What Impulse Response Do Instrumental Variables Identify?

TL;DR

This paper shows that LP-IV identification in macro economics yields impulse-response estimates that are affine combinations of the underlying component shocks when the shock is composite. The authors derive that the LP-IV estimand can equal a weighted sum of component responses with instrument-specific weights that may be negative, complicating causal interpretation, and that standard 2SLS with multiple instruments can exacerbate this issue. To recover informative conclusions, they develop two identification strategies: use disaggregated data with restrictions on inter-component effects to identify component impulses, and impose sign restrictions on the instrument–shock correlations to bound those impulses; they further establish sharpness results and discuss unbounded identified sets under weak stationarity. The empirical applications—decomposing the U.S. government spending multiplier into defense and non-defense components and disentangling pure monetary shocks from central bank information shocks—illustrate how LP-IV can yield informative bounds and reveal potential biases in conventional IV analyses. Overall, the work provides a principled framework for set-identified impulse response analysis with multiple IVs, offering practical tools for robust macroeconomic inference and policy evaluation.

Abstract

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.

Figures (15)

• Figure 1: Identified Sets for Sign Restrictions on $\alpha_{1}$ and $\alpha_{2}$
• Figure 2: Intersection of Identified Sets
• Figure 3: Cumulative Aggregate Spending Multipliers Across Different Horizons with 90% Confidence Bands. Identified by RZ news shock (solid), BP defense shock (dash-dotted), both shocks using LP-2SLS (dash with x)
• Figure 4: Cumulative Sectoral Multipliers Across Different Horizons with 90% Confidence Bands. Non-defense spending multiplier ($*$), defense spending multiplier ($\circ$)
• Figure 5: Impulse Response of Sectoral Spending to the IV Shocks with 90% Confidence Bands: Defense spending (dash-dotted), Non-defense spending (solid)

Theorems & Definitions (17)

• Proposition 1
• Remark 1
• Remark 2
• Corollary 1
• Proposition 2
• Remark 3
• Corollary 2
• Proposition 3
• Remark 4
• Proposition 4