • Figure 1: Efficiency ranking of three different estimators of the fixed impulse response $\beta(\rho,h) = \rho^h$ in the homoskedastic AR(1) model: lag-augmented LP ($\text{LP}_\text{LA}$), non-augmented LP ($\text{LP}_\text{NA}$), and lag-augmented AR ($\text{AR}_\text{LA}$). Gray area: combinations of $(|\rho|,h)$ for which $\text{LP}_\text{LA}$ is more efficient than $\text{AR}_\text{LA}$. Thatched area: $\text{LP}_\text{LA}$ is more efficient than $\text{LP}_\text{NA}$. See \ref{['sec:comp_details_releff']} for analytical derivations of the indifference curves (thick lines).

Theorems & Definitions (17)

  • Definition 1: VAR parameter space
  • Proposition 1
  • proof
  • Lemma A.1: Central limit theorem for $\xi_{i,t}(A,h) (w'u_t)$
  • proof
  • Lemma A.2: Consistency of standard errors.
  • proof
  • Lemma A.3: Convergence rates of estimators
  • proof
  • Lemma A.4: OLS numerator
  • ...and 7 more