When and Why State-Dependent Local Projections Work

TL;DR

This paper clarifies the estimands of state-dependent local projections, compares them with VARs, and discusses implications for interpretation and potential misinterpretations in instrumental variable settings.

Contribution

It provides a general characterization of state-dependent LPs, distinguishes them from VARs, and offers guidance on correct interpretation and potential pitfalls.

Findings

State-dependent LPs recover weighted averages of causal effects.

State-dependent LPs and VARs target different estimands.

In LP-IV, weighting can create nonzero interactions even without state dependence.

Abstract

This paper studies state-dependent local projections (LPs). First, I establish a general characterization of their estimand: under minimal assumptions, state-dependent LPs recover weighted averages of causal effects. This holds for essentially all specifications used in practice. Second, I show that state-dependent LPs and VARs target different estimands and propose a simple VAR-based estimator whose probability limit equals the LP estimand. Third, in instrumental variable (LP-IV) settings, state-dependent weighting can generate nonzero interaction terms, even when the effects are not state-dependent. Overall, this paper shows how to correctly interpret state-dependent LPs, clarifying their connection to VARs and highlighting a key source of LP-IV misinterpretation.