Local Projections and VARs Estimate the Same Impulse Responses

We prove that local projections (LPs) and Vector Autoregressions (VARs) estimate the same impulse responses. This nonparametric result only requires unrestricted lag structures. discuss several implications: (i) LP VAR estimators are not conceptually separate procedures; instead, they simply two dimension reduction techniques with common estimand but different finite?sample properties. (ii) VAR?based structural identification—including short?run, long?run, or sign restrictions—can equivalently be performed using LPs, vice versa. (iii) Structural estimation an instrument (proxy) can carried out by ordering first in a recursive VAR, even under noninvertibility. (iv) Linear VARs as robust to nonlinearities linear LPs.