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arxiv: 2605.05404 · v2 · pith:R7DSHWQUnew · submitted 2026-05-06 · 💰 econ.EM

Causal State-Dependent Local Projections

Joel M. David , Raffaella Giacomini , Xiyu Jiao , Weining Wang This is my paper

Pith reviewed 2026-05-20 23:07 UTC · model grok-4.3

classification 💰 econ.EM
keywords state-dependent local projectionscausal impulse responsesmicro-macro panelssieve estimationmonetary policy shocksheterogeneous agentslinear conditional mean
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0 comments X

The pith

State-dependent local projections recover causal impulse responses if the conditional mean is linear in the aggregate shock at each horizon.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that state-dependent local projections identify causal effects of aggregate shocks varying with observable states when the outcome's conditional mean is linear in the shock. This linearity condition holds in standard heterogeneous-agent and macro-finance models solved by first-order perturbation. Common linear-interaction specifications do not deliver causal objects, prompting a new sieve-based estimator that supports flexible dependence and valid inference on panel data. If correct, the approach lets researchers measure how monetary policy or similar shocks affect firms differently by state and then aggregate those responses reliably. The result strengthens causal claims for these methods in settings that combine micro and macro variation more than in pure time-series data.

Core claim

Local projections recover causal impulse responses under the sufficient condition that the conditional mean is linear in the aggregate shock at each horizon, and that this condition holds in a broad class of canonical micro-macro environments, including first-order perturbation solutions of heterogeneous-agent macro and macro-finance models. The commonly used linear interaction LPs generally fail to recover causal objects. A sieve-based LP estimator recovers the causal responses and delivers valid pointwise and uniform inference in micro-macro panels.

What carries the argument

The sufficient condition that the conditional mean of the outcome is linear in the aggregate shock at each horizon; this makes the local-projection coefficients equal to the causal state-dependent impulse responses.

If this is right

  • In first-order perturbation solutions of heterogeneous-agent models the linearity condition is satisfied, so local projections are causal there.
  • Linear interaction local projections generally fail to recover causal state-dependent responses.
  • The sieve-based estimator recovers causal objects and supplies valid pointwise and uniform inference on micro-macro panels.
  • Allowing flexible state dependence materially alters estimated heterogeneous firm investment responses and the implied aggregate effects of monetary policy shocks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same linearity condition could be checked or imposed when applying state-dependent methods to fiscal or trade shocks.
  • If the condition holds approximately in a wider set of models, state-dependent projections may remain useful even when exact linearity is violated.
  • The estimator opens the door to testing state dependence directly in other aggregate-shock settings that feature both micro and macro data.
  • Similar sufficient conditions might be derived for state-dependent vector autoregressions or other reduced-form approaches.

Load-bearing premise

The conditional mean of the outcome is linear in the aggregate shock at each horizon.

What would settle it

A simulation or empirical setting in which the conditional mean is clearly nonlinear in the aggregate shock at some horizon, yet the local-projection estimates still match the true causal responses computed from the known data-generating process.

Figures

Figures reproduced from arXiv: 2605.05404 by Joel M. David, Raffaella Giacomini, Weining Wang, Xiyu Jiao.

Figure 1
Figure 1. Figure 1: Monte Carlo comparison of sieve and linear IRFs under the cubic DGP view at source ↗
Figure 2
Figure 2. Figure 2: Impulse Response of Firm-Level Investment to a Monetary Policy Shock view at source ↗
Figure 3
Figure 3. Figure 3: Differential Investment Responses to a Monetary Policy Shock view at source ↗
Figure 3
Figure 3. Figure 3: Differential Investment Responses to a Monetary Policy Shock [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Aggregate Investment Response to a Monetary Policy Shock view at source ↗
Figure 5
Figure 5. Figure 5: True nonlinear function gh(z) and its derivative g ′ h (z) 34 view at source ↗
Figure 6
Figure 6. Figure 6: Weight function ωh(z) Under the true model (30) with gh(z) and Zi,t−1, Xt as specified above, Lemma 1 gives βh = Z 3 1 ωh(z) g ′ h (z) dz = − 1 28 . The issue is that ωh(z) is negative over part of its support ( view at source ↗
Figure 7
Figure 7. Figure 7: Monte Carlo comparison of sieve and linear IRFs under the Fourier DGP view at source ↗
Figure 8
Figure 8. Figure 8: Monte Carlo comparison of sieve IRFs across selector choices under the cubic DGP over view at source ↗
read the original abstract

State-dependent local projections (LPs) are widely used to estimate how impulse responses to exogenous aggregate shocks vary as a function of observable state variables, yet their causal interpretation remains unclear. We show that LPs recover causal impulse responses under the sufficient condition that the conditional mean is linear in the aggregate shock at each horizon, and that this condition holds in a broad class of canonical micro-macro environments, including first-order perturbation solutions of heterogeneous-agent macro and macro-finance models. We further show that the commonly used linear interaction LPs generally fail to recover causal objects. We therefore develop a sieve-based LP estimator that recovers the causal responses and delivers valid pointwise and uniform inference in micro-macro panels. Empirically, allowing for flexible state dependence materially changes both the pattern of heterogeneous firm investment responses and their aggregate implications for the transmission of monetary policy shocks. Our findings thus place state-dependent LPs on firmer causal footing in micro-macro settings than in purely aggregate ones, provided state dependence is estimated nonparametrically.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that state-dependent local projections recover causal impulse responses under the sufficient condition that the conditional mean of the outcome is linear in the aggregate shock at each horizon, and that this condition holds in a broad class of canonical micro-macro environments including first-order perturbation solutions of heterogeneous-agent macro and macro-finance models. It shows that commonly used linear interaction LPs generally fail to recover causal objects, develops a sieve-based LP estimator that recovers the causal responses with valid pointwise and uniform inference in micro-macro panels, and presents empirical evidence that flexible state dependence materially changes heterogeneous firm investment responses to monetary policy shocks and their aggregate implications.

Significance. If the central results hold, the paper provides a rigorous causal foundation for state-dependent LPs in micro-macro settings, where these methods are widely applied but their interpretation has been unclear. The identification of the linearity sufficient condition, the demonstration that it obtains under standard first-order perturbations, and the development of a sieve estimator with valid inference represent a substantive advance over purely aggregate LP applications. The empirical application further illustrates that nonparametric state dependence alters substantive conclusions about transmission mechanisms.

major comments (2)
  1. [§3] §3 (Theoretical Results), around the statement of the sufficient condition: The claim that E[y_{t+h} | shock_t, state_t] is exactly linear in the realized aggregate shock under first-order perturbation solutions of HA models requires explicit verification. Even with linear policy rules, the law of motion for the cross-sectional distribution and endogenous aggregate states can induce nonlinearity in the h-step-ahead conditional expectation when the observable state vector includes lagged aggregates or distribution moments. Please provide the specific derivation showing why aggregation and state evolution preserve exact linearity without additional restrictions on the perturbation order or the included state variables.
  2. [§4] §4 (Sieve Estimator), Eq. (corresponding to the nonparametric LP specification): The uniform inference result for the sieve estimator appears to rely on the maintained linearity condition holding exactly. If the condition is only approximate (as may occur with higher-order terms or aggregation), the bias term in the asymptotic expansion needs to be characterized explicitly to confirm that the confidence bands remain valid.
minor comments (2)
  1. [Abstract] The abstract states that the linearity condition 'holds in a broad class of canonical micro-macro environments'; a brief enumeration of the precise model classes (e.g., which moments of the distribution are treated as states) would improve clarity.
  2. [Empirical Application] In the empirical section, the comparison between linear-interaction and sieve LPs would benefit from reporting the exact functional form of the sieve basis and the chosen dimension to allow replication.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments help clarify key aspects of the theoretical results. We respond to each major comment below and indicate planned revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Theoretical Results), around the statement of the sufficient condition: The claim that E[y_{t+h} | shock_t, state_t] is exactly linear in the realized aggregate shock under first-order perturbation solutions of HA models requires explicit verification. Even with linear policy rules, the law of motion for the cross-sectional distribution and endogenous aggregate states can induce nonlinearity in the h-step-ahead conditional expectation when the observable state vector includes lagged aggregates or distribution moments. Please provide the specific derivation showing why aggregation and state evolution preserve exact linearity without additional restrictions on the perturbation order or the included state variables.

    Authors: We thank the referee for this observation. Under first-order perturbations, individual policy rules are linear in the aggregate shock, predetermined states (including lagged aggregates and distribution moments) enter linearly, and aggregates are linear functionals of the cross-sectional distribution. Because the shock enters only through these linear channels and the h-step evolution under the first-order approximation introduces no higher-order terms in the shock, the conditional mean E[y_{t+h} | shock_t, state_t] remains exactly linear. We will add an explicit step-by-step derivation of this preservation result to the appendix in the revision, confirming that standard first-order perturbation assumptions suffice without further restrictions. revision: yes

  2. Referee: [§4] §4 (Sieve Estimator), Eq. (corresponding to the nonparametric LP specification): The uniform inference result for the sieve estimator appears to rely on the maintained linearity condition holding exactly. If the condition is only approximate (as may occur with higher-order terms or aggregation), the bias term in the asymptotic expansion needs to be characterized explicitly to confirm that the confidence bands remain valid.

    Authors: Our uniform inference results are derived under the exact linearity condition, which we show obtains in the first-order perturbation class. To address the referee's concern about approximate cases, we will add to the revision an explicit characterization of the leading bias term in the asymptotic expansion when linearity holds only up to o_p(1) approximation error. Under standard conditions on the size of higher-order terms, this bias is asymptotically negligible relative to the convergence rate of the sieve estimator, preserving the validity of the uniform confidence bands. We will also include a brief discussion of practical implications when the condition is approximate. revision: yes

Circularity Check

0 steps flagged

Derivation of causal recovery conditions for state-dependent LPs is self-contained

full rationale

The paper derives a sufficient condition (linearity of the conditional mean of the outcome in the aggregate shock at each horizon) under which local projections recover causal impulse responses. It then verifies that this linearity condition holds in first-order perturbation solutions of heterogeneous-agent macro and macro-finance models via direct analysis of the linearized law of motion and aggregation. These steps rely on explicit, independently stated model assumptions and standard perturbation techniques rather than any definitional equivalence, fitted parameter renamed as prediction, or load-bearing self-citation chain. The central claim is therefore not forced by construction from its inputs and remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the linearity condition in the conditional mean and on the properties of first-order perturbation solutions in heterogeneous-agent models; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The conditional mean is linear in the aggregate shock at each horizon
    Stated as the sufficient condition under which LPs recover causal impulse responses.
  • domain assumption First-order perturbation solutions of heterogeneous-agent macro and macro-finance models satisfy the linearity condition
    Invoked to claim the condition holds in a broad class of canonical environments.

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