Seminario Corrected Local Projections for Structural Impulse Response Functions

Abstract

Local projection (LP) and local projection IV (LP-IV) estimators are widely used to estimate structural impulse response functions (IRFs) in macroeconomics, but their variance grows with the projection horizon due to the accumulation of moving average (MA) components in the LP error. This paper makes three main contributions. First, we derive the exact error structure of structural LP and LP-IV estimators. Unlike the reduced-form LP error, which follows an MA(h−1) process, we show that the structural LP and LP-IV errors follow an MA(h) process. The additional MA term arises from the contemporaneous structural identification step and reflects the contribution of non-recoverable structural shocks under partial invertibility. Second, exploiting this MA(h) error structure, we introduce the corrected LP (CLP) and corrected LP-IV (CLP-IV) estimators, which subtract the intermediate MA accumulation terms from the LP and LP-IV dependent variables prior to estimation. Existing corrected LP estimators establish efficiency gains only for reduced-form IRFs in AR(1) models; we prove that CLP and CLP-IV are strictly more efficient than LP and LP-IV respectively for SVAR models with a fixed or infinite number of lags, under partial invertibility of the shock structure, and with a potentially mismeasured instrument — under mild mean independence conditions on the structural shocks that allow for heteroskedasticity. Third, we show that CLP and CLP-IV also achieve second-order bias reduction as a byproduct of the MA correction, without requiring explicit bias correction formulas. Existing results on LP bias are derived under conditional homoskedasticity for the case where the structural shock is directly observed; we extend the bias analysis to LP-IV and to heteroskedastic errors, and show that the MA correction in CLP and CLP-IV removes the dominant bias-inducing terms simultaneously with the variance reduction.