What Impulse Response Do Instrumental Variables Identify?

The local projection-instrumental variable (LP-IV) literature has been largely silent on cases in which impulse responses are set-identified, arising when the shock of interest is composite and instruments are correlated with multiple components. We demonstrate that LP-IV estimands constructed using one instrument at a time identify affine combinations of impulse responses to structural shock components with instrument-specific and potentially negative weights, challenging standard causal interpretation. The two-stage least squares compounds the identification problem. However, we show that individual LP-IV estimands characterize the identified set when sign restrictions on the correlations between instruments and structural shock components are imposed. Under weak stationarity, these identified sets are sharp and cannot be further narrowed in key cases. Two empirical examples--decomposing the U.S. government spending multiplier and disentangling pure monetary shocks from central bank information shocks--illustrate the usefulness of our approach.