Dynamic Identification Using System Projections and Instrumental Variables

We propose System Projections with Instrumental Variables (SP-IV) to estimate dynamic structural relationships. SP-IV replaces lag sequences of instruments in traditional IV with lead sequences of endogenous variables. SP-IV permits identification over many time horizons and allows the inclusion of controls, which weakens exogeneity requirements and improves effective instrument strength. SP-IV also enables the estimation of structural relationships between impulse responses obtained from local projections or vector autoregressions. We provide a bias-based test for instrument strength and present inference procedures under strong and weak identification. SP-IV outperforms competing estimators of the Phillips Curve parameters in simulations. We estimate the Phillips Curve implied by the main business cycle shock of Angeletos et al. (2020), and find evidence for forward-looking behaviour. The data is consistent with weak but also relatively strong cyclical connections between inflation and unemployment.