Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
95,142 characters · 19 sections · 36 citation commands
Causal State-Dependent Local Projections
JEL Classification: C14, C12, C23, E32 \\ Keywords: Impulse Response Functions, Identification, Nonparametric Estimation, Uniform Inference, Micro-Macro Panels, Heterogeneous-Agent Models
State-dependent local projections (LPs) are widely used in empirical practice, yet their causal interpretation is unclear, even under standard shock exogeneity assumptions. kolesar2024dynamic show that scalar LPs (regressing outcomes on the shock) can admit a causal interpretation under shock exogeneity, even when the data-generating process (DGP) is nonlinear. gonccalves2024state, by contrast, show that state-dependent LPs (allowing the effect of the shock to vary with the state through interactions) can be severely biased in environments where the state is endogenous to the shock. As emphasized by JordaTaylor2025, extending causal interpretations beyond scalar LPs to nonlinear or state-dependent settings remains an open challenge, despite their widespread empirical use. This determines whether widely used estimates of state-dependent responses to aggregate shocks can be given a causal interpretation.
This paper addresses this gap by studying the identification, estimation, and inference of state-dependent impulse responses using LPs. It makes three main contributions. First, we establish that state-dependent LPs recover causal impulse responses under a sufficient condition on the conditional mean. We show that this condition is naturally satisfied in a broad class of canonical micro--macro environments, but is less plausible in purely aggregate environments, where the concerns emphasized by gonccalves2024state remain relevant. This condition is a property of the underlying DGP, while estimation proceeds via standard LPs under shock exogeneity. Second, we clarify the role of LP specification choices for causal interpretation when the sufficient condition is satisfied. While the choice of state variable does not by itself invalidate causal interpretation, imposing linear state dependence generally does, except in the knife-edge case in which state dependence is exactly linear in the underlying DGP. Third, we develop a practical sieve-based nonparametric approach to state-dependent impulse responses in LPs that - when the sufficient condition is satisfied - estimates the causal impulse response function and delivers valid pointwise and uniform confidence bands in micro--macro panels.
The central takeaway for empirical researchers is that state-dependent LPs admit a causal interpretation under restrictions on the underlying DGP that are satisfied in a broad class of canonical micro--macro environments. These restrictions do not require specifying the DGP: with an exogenous shock, state-dependent LPs recover causal impulse responses directly, but doing so generally requires moving beyond commonly used linear interaction specifications and adopting a nonparametric approach to modeling state dependence.
Throughout the paper, we focus on micro--macro settings, as these are the environments in which identifying conditions are most plausibly satisfied. State-dependent LPs are widely used in such settings to study how aggregate shocks affect heterogeneous households and firms. In practice, researchers implement these specifications by regressing a micro outcome $Y_{i,t+h}$ on an aggregate shock $X_t$ interacted with micro-level characteristics $Z_{i,t-1}$, such as balance sheet variables, financial constraints, or other predetermined state variables, typically imposing linear state dependence (i.e., including $Z_{i,t-1} X_t$ as a regressor).
Formally, the sufficient condition that delivers a causal interpretation of state-dependent LPs is that the horizon-$h$ conditional mean of the micro outcome is linear in the aggregate shock with a functional coefficient $g_h(Z_{i,t-1})$ depending on the state:
where $\mathcal{F}_{t-1}$ denotes the information available at time $t-1$. Under this condition, $g_h(z)$ equals the total causal impulse response at horizon $h$ for units with initial state $z$. This condition is a restriction on the underlying DGP and does not require specifying or estimating the conditional mean. The key insight is that linearity in the aggregate shock allows the impulse response to be represented as a state-dependent coefficient in a LP, so that identification follows directly under shock exogeneity. The nuisance component $r_{i,h}(\mathcal{F}_{t-1})$ of the conditional mean need not be modeled. Importantly, identification does not depend on the precise choice of state $Z$: as long as it is predetermined, we show that a misspecified state still yields a valid causal interpretation, but as a weighted average over the underlying heterogeneity that is not captured. If condition (ref) fails, however, state-dependent LP coefficients are generally not causal.
We then characterize economically relevant classes of micro–macro DGPs under which condition (ref) holds and fails. In environments where the aggregate shock enters linearly and propagation dynamics are time-invariant, state-dependent LPs recover causal impulse responses. This structure arises in a broad class of widely used models. In particular, local log-linear solutions of Heterogeneous-Agents (HA) and Heterogeneous Agents New Keynesian (HANK) models and macro–finance models imply that aggregate shocks generate state-dependent exposure across units while leaving the underlying propagation mechanisms unchanged. This implies that the sufficient condition is satisfied in these models. Concretely, a contractionary monetary shock may reduce consumption more for highly leveraged households, depress investment more for highly leveraged firms, and lower wealth more for holders of long-duration assets, even though the structural laws governing consumption smoothing, capital accumulation, and wealth dynamics remain unchanged. When this structure is motivated by linearized solutions, it requires that the first-order approximation remains accurate over the range of aggregate shocks considered. In these environments, state-dependent LPs can recover causal impulse responses directly, without requiring a full specification of the underlying model.
Departures from this structure can arise in a number of ways. One occurs when the relationship between outcomes and the aggregate shock is nonlinear, as in models with occasionally binding constraints, regime switching, or in global nonlinear solutions of heterogeneous-agent models. Another, more subtle, violation arises when shocks affect not only exposure but also the propagation dynamics of micro outcomes, so that the total response, combining direct and indirect effects, cannot be summarized by a coefficient that depends only on the state. In these cases, state-dependent LP coefficients do not correspond to causal impulse responses, as illustrated by the aggregate setting in gonccalves2024state. The micro–macro model of almuzara2025nonlinear provides a related example that falls outside the class we characterize. Recovering causal impulse responses then requires allowing for fully nonlinear responses to the shock.
That the condition holds in canonical micro--macro environments does not, however, justify the linear state-dependent LP specifications commonly used in empirical work. Even when the sufficient condition holds, recovering the causal impulse response generally requires nonparametric estimation of $g_h(\cdot)$. Unless the true state dependence is exactly linear, we show that the commonly used linear specifications capture projection objects that do not generally have a causal interpretation. We therefore develop an easy-to-implement sieve estimation approach that directly recovers the causal state-dependent impulse response function and delivers valid pointwise and uniform inference.
Monte Carlo evidence illustrates the practical implications of our approach. Allowing for nonparametric state dependence is quantitatively important, especially at short horizons and for larger shocks, where linear specifications can substantially distort the impulse response function. Uniform confidence bands exhibit performance consistent with the nonparametric inference literature with data-driven tuning, including some undercoverage in smaller samples, but coverage improves as the time dimension increases. In addition, intermediate terms - future shock–state interactions that enter the LP error and are typically omitted because they do not affect identification - can be exploited to obtain substantially tighter confidence bands.
Applying our approach to firm-level investment responses to monetary policy shocks, we revisit the widely used LP in ottonello2020financial, which imposes linear state dependence in firms’ financial conditions, e.g., distance to default. We show that their estimated positive slope is in large part a misspecification artifact: our approach reveals a significantly nonlinear, hump-shaped response, with the largest effects concentrated among firms near the mean of the distribution rather than at the least risky right tail. This pattern of firm-level responsiveness has important quantitative implications: because most firms are concentrated in regions of the distribution where the nonlinear response exceeds the linear approximation, the latter effectively averages across states and attenuates the estimated effects. As a result, our nonparametric estimates imply substantially larger aggregate effects of monetary policy shocks on firms' investment.
Our framework connects and clarifies several strands of the literature. We speak directly to the empirical literature that studies heterogeneous responses to aggregate shocks using LPs that interact shocks with micro characteristics (e.g., cloyne2023monetary, ottonello2020financial, jeenas2023firm). Our message is as follows. State-dependent LPs can be given a causal interpretation in micro--macro settings where shocks affect units differently through their characteristics but do not change how the economy propagates shocks over time, as in standard linearized heterogeneous-agent and macro-finance models. In these environments, researchers can continue to use state-dependent LPs to estimate heterogeneous effects. However, commonly used linear interaction specifications generally do not recover causal impulse responses, even when the causal interpretation is valid. Instead, state dependence should be modeled nonparametrically, which can be implemented with a simple modification of standard LP regressions. By contrast, when shocks affect propagation itself, as is more likely in aggregate time-series applications, a causal interpretation is less credible.
Our results reinterpret and extend the recent econometric literature on LPs. kolesar2024dynamic show that scalar LPs admit a causal interpretation as weighted averages of marginal effects under shock exogeneity, even when the true DGP is nonlinear. Bousquet2025_UncoveringNonlinearities specializes this result to settings with nonlinearities in the shock. We show that this robustness does not generally extend to state-dependent LPs once heterogeneity is parametrically restricted. Under additional structure on the DGP, however, state-dependent LPs recover the causal impulse response function itself - rather than a weighted average - provided that the state dependence is estimated nonparametrically.
gonccalves2024state show that state-dependent LPs need not recover causal impulse responses when the state is endogenous. Our analysis complements their insight by showing that the crucial mechanism for such failure is nonlinearity of the shock and/or shock-dependent propagation. While both are plausible in aggregate time series, we show that in micro--macro panels there exists a large and economically relevant class of DGPs in which shocks enter linearly and propagation remains stable. In these environments state-dependent LPs are valid even when the state is endogenous (again, provided that state dependence is estimated nonparametrically).
gonccalves2024_wp recover impulse responses in nonlinear environments by estimating the full response of the outcome as a function of the shock and then constructing impulse responses as differences in this response function across shock realizations. By contrast, we restrict attention to environments in which the conditional mean is linear in the aggregate shock with a state-dependent coefficient. This structure allows us to recover impulse responses directly as the coefficient function in a LP, without estimating the full response surface or performing a second-step transformation, and under weaker martingale-difference exogeneity of the shock rather than i.i.d.\ shocks.
From an econometric perspective, our estimator can be viewed as a functional coefficient regression in which the coefficient on the aggregate shock is identified by exogeneity. This structure allows us to recover the causal object directly without specifying or estimating the remaining components of the conditional mean, distinguishing our approach from standard implementations that require explicit modeling of nuisance components.
Relative to work on inference in micro--macro panels with aggregate shocks, our contribution is complementary but distinct. almuzara2025nonlinear study inference for scalar LPs, but their framework does not extend to the state-dependent impulse responses we consider. We instead provide pointwise and uniform inference for state-dependent LPs, building on recent advances in sieve and high-dimensional time-series methods (e.g., Chernozhukov2014gaussian, Zhang2017Guassian, lu2020kernel, chen2024inference) and carefully adapting them to the macro--micro LP setting, which raises nonstandard challenges for inference.
Our empirical application is related to paranhos2025firms, who proposes a nonparametric extension of Ottonello and Winberry (2020) using random forests to flexibly estimate heterogeneous impulse responses. While her approach relaxes functional form assumptions on how responses vary across firms, it takes as given that such state-dependent impulse responses admit a causal interpretation. In contrast, our contribution is to characterize the conditions under which these objects are identified in the first place. We further show that, even when these conditions hold, a nonparametric representation of state dependence is generally required for the resulting impulse responses to retain a causal interpretation. Within such settings, her approach can be viewed as an alternative way of estimating the same object. Our sieve-based implementation, however, remains closer to standard LP practice and delivers uniform inference for the full impulse response function.
Overall, our paper clarifies when state-dependent LPs admit a causal interpretation and provides practical tools for estimation and inference of heterogeneous responses to aggregate shocks, with particular relevance for micro--macro settings.
This illustrative section defines the state-dependent causal impulse response of interest and presents the key condition under which state-dependent LPs admit a causal interpretation. It then uses three stylized DGPs and three different specifications of LPs to clarify when a causal interpretation holds and when it fails.
Consider the following stylized nonlinear DGPs for a scalar micro outcome $Y_{it}$, exogenous aggregate shock $X_t$, and predetermined micro state $Z_{i,t-1}$:
where $g(\cdot)$ is nonlinear in $Z$, $f(\cdot,\cdot)$ is nonlinear in $X$, and in DGP 2 the propagation coefficient $\rho_t$ depends on the aggregate shock, so that perturbations to $X_t$ affect future propagation.
These DGPs isolate three distinct channels through which aggregate shocks affect outcomes: heterogeneous exposure to a shock that enters linearly, but with constant propagation (DGP 1), shock-dependent propagation (DGP 2), and nonlinear dependence on the shock (DGP 3). As we show below, only in the first case state-dependent LPs recover causal impulse responses.
Our object of interest is the finite-$\delta$ impulse response at horizon $h$ for a given state, \[ \text{IRF}_h(\delta \mid z) = \mathbb{E} \big[ Y_{i,t+h}(\delta) - Y_{i,t+h} \mid Z_{i,t-1}=z \big] = \mathbb{E}\!\left[ \mathbb{E}\big( Y_{i,t+h}(\delta) - Y_{i,t+h} \mid \mathcal F_{t-1},X_t \big) \;\middle|\; Z_{i,t-1}=z \right], \] where $Y_{i,t+h}(\delta)$ is the potential outcome when the shock at time $t$ is perturbed from $X_t$ to $X_t+\delta$.
The central question is whether the total effect of a perturbation to $X_t$ at horizon $h$ can be expressed as a function of the initial state multiplying the shock. If so, the impulse response admits a representation \[ \text{IRF}_h(\delta \mid z)=g_h(z)\,\delta, \] where $g_h(z)$ captures both the direct effect of the shock and all indirect effects arising from its propagation over time. This multiplicative structure is the key condition under which LPs recover state-dependent causal impulse responses.
Under DGP 1, a perturbation $\delta$ changes $Y_{it}$ by $g(Z_{i,t-1})\delta$. Because propagation is linear and time-invariant, the key condition is satisfied: \[ \text{IRF}_h(\delta\mid z) = \rho^h g(z)\,\delta. \] This extends to more general lag structures: if the shock enters linearly and propagation dynamics are time-invariant, the impulse response admits the multiplicative form $\text{IRF}_h(\delta\mid z)=g_h(z)\delta$. All dynamic feedback effects are absorbed into the single coefficient $g_h(z)$.
Under DGP 2, the key condition fails because a perturbation to $X_t$ affects future propagation. At $h=1$, \[ \mathbb{E}\!\left[ Y_{i,t+1}(\delta)-Y_{i,t+1} \mid \mathcal F_{t-1},X_t \right] = \mathbb{E}\!\left[ \rho_{t+1}(\delta)\,Y_{it}(\delta) - \rho_{t+1}\,Y_{it} \mid \mathcal F_{t-1},X_t \right], \] so the response depends on both the perturbed outcome $Y_{it}(\delta)$ and the perturbed propagation coefficient $\rho_{t+1}(\delta)$. Because both objects vary with the shock, the effect involves their interaction and is not, in general, proportional to $\delta$ with a coefficient depending only on the initial state. This environment is a micro--macro analogue of the aggregate DGP considered by gonccalves2024state where state-dependent LPs fail. Our formulation highlights that this failure arises from shock-dependent propagation, rather than from endogeneity of the state per se.
Under DGP 3, nonlinearity in the shock means that even at $h=0$: \[ Y_{it}(\delta)-Y_{it} = f(Z_{i,t-1},X_t+\delta)-f(Z_{i,t-1},X_t), \] which depends on the level of $X_t$ and is not proportional to $\delta$. No representation of the form $g_h(Z_{i,t-1})\delta$ for the impulse response generally exists and the key condition fails.
In sum, only in environments like DGP 1 can the impulse response at each horizon be expressed as a state-dependent coefficient on the shock.
We now discuss what different LP specifications recover in these different DGPs. Consider:
where $\phi(\cdot)$ is a vector of basis functions allowing for flexible state-dependence.
kolesar2024dynamic show that scalar LPs as (ref) retain a clean average causal interpretation even when the DGP is nonlinear, and thus the result applies to all three DGPs.
By contrast, an LP with linear state dependence as (ref) forces the regression to approximate $g_h(z)$ by its best linear projection. Unless $g_h(\cdot)$ is exactly linear, we show that the interaction slope $\beta_h$ does not coincide with the impulse response at any particular state or with any well-defined aggregation of state-specific impulse responses.
This reveals a fundamental asymmetry. The robustness of scalar LPs relies on averaging across heterogeneous effects. Once one conditions on the state, this logic no longer protects the estimator from functional-form misspecification. Modeling heterogeneity is therefore more demanding than estimating average effects. Many empirical papers interpret the coefficient on a linear shock–state interaction as the causal effect for high- versus low-state units. Our analysis shows that such an interpretation is generally not justified when state dependence is nonlinear.
Allowing flexible nonparametric dependence on the state $Z_{i,t-1}$ in our proposed LP specification - such as (ref) - instead permits the regression to recover directly the causal impulse response function $g_h(z)$ if the data are generated by DGP 1.
In DGP 2 and DGP 3, by contrast, the impulse response does not admit a multiplicative representation in the first place and state-dependent LP coefficients (both linear and nonparametric) are generally projections without a clean causal interpretation.
Table 1 summarizes the central message of the paper.
In the remainder of the paper, we (i) derive a sufficient condition under which state-dependent LPs recover causal impulse responses without specifying the full DGP; (ii) characterize when this condition holds or fails, admitting environments with state-dependent exposure and constant propagation (such as DGP 1) while excluding those with shock-dependent propagation or nonlinear-in-shock effects (such as DGP 2--3); and (iii) show that standard linear state-dependent LPs do not recover this object except in a knife-edge case, motivating a nonparametric approach. We argue that this admissible class includes many canonical micro--macro DGPs and develop a nonparametric approach for estimation and inference of causal state-dependent impulse responses via LPs in these settings.
We now discuss identification and the conditions under which it holds or fails.
This section provides a general characterization of when LPs recover state-dependent causal impulse responses. Rather than specifying a full structural model or DGP, we impose a sufficient condition on the horizon-specific conditional mean that delivers a causal interpretation of state-dependent LPs. We then show that, even under this condition, standard linear state-dependent LPs generally fail to recover the causal object, motivating our nonparametric approach.
We observe a panel $\{Y_{it},X_t,Z_{it},W_{it}\}$ for $i=1,\dots,N$ and $t=1,\dots,T$, where $Y_{it}$ is a micro outcome, $X_t$ is an aggregate shock, $Z_{it}$ is a micro state, and $W_{it}$ collects control variables that may be included in the LP specification. Let $\mathcal F_{t-1}$ denote the information available at time $t-1$, \[ \mathcal{F}_{t-1} = \sigma\big(\{Y_{is},Z_{is},W_{is}\}_{i,s\le t-1},\,\{X_s\}_{s\le t-1}\big). \] Our object of interest is the state-dependent impulse response of $Y$ to a one-time perturbation of the realized shock at date $t$. Let $Y_{i,t+h}(\delta)$ denote the potential outcome when the realized shock at date $t$ is increased from $X_t$ to $X_t+\delta$, holding all other shocks at their realized values. The finite-$\delta$ causal effect conditional on the initial micro state is \[ \mathrm{IRF}_Y(h,\delta\mid Z_{i,t-1}=z) := \mathbb{E}\big[Y_{i,t+h}(\delta)-Y_{i,t+h}\,\big\vert\,Z_{i,t-1}=z\big]. \]
For each horizon $h\ge 0$, we consider the LP
where $g_h(\cdot)$ is an unknown function and $W_{i,t-1}$ are controls. The empirical object of interest is the function $g_h(\cdot)$, interpreted as the horizon-$h$ impulse response per unit shock conditional on $Z_{i,t-1}=z$.
The following high-level assumptions suffice for identification.
Assumptions (ref)--(ref) are standard: the state and controls are predetermined, and the aggregate shock is exogenous. Assumption (ref) is the high-level analogue of the multiplicative structure highlighted in Section (ref), and is the key condition for identification. It requires that the total effect of the date-$t$ shock $X_t$ on the horizon-$h$ conditional mean - including all direct and indirect channels operating through the endogenous evolution of the system - be linear in $X_t$, with a slope that depends only on the initial micro state $Z_{i,t-1}$. The nuisance component $r_{i,h}(\mathcal F_{t-1})$ is unspecified and unrestricted and need not coincide with the linear projection onto the controls $W_{i,t-1}$ used in the LP specification (ref).\footnote{The controls $W_{i,t-1}$ are included in the LP as a projection device, whereas $r_{i,h}(\mathcal F_{t-1})$ denotes the unknown conditional mean component. Note that identification does not require correct specification of $r_{i,h}(\mathcal F_{t-1})$. Instead, controls serve to partial out predetermined variation, improving efficiency and defining the conditioning set for the impulse response. As a result, different control sets correspond to different conditional impulse responses, each of which is consistently estimated under Assumption 3. Unit fixed effects can be absorbed into $r_{i,h}(\mathcal F_{t-1})$ without affecting identification but shift the variation used to estimate $g_h(\cdot)$ toward within-unit variation in exposure to the aggregate shock over time.} In particular, it may contain arbitrary lagged dynamics, fixed effects, and nonlinear dependence on past aggregate and idiosyncratic variables, provided these components are $\mathcal F_{t-1}$-measurable.
Note that, in many empirical applications, researchers instead use cumulative changes $Y_{i,t+h} - Y_{i,t}$ as the dependent variable in LPs. Under Assumption (ref), the corresponding estimand is $g_h(z) - g_0(z)$. Thus, cumulative LPs recover cumulative causal impulse responses under the same conditions.
Under Assumptions 1--3, $g_h(\cdot)$ is identified: \[ g_h(z) = \frac{\mathbb E[X_t Y_{i,t+h}\mid Z_{i,t-1}=z]} {\mathbb E[X_t^2\mid Z_{i,t-1}=z]}. \]
The following result clarifies that a causal interpretation is still available, even when the state variable $Z_{i,t-1}$ does not coincide with the true state governing heterogeneity.
\noindentProposition 1 (State choice preserves causal interpretation). Suppose Assumptions (ref)--(ref) hold and that, for some $s_{i,t-1}\in\mathcal F_{t-1}$, \[ \mathbb E[Y_{i,t+h}\mid \mathcal F_{t-1},X_t] = \tilde g_h(s_{i,t-1})X_t+r_{i,h}(\mathcal F_{t-1}). \] Let $Z_{i,t-1}$ be any $\mathcal F_{t-1}$-measurable state, and define \[ g_h(z) = \frac{\mathbb E[X_tY_{i,t+h}\mid Z_{i,t-1}=z]} {\mathbb E[X_t^2\mid Z_{i,t-1}=z]}, \qquad \mathbb E[X_t^2\mid Z_{i,t-1}=z]>0.\footnote{With predetermined controls, the same result holds after partialling them out.} \] Then \[ g_h(z) = \mathbb E\!\left[ w_h(s_{i,t-1};z)\tilde g_h(s_{i,t-1}) \mid Z_{i,t-1}=z \right], \quad w_h(s;z) = \frac{\mathbb E[X_t^2\mid s_{i,t-1}=s,Z_{i,t-1}=z]} {\mathbb E[X_t^2\mid Z_{i,t-1}=z]}. \] The weights satisfy $w_h(s;z)\ge0$ and $\mathbb E[w_h(s_{i,t-1};z)\mid Z_{i,t-1}=z]=1$. Thus, $g_h(z)$ is a convex (variance-weighted) average of the underlying causal responses among observations with $Z_{i,t-1}=z$. For simplicity, we present the analysis for a scalar state variable, but the results extend directly to multivariate states.
The causal object of interest is the function $g_h(\cdot)$ itself. A natural estimator is therefore nonparametric. By contrast, much of the applied literature assumes that $g_h(\cdot)$ is linear and interprets the slope coefficient as a state-dependent effect. The next result shows that this interpretation fails in general.
\noindentProposition 2 (Why linear state-dependent LPs generally fail). Suppose Assumptions 1--3 hold and that $g_h(\cdot)$ is differentiable almost everywhere. The coefficient on the linear interaction term in a state-dependent LP (e.g., equation (ref) in Section (ref)) can be written as \[ \beta_h = \int g_h'(z)\, \omega_h(z)\, dz, \] where $g_h'(z)$ denotes the derivative of the impulse response function, and $\omega_h(z)$ is a weighting function that depends on the distribution of the data, satisfies $\int \omega_h(z)\,dz=1$, but is not restricted to be nonnegative. Thus, the linear interaction coefficient does not, in general, correspond to a well-defined causal effect. Appendix (ref) provides a more detailed and formal characterization of this result.
In contrast to the scalar LP case studied by kolesar2024dynamic, where the estimand is a positively weighted average of heterogeneous causal effects and admits a clear subpopulation interpretation, the weights here need not be positive. As a result, the linear-interaction coefficient does not aggregate causal effects across units but instead combines marginal effects from different regions of the state space with potentially opposing signs. Economically, this breaks the usual interpretation of the coefficient as describing how responses vary with the state: a positive (or negative) estimate need not imply that higher-$z$ units respond more (or less) to the shock.
Assumption 3 is a restriction on the horizon-$h$ conditional mean of the outcome, rather than a full characterization of the DGP. The assumption is not testable without further structure, but it is implied by economically meaningful classes of DGPs. This subsection provides an economic interpretation of this condition and highlights classes of canonical micro--macro DGPs in which it is satisfied. Importantly, this characterization is not used for estimation; rather, it clarifies the environments in which state-dependent LPs recover economically meaningful causal impulse responses.
As we show below, a transparent case in which Assumption 3 is satisfied is when the effect of aggregate shocks operates through state-dependent exposure of units to a shock that enters linearly, while propagation dynamics are time-invariant.
(i) Log-linearized heterogeneous-agent models.
Assumption 3 is naturally satisfied by first-order (log-)linear solutions of HA/HANK models. For any household type (grid point) $i$ with predetermined idiosyncratic state $s_{i,t-1}$, the linearization implies that the conditional mean of a micro outcome is affine in the unanticipated aggregate shock, with a type-specific coefficient that may depend flexibly on the micro state. Concretely, perturbation methods used in quantitative HANK, including the Reiter approach formalized by BayerLuetticke2020, write the heterogeneous-agent equilibrium as a nonlinear functional difference equation and approximate it around a stationary equilibrium without aggregate risk. The first-order solution is a linear state-space system: the law of motion for the relevant aggregate and distributional states, and the mapping from these states to individual outcomes, are governed by fixed coefficients that do not depend on the realization of the shock.\footnote{See BayerLuetticke2020 for the perturbation logic and the resulting linear state-space representation: the linearized law of motion for the high-dimensional state vector is governed by a fixed matrix $O$ and a fixed mapping from states to controls $G$ BayerLuetticke2020.}
Mapping to our notation, let $Y_{it}$ be any household-level outcome (e.g.\ consumption or earnings), let $X_t$ denote the aggregate shock, and let $Z_{i,t-1}$ be a chosen predetermined micro characteristic. At impact, the linearized solution implies \[ \mathbb{E}\!\left[Y_{i,t}\mid \mathcal{F}_{t-1},X_t\right] = g_0(s_{i,t-1})X_t+r_{i,0}(\mathcal{F}_{t-1}). \] Because all subsequent propagation is governed by the same fixed linear law of motion and policy rules, the effect of the shock at date $t$ remains linear as the system is iterated forward. Hence, for each horizon $h$, \[ \mathbb{E}\!\left[Y_{i,t+h}\mid \mathcal{F}_{t-1},X_t\right] = g_h(s_{i,t-1})X_t+r_{i,h}(\mathcal{F}_{t-1}). \] Thus, Assumption 3 holds exactly within the linearized model. The restriction is not that heterogeneity is linear in the micro state: $g_h(\cdot)$ may be highly nonlinear because it reflects the nonlinear household problem evaluated at different points of the state space. Rather, the restriction is that the shock affects outcomes through a linearized system whose propagation coefficients are fixed and therefore not themselves altered by the shock.
This structure can be given a microfoundation using the sufficient-statistics representation of Auclert2019. In his partial-equilibrium micro block, an individual's consumption response to an aggregate shock can be decomposed, to first order, into a substitution effect and a wealth (revaluation) effect given by the product of the individual's marginal propensity to consume (MPC) and a balance-sheet exposure term Auclert2019. In our notation, this implies that the response can be written as a loading on the shock that depends on MPCs and balance-sheet positions, which are unrestricted (and potentially nonlinear) functions of the micro state. This decomposition characterizes the cross-sectional structure of the response, while the linearized dynamics that are standard in perturbation-based HANK models ensure that propagation across horizons is governed by time-invariant coefficients, so that the effect at horizon $h$ can be summarized by $g_h(s_{i,t-1})$. This perspective also motivates our empirical choice of a low-dimensional state $Z_{i,t-1}$: while the full micro state $s_{i,t-1}$ is high-dimensional, responses depend primarily on MPCs and balance-sheet exposures, which are low-dimensional functions of predetermined characteristics such as liquid wealth. Conditioning on $Z_{i,t-1}$ therefore targets the key sources of heterogeneity emphasized by the model.
The preceding discussion is exact for the linearized equilibrium and approximate for the underlying nonlinear economy. In perturbation-based HANK solutions, the equilibrium is approximated around a stationary steady state, typically without aggregate risk. Our object of interest is a shock perturbation: the realized aggregate shock is changed from $X_t$ to $X_t+\delta$, and the finite-$\delta$ impulse response compares outcomes under these two shock realizations. Within the linearized system, this comparison is exact for any $\delta$, because the approximating equilibrium mapping is linear in the shock by construction. Thus, replacing $X_t$ by $X_t+\delta$ changes outcomes proportionally to $\delta$, so Assumption 3 holds exactly within the linearized model.
When the linearized system is used as an approximation to the underlying nonlinear economy, the relevant requirement is that the first-order approximation remain accurate over the range of shock realizations being compared, including the paths induced by $X_t$ and by $X_t+\delta$. This is a local approximation requirement, not a restriction on the size of $\delta$ per se. Moderate perturbations may be well approximated when the induced path remains in a region where the equilibrium mapping is smooth and close to linear. By contrast, the approximation can deteriorate when the perturbation moves the economy into regions where nonlinearities are quantitatively important, for example because constraints become binding for many agents, aggregate constraints such as the zero lower bound bind, or the shock induces large reallocations across agents with different exposures. In such cases, the counterfactual path associated with $X_t+\delta$ is no longer well summarized by the linearized dynamics around the steady state.\footnote{If $\mathbb E[Y_{i,t+h}\mid \mathcal F_{t-1},X_t]$ is twice differentiable in $X_t$, a second-order Taylor expansion implies that the finite-shock response to a perturbation $\delta$ equals its linear approximation plus a remainder of order $O(\delta^2)$, governed by the curvature of the conditional mean over the relevant range.}
(ii) Macro and macro-finance models.
Consider a panel VAR for the joint process $(Y_{it},W_{it})^{\top}$ with state-dependent exposure to the aggregate shock and time-invariant propagation:
where $A_i(L)$ is a lag polynomial with possibly heterogeneous but time-invariant coefficients. The functions $\lambda_Y(\cdot)$ and $\lambda_W(\cdot)$ govern exposure of all variables to the aggregate shock and may be constant or arbitrary measurable functions of the predetermined state $Z_{i,t-1}$.
Because the system is linear in $X_t$ and the propagation coefficients are fixed, iterating (ref) forward implies that, for each horizon $h$, \[ \mathbb{E}\!\left[Y_{i,t+h}\mid \mathcal F_{t-1},X_t\right] = r_{i,h}(\mathcal F_{t-1}) + g_h(Z_{i,t-1})\,X_t, \] for some measurable function $g_h(\cdot)$. The function $g_h(Z_{i,t-1})$ summarizes both the direct effect of $X_t$ on $Y_{it}$ (through $\lambda_Y$) and all indirect effects operating through the endogenous post-$t$ path of $W$ and $Y$ (through $\lambda_W$ and the propagation $A_i(L)$).
This structure allows rich state dependence in exposure while ruling out changes in the dynamics governing the evolution of $(Y,W)$ following the shock. Under this exposure--propagation separation, Assumption 3 holds and state-dependent LPs recover causal impulse responses.
This structure is consistent with leading heterogeneous-firm monetary models. In ottonello2020financial, monetary policy shocks shift the intertemporal price system and firms' financing conditions. In this setting, $Y_{it}$ is firm investment or capital growth, $X_t$ is the monetary shock, and $Z_{i,t-1}$ is a predetermined financial state (e.g., net worth, leverage, or distance-to-default). The model generates strong nonlinearity in exposure because firms' marginal conditions depend on default risk and exhibit threshold effects (see their Figure 2), so the function $\lambda(\cdot)$ need not be linear. Importantly, while the financial state evolves endogenously after the shock, the structural forces governing propagation (e.g., depreciation and adjustment costs) remain time-invariant. As a result, the shock shifts the level of $Y_{i,t+h}$ without altering its propagation dynamics.
A closely related implication emerges in cui2025risk. Their mechanism generates nonlinear exposure: the effect of an interest-rate shock on risk-taking depends on a predetermined leverage constraint, and the threshold for adopting risky projects can be hump-shaped in the interest rate. In our notation, $Y_{it}$ denotes firm-level risk-taking or investment, $X_t$ the interest-rate shock, and $Z_{i,t-1}$ the predetermined financial constraint. The nonlinearity arises from equilibrium choice conditions, rather than from changes in the propagation dynamics of $Y$ following the shock.
(i) Nonlinear dependence on the shock. Assumption 3 excludes environments in which the conditional mean is nonlinear in the shock, e.g., \[ \mathbb{E}[Y_{i,t+h}\mid \mathcal F_{t-1},X_t] = f_h(Z_{i,t-1},X_t)+r_{i,h}(\mathcal F_{t-1}). \] Examples include regime-switching policy, zero-lower-bound nonlinearities, or large discrete interventions. In these cases, the effect of a perturbation depends on the magnitude or sign of $X_t$.
(ii) State-dependent propagation. Assumption 3 also fails when shocks affect the propagation of the outcome itself. In such environments, shocks alter not only contemporaneous outcomes but also the law of motion governing their future dynamics through their effect on endogenous states, so that the horizon-$h$ response cannot be summarized as a function of the initial state alone. Such mechanisms arise, for example, in sovereign default models such as almuzara2025nonlinear, where aggregate shocks affect borrowing conditions and default risk, which in turn shape future dynamics. In these environments, shocks propagate through endogenous states, so that the law of motion itself depends on the shock.
(iii) Fully nonlinear HA/HANK models. The previous mechanisms arise naturally in fully nonlinear heterogeneous-agent environments beyond first-order (log-linear) approximations. In such models, aggregate shocks can move the economy across regions of the state space where the local dynamics differ, for example due to occasionally binding constraints, kinked adjustment costs, or endogenous regime shifts. As a result, shocks affect not only the level of $Y_{it}$ but also its propagation. Formally, if \[ Y_{i,t+1} = f\left(Y_{it},Z_{it},X_t\right), \] and $\partial f/\partial Y$ depends on states that are themselves affected by the shock, then the propagation of $Y$ becomes shock-dependent. In that case, the horizon-$h$ response depends on the entire nonlinear state path induced by $X_t$, and the conditional mean generally cannot be written in the multiplicative form of Assumption 3.
Such violations of Assumption 3 are economically plausible in aggregate time-series settings, where shocks can alter persistence. The critique of gonccalves2024state that state-dependent LPs do not recover finite-$\delta$ impulse responses is therefore valid in these settings. Our condition targets a different empirical setting: micro--macro panels in which cross-sectional heterogeneity operates through state-dependent exposure to common aggregate shocks that enter linearly, while the propagation of individual outcomes is governed by time-invariant structural primitives (e.g., depreciation, adjustment costs, or idiosyncratic risk processes). In that environment, Assumption 3 isolates state dependence in exposure from propagation and restores a transparent causal interpretation of state-dependent LPs (provided the state dependence is estimated nonparametrically, which we discuss next).
This section develops a sieve estimator of the state-dependent impulse response function $g_h(\cdot)$ and associated inference procedures. The parameter of interest is the coefficient function $g_h(z)$ defined by the conditional mean restriction established in Section (ref). Section (ref) shows that this function is the causal object of interest and that, in general, recovering it requires nonparametric estimation.
The resulting LP estimation problem has a structure in which the aggregate shock enters with a functional coefficient $g_h(Z_{i,t-1})$, while additional regressors (such as predetermined controls) enter as finite-dimensional components.
Our setting builds on standard econometric tools - sieve approximation, HAC covariance estimation, and multiplier bootstrap methods for uniform inference -but applies them in a LP framework with features that are not covered by off-the-shelf results.
On the one hand, the LP structure with an exogenous aggregate shock is particularly convenient: because $X_t$ is a martingale difference, the nuisance component of the conditional mean is orthogonal to the regressor of interest. As a result, $g_h(\cdot)$ can be identified without specifying the rest of the conditional mean, unlike in standard regression settings.
On the other hand, the LP environment introduces several nonstandard features that require adapting existing sieve methods. First, the object of interest is a functional coefficient multiplying an aggregate shock, rather than a generic conditional mean. Second, overlapping outcomes across horizons induce time-series dependence in the regression errors. Third, the aggregate shock generates cross-sectional dependence across units. Finally, inference targets the entire function $g_h(\cdot)$, rather than a finite-dimensional parameter, making uniform inference essential.
Taken together, these features imply that while our approach relies on standard building blocks, their combination in a macro--micro LP setting is nonstandard and requires careful adaptation and verification of the underlying asymptotic arguments.
Relative to the general approach to nonlinear impulse responses considered by gonccalves2024state, we restrict attention to the class of environments characterized in Section (ref), in which the shock enters the conditional mean linearly and heterogeneity operates through a state-dependent functional coefficient. Their approach recovers impulse responses by first estimating the response of the outcome as a function of the shock, allowing for general nonlinear dependence, and then constructing impulse responses as differences in this response function evaluated at different shock realizations and averaged over the distribution of the shock. While differencing removes nuisance components in population, the procedure involves estimating a nonparametric response function and then transforming it, so that estimation error from the first step carries over into the second-step construction of the impulse response. By contrast, we exploit the linear-in-shock structure implied by our identifying restriction to eliminate the nuisance component at the level of the identifying moment and estimate the coefficient function directly, without requiring a second-step transformation. The two approaches also differ in their assumptions on the shock process: their construction relies on i.i.d.\ shocks to justify averaging over the shock distribution, whereas our approach only requires martingale-difference exogeneity.
The remainder of this section formalizes the estimator and inference procedures. Full technical details and proofs are collected in Appendix (ref) and Appendix (ref).
For each horizon $h \in \{0,1,\ldots,H\}$, we estimate $g_h(\cdot)$ using a sieve approximation. Let $\{\phi_{j,J}(z)\}_{j=1}^J$ denote a basis of cubic B-splines defined on the support of $Z_{i,t-1}$, where interior knots are located at equally spaced empirical quantiles and with a constant term included. For a given sieve dimension $J$, approximate
Substituting (ref) into the LP yields the estimation equation
Let $D_{it}(J)$ denote the vector of regressors in (ref). The OLS estimator is
with $\widehat{b}_h(J)$ denoting the subvector corresponding to the sieve coefficients. The associated estimator of the impulse response function is
We select the sieve dimension $J$ using the Akaike information criterion (additional selection criteria are examined in the simulation section and in the Appendix and perform similarly):
where $\widehat{u}_{i,t+h}(J)$ denote the OLS residuals from (ref) and $K_J = J + \dim(W_{i,t-1})$ is the total number of estimated parameters. The selected number of sieves is
In practice, we take the candidate set $\mathcal J=\{4,\ldots,20\}$, with the upper bound serving as a finite-sample safeguard against an overly rich basis. Here $J=4$ gives a global cubic polynomial, while larger values of $J$ add interior knots to the cubic B-spline basis and allow for more flexible nonlinearities. The final estimator is
The following algorithm summarizes the steps involved in estimating the impulse response function. \paragraph{Algorithm 1 (Sieve LP estimator of the impulse response function).} For each horizon $h = 0,\ldots,H$:
We consider both pointwise and uniform inference for the impulse response function $g_h(\cdot)$.
Uniform inference is important because the empirical object is often the entire state-dependent response function, rather than the response at a single preselected state. Economic conclusions frequently depend on features of the whole curve - such as whether responses are monotone, hump-shaped, concentrated in the tails, or significantly different across regions of the state distribution. Pointwise intervals can give a misleading impression when researchers inspect many values of the state simultaneously. By contrast, uniform confidence bands allow for valid statements about the shape of the response function over the relevant support, while controlling coverage jointly across states. In the types of applications considered here, this is particularly important for distinguishing a genuinely nonlinear transmission pattern from sampling variation around a linear approximation.
Let $D_{it}$, $\widehat b_h$, $\widehat u_{i,t+h}$ denote, respectively, the vector of regressors, the sieve coefficient estimator, OLS residual in (ref) corresponding to the AIC-selected number of basis functions $\widehat{J}_h$. Under regularity conditions, the estimator admits the linear representation
for a suitable influence function $\psi_{h,i,t}$ that reflects the projection of the sieve regressors on the control variables. The precise form of $\psi_{h,i,t}$ and the regularity conditions are given in Appendices (ref) and (ref).
Define the sample second moment matrix \[ \frac{1}{N(T-h)} \sum_{i=1}^N\sum_{t=1}^{T-h} D_{it}D_{it}^{\top} =
, \] where the first block corresponds to $X_t\phi_{\widehat J_h}(Z_{i,t-1})$ and the second to $W_{i,t-1}$. Form the partialled-out sieve regressor
and the score process
We estimate the long-run covariance matrix of the score process by
where $w_k=1-k/(L+1)$ and $L = \lfloor 4 [N(T-h)/100]^{2/9} \rfloor$. The covariance matrix of $\widehat b_h$ is estimated by
where
The pointwise variance of $\widehat g_h(z)$ is
Thus, for fixed $z$ on the empirical support of $Z_{i,t-1}$, the $(1-\alpha)$ pointwise confidence interval is
where $z_{1-\alpha/2}$ is the standard normal critical value.
For uniform inference, let $\mathcal Z_G$ be a fine grid covering the empirical support of $Z_{i,t-1}$. For each bootstrap draw $b=1,\ldots,B$, generate \[ \xi_h^{(b)}\sim \mathcal N(0,I_{\widehat J_h}), \] and form the Gaussian process
Compute the supremum statistic
and let $c_{h,1-\alpha}$ be the empirical $(1-\alpha)$ quantile of $\{ T_h^{(b)} \}_{b=1}^B$. The $(1-\alpha)$ uniform confidence band is
The distinction is that $\mathcal I_h(z)$ provides coverage at a fixed state $z$, while $\mathcal C_h(\cdot)$ provides simultaneous coverage over all grid points in $\mathcal Z_G$.
The following algorithms summarize the construction of pointwise confidence intervals and uniform confidence bands.\footnote{For cumulative specifications where the outcome is $Y_{i,t+h} - Y_{i,t}$, the corresponding estimand is $g_h(z) - g_0(z)$. Inference proceeds analogously by estimating the LP with the transformed outcome.}
\paragraph{Algorithm 2 (Pointwise confidence intervals).} For each horizon $h = 0,\ldots,H$:
\paragraph{Algorithm 3 (Uniform confidence bands).} For each horizon $h = 0,\ldots,H$:
We summarize the main theoretical result; formal conditions and proofs are provided in Appendix (ref).
All simulations are based on a DGP satisfying Assumption 3. We run $300$ Monte Carlo replications of data generated as
where $X_t \sim \mathcal{N}(0,1)$ and $\varepsilon_{it} \sim \mathcal{N}(0,1)$. The state variable evolves as \[ Z_{it} = \mu_i + \xi_{it}, \quad \mu_i \sim \mathcal{N}(0,3), \quad \xi_{it} = 0.8\, \xi_{i,t-1} + \sqrt{1-0.8^2}\, v_{it}, \quad v_{it} \sim \mathcal{N}(0,1). \] We discard the first $500$ observations and retain $T$. Results are robust to including fixed effects (not reported).
We focus on the cubic specification
The corresponding horizon-$h$ causal impulse response is
We estimate the nonparametric state-dependent LP \[ Y_{i,t+h} = g_h (Z_{i,t-1})X_t + \gamma_h Y_{i,t-1} + u_{i,t+h}, \quad h = 0,\ldots,H, \] using the spline sieve estimator described in Section (ref). The estimated impulse response is
We select the sieve dimension $J$ using AIC, GCV, and LASSO, alongside an oracle choice ($J=4$). For AIC and GCV, we search over $\mathcal{J}=\{4,\ldots,20\}$, with GCV given by \[ \mathrm{GCV}_h(J) = \frac{\sum_{i,t}\widehat u_{i,t+h}(J)^2}{N(T-h)\left(1-\frac{K_J}{N(T-h)}\right)^2}, \quad \widehat J_h = \arg\min_{J\in\mathcal J}\mathrm{GCV}_h(J). \] For LASSO, we set a maximal dimension $J_{\mathrm{lasso}}=50$ and estimate \[ \widehat{\theta}_{h} = \arg\min_{\theta_h} \left\{ \sum_{i,t}\bigl(Y_{i,t+h} - D_{it}(J_{\mathrm{lasso}})^\top \theta_h\bigr)^2 + \lambda_h \|\theta_h\|_1 \right\}, \] where $\lambda_h$ is chosen by cross-validation. The selected dimension $\widehat J_h$ is the number of nonzero sieve coefficients.
We evaluate the performance of the sieve estimator for the state-dependent impulse response and its gains relative to standard linear state-dependent LPs. Under the DGP, which satisfies Assumption 3, the causal impulse response is well defined as $g_h(z)$, so the simulation isolates the role of functional-form restrictions. We set $N=500$, $T=200$, and select the number of basis functions using AIC.
For comparison, we estimate the linear state-dependent LP \[ Y_{i,t+h} = (\alpha_h + \beta_h Z_{i,t-1})X_t + \gamma_h Y_{i,t-1} + u_{i,t+h}, \] which implies
Figure (ref) reports Monte Carlo averages and 10% to 90% percentile bands for the sieve and linear IRFs, together with the true response, for a unit shock ($\delta=1$) over a grid of $500$ points in $[-4.65,4.65]$.
Figure (ref) highlights the main gain from sieve estimation relative to the linear state-dependent LP. When the true response varies nonlinearly with the state, the linear specification yields a distorted profile with substantial bias, particularly in the tails. By contrast, the spline sieve closely tracks the true IRF across the entire range of $z$, with tight Monte Carlo dispersion. The two estimators can also imply opposite signs in some regions, leading to qualitatively different economic conclusions.
As the horizon increases and the response becomes more linear, the discrepancy between the two approaches attenuates, although the sieve estimator continues to recover the shape more accurately. This reflects both improved approximation and the fact that the sieve targets the causal state-dependent IRF, whereas the linear specification recovers a projection that generally differs from the causal response.
These results are not driven by the choice of basis. Appendix (ref) shows that the same conclusions hold when the true $g(z)$ is generated by a Fourier series but estimated using splines (Figure (ref)).
We next quantify the gains of nonparametric estimation using the root integrated mean squared error (RIMSE). Table (ref) compares the sieve IRF to the linear IRF for different horizons $h$ and impulse sizes $\delta \in \{0.5\sigma_X,\sigma_X,2\sigma_X\}$, where $\sigma_X$ is the standard deviation of the shock $X_t$ (here equal to 1).
The results reinforce the visual evidence. The sieve estimator delivers substantially lower RIMSE than the linear specification. The gains are largest at short horizons, where nonlinearities are most pronounced, and remain meaningful even at longer horizons despite the impulse response becoming almost linear. Importantly, the gains also widen with $\delta$, indicating that the cost of misspecifying nonlinear state dependence becomes more pronounced for larger shocks. Overall, the findings confirm that accounting for nonlinearities is important for accurately recovering the impulse response function.
We evaluate the finite-sample performance of sieve-based uniform confidence bands across sample sizes $(N,T) \in \{100,500\} \times \{40,120,200\}$ and horizons $h \in \{0,4,8,12\}$.
For the spline sieve, we consider an oracle specification ($J=4$) and data-driven choices based on AIC, GCV, and LASSO. The main text focuses on AIC; results for all selectors are reported in the appendix.
Given $J$, we construct uniform confidence bands for $g_h(z)\delta$ (with $\delta=1$) using Algorithm 3 with $B=2000$ bootstrap draws and nominal coverage $1-\alpha=0.95$. Performance is evaluated over a grid of $500$ points on $[\min\{Z_{it}\}, \max\{Z_{it}\}]$ using coverage (fraction of replications in which the true IRF lies within the band over the grid) and average band width.
Before turning to overall band performance, we examine the role of including intermediate shocks in the LP. For $h \ge 1$, $Y_{i,t+h}$ depends on future interaction terms $g_j(Z_{i,t+j-1})X_{t+j}$ for $j=1,\ldots,h$.
We compare specifications that include or omit these terms, approximating each $g_j(\cdot)$ using the same sieve dimension. Under exogeneity, the intermediate terms are orthogonal to $X_t$, so their omission does not affect identification, but including them improves efficiency.\footnote{Orthogonality follows from the multiplicative structure and the exogeneity of future shocks: $X_{t+j}$ is mean independent of past information, including $X_t$, while $g_j(Z_{i,t+j-1})$ is measurable with respect to that information.}
Table (ref) shows that the two specifications coincide at $h=0$, where no intermediate shocks are present. For $h \ge 4$, coverage is similar, but including intermediate shocks yields substantially tighter bands, indicating a sizable efficiency gain. We therefore include intermediate shocks in all subsequent simulations and in the empirical application.
Table (ref) reports results for $N \in \{100,500\}$ and $T \in \{40,120,200\}$ at horizons $h \in \{0,4\}$ under AIC selection; additional results are reported in Appendix (ref). Performance improves with $T$, as coverage increases and bands narrow, while differences across $N$ are limited. At longer horizons, coverage declines modestly and bands widen.
Overall, the AIC-based bands exhibit some undercoverage relative to the nominal level. Appendix (ref) shows that coverage is close to nominal for the oracle selector, implying that the gap is driven by smoothing-parameter selection rather than by the band construction itself. This is consistent with the nonparametric literature, where data-driven tuning introduces additional variability that can lead to undercoverage in finite samples, especially for uniform inference.
We assess the sensitivity of sieve estimation to the choice of selector for the number of basis functions at horizon $h=4$. Estimation results are qualitatively similar across selectors and are therefore not reported. Table (ref) reports the corresponding inference results. Oracle, AIC, and GCV yield similar coverage with relatively tight bands, while LASSO achieves slightly higher coverage at the cost of substantially wider bands. Overall, sieve estimation is robust to selector choice, although different selectors imply a coverage–width trade-off.
To demonstrate the empirical relevance of our nonparametric state-dependent LP, we apply our methodology to study the effect of monetary policy shocks on firm investment in the spirit of ottonello2020financial, cloyne2023monetary and jeenas2023firm, among others. Specifically, we study how the investment response to these macroeconomic shocks varies across heterogeneous firms, with firm-level financial conditions serving as the state variable. This setting fits our framework to the extent that monetary policy shocks operate through heterogeneous balance-sheet exposure, while propagation dynamics are well approximated as stable over the relevant horizon.
Following ottonello2020financial (OW), we use distance to default (D2D) -- an estimate of the probability of default using leverage ratios and equity price volatility -- as our primary measure of firm financial conditions.\footnote{D2D is equal to the (log) ratio of firm value to debt (an inverse measure of leverage), appropriately adjusted for the volatility of firm value. It can be interpreted as the number of standard deviations the (log) ratio of value to debt must fall below its mean for the firm to default on its debt. Although OW also study leverage, we focus on D2D for two main reasons: first, there is a large empirical literature in corporate finance that documents the performance of D2D as an indicator of the likelihood that a firm will declare bankruptcy and/or default on a bond and the key economic importance of D2D in determining default intensities duffie2007multi,schaefer2008structural,duffie2011measuring,atkeson2017measuring; second, OW find weaker results using leverage, perhaps because it is a less precise indicator of firms' financial soundness.} Firms with high D2D are at lower risk of default and hence safer, i.e., more financially sound. Firms with low D2D are the opposite. For ease of comparability, we use the same monetary policy shocks as OW, identified from high-frequency changes in Federal Funds rate futures in a narrow window around FOMC announcements. As in OW, we measure firm investment using Compustat quarterly data as the change in the log book value of the firm's capital stock, denoted $k_{i,t} \equiv \log K_{i,t}$, and compute D2D following the approach in gilchrist2012credit. Throughout, we standardize D2D so that the units are in standard deviations relative to the sample mean, as do OW. We follow the same sample selection criteria as in OW.
We estimate the following LP specification:
Here, the dependent variable measures the firm's cumulative (net) investment from quarter $t$ through $t+h$; $\alpha_{sth}$ denotes sector-time-horizon fixed-effects; $g_h\left(Z_{i,t-1}\right)$ is our nonlinear function of the firm's financial condition (D2D); $X_t$ is the monetary policy shock; $\Delta k_{i,t-1}$ is the firm's lagged one-period investment rate; $W_{i,t-1}$ is a set of firm-level controls; and $u_{i,t+h}$ is an error term.\footnote{As in OW, we include as controls $Z_{i,t-1}$, total assets, sales growth, current assets as a share of total assets, a fiscal quarter indicator, and the interaction of $Z_{i,t-1}$ with lagged GDP growth. Our specification differs from OW in three ways: first, we control for the firm's lagged investment rate; second, they use the firm's demeaned financial position relative to its average level; third, they include firm fixed effects. None of these differences qualitatively changes our conclusions. Finally, we include time fixed effects only to facilitate comparison with OW, while noting that their inclusion may raise additional identification issues that are beyond the scope of this paper. .}
We estimate two versions of (ref) that differ only in the specification of the $g\left(\cdot\right)$ function. First, we impose a linear function such that $g_h\left(Z_{i,t-1}\right) = \beta_h Z_{i,t-1}$. Second, we implement the flexible nonlinear approach developed in Section (ref). We display our results in Figure (ref), which plots the impulse response function -- the investment response to the monetary policy shock -- on impact and at horizons of four, eight and 12 quarters following the shock. Each panel in the figure displays the investment response across the entire distribution of firms as a function of their pre-shock financial condition, i.e., $g_h \left(Z_{i,t-1}\right)$, accompanied by uniform confidence bands around the nonlinear estimate constructed using Algorithms 1 and 3. The shock is normalized as a 25 basis point surprise interest rate cut.
Panel A of Figure (ref) sharply illustrates our main findings. Imposing a linear specification yields a positive impact coefficient, which implies a monotonic upward-sloping relationship between a firm's investment response and financial condition -- less risky, more financially sound firms with higher D2D are more responsive to monetary policy shocks. The result corroborates the influential findings in OW, who also estimate a positive coefficient. In contrast, the nonlinear IRF shows quite a different pattern: a non-monotonic hump-shaped response that peaks at approximately the mean of the distribution, rather than at the least risky right tail. Thus, although responsiveness is increasing for firms that are close to default, as found in previous work, the relationship turns negative around the mean, implying that there is a large portion of the distribution where riskier, lower distance to default firms are more responsive to monetary policy shocks, not less.\footnote{Due to right-skewness in the distribution, the median D2D is slightly below the mean.}
Although our econometric model does not reveal the precise economic forces driving these patterns, one interpretation is that firms close to default are investment constrained and respond little, firms near the middle of the distribution are most sensitive to interest rates, and the safest firms less so. Thus, our method uncovers a richer, more nuanced relationship between shock sensitivity and financial conditions that is masked by the linear specification, underscoring the importance of allowing for more general state-dependence when analyzing microeconomic responses to macroeconomic shocks.\footnote{Our results are qualitatively unchanged under various modifications to our main specification; for example, including firm fixed-effects somewhat dampens responsiveness across the distribution, but the hump-shaped pattern we uncover remains.}
Panels B--D of Figure (ref) report IRFs at four-quarter intervals up to 12 quarters after the shock. The non-monotonicity becomes more pronounced over time: firms near the center of the D2D distribution exhibit increasingly larger cumulative investment responses than those at either tail.\footnote{Although not shown, the cumulative IRF changes little beyond 12 quarters, indicating that the effects of the shock have largely dissipated.} In contrast, the linear specification misses these dynamics and suggests that the response of high-D2D firms continues to grow relative to that of firms closer to default.
To illustrate the economic implications of response heterogeneity, Figure (ref) plots the differential in cumulative impulse responses across selected points of the D2D distribution under the linear and nonlinear specifications, which directly captures how the transmission of shocks varies across firms over the post-shock horizon. We report pointwise confidence intervals using Algorithm 2, since the objects of interest are differences in impulse responses at specific states.\footnote{Specifically, we consider $\widehat g_h(Z_a) - \widehat g_h(Z_b)$, where $Z_a$ and $Z_b$ denote two fixed points of the state distribution, such as the 50th and 5th percentiles, or the 95th and 50th percentiles, of $Z_{i,t-1}$. Accordingly, instead of using the pointwise variance estimator in (ref) directly, we construct pointwise confidence intervals using the variance $( \phi_{\widehat J_h}(Z_a) - \phi_{\widehat J_h}(Z_b) )^{\top} \widehat V_h ( \phi_{\widehat J_h}(Z_a) - \phi_{\widehat J_h}(Z_b) )$ in Algorithm 2.} In Panel A, we compare a firm at the 50th percentile of the D2D distribution to one at the 5th percentile. Although both sets of estimates predict that the firm at the 50th percentile will invest more than the firm at the 5th percentile in response to the monetary policy shock, the linear estimates significantly underpredict the difference; for example, after 16 quarters, the linear estimates yield a cumulative investment differential of just under 5%, whereas the nonlinear estimates suggest a cumulative differential exceeding 20%. In Panel B we similarly compare a firm at the 95th percentile of the state distribution to the 50th; in this case, the two sets of estimates yield virtually opposite results: the linear estimates predict a positive differential of almost 10% at 16 quarters following the shock, whereas the nonlinear estimates show that this difference is in fact negative and close to -10%. Thus, the linear specification underestimates the strength of the positive relationship between responsiveness and D2D in the left tail of the distribution, and misses entirely on the negative relationship in the right tail. Similar results hold at the shorter horizons as well. In short, imposing a linear specification can lead to misleading conclusions regarding the investment response of firms across the financial distribution.
Our results have important quantitative implications for the role of financial heterogeneity in the transmission of monetary policy: because most firms are concentrated in regions of the distribution where the nonlinear response exceeds the linear approximation, the latter effectively averages across states and attenuates the estimated impact of monetary policy shocks.
The response of aggregate investment to the monetary policy shock operating through financial heterogeneity is equal to the average of the firm-level responses weighted by their shares of the aggregate capital stock at the time of the shock, i.e.,
where $\Delta k_{t+h} = \log K_{t+h} - \log K_{t-1}$ and $K_t = \sum_i K_{it}$ denotes the aggregate capital stock.\footnote{This calculation captures only the incremental impact of the shock operating via heterogeneity in firms' financial conditions; the impact of the shock that is common across firms is picked up by the sector-time fixed-effects in equation (ref).} As the expression highlights, aggregate responsiveness depends not only on individual responses, but the joint distribution of those responses with firm shares of aggregate capital.\footnote{A similar result is used in david2023rise to decompose the impact of monetary policy on aggregate investment across tangible and intangible capital.}
We calculate aggregate responsiveness upon shock impact under both the linear and nonlinear specifications of $g_h\left(\cdot\right)$ using the estimates from Panel A of Figure (ref) and the distribution of firm capital on a quarter-by-quarter basis. We plot the results in Figure (ref). There are two main takeaways: first, the linear specification significantly underestimates the sensitivity of aggregate investment to monetary policy. On average, the linear specification implies that financial heterogeneity adds about 0.09 percentage points (p.p.) to the responsiveness of aggregate investment. In contrast, the nonlinear specification implies that financial heterogeneity adds about 1.0 p.p., roughly an order of magnitude larger.\footnote{The result is consistent with previous work finding that aggregate non-residential private fixed investment tends to be highly sensitive to monetary policy relative to other categories of GDP david2023rise.} The finding implies that not only does the linear specification miss on important aspects of the distribution of micro-level responses, but also that those micro-level biases affect “bottom-up” calculations of the implications of micro-level responses for the macro-level response. The result stems largely from the fact that most firms are concentrated in regions of the distribution where the nonlinear response exceeds the linear approximation and hence the linear specification attenuates the estimated aggregate impact of the macroeconomic shock.
The second key takeaway from Figure (ref) is that the time-series properties of aggregate responsiveness are quite different across the two specifications; indeed, they have a negative correlation of about -0.7. In other words, the linear specification implies that monetary policy is generally less effective exactly when the nonlinear specification says it will be more so. As a particularly stark example, take the Great Financial Crisis (GFC) and its immediate aftermath from late 2007 through the end of 2009. When firms' financial conditions deteriorated and more firms moved closer to default, the positive coefficient from the linear LP predicts that monetary policy would be less effective in stimulating investment; in contrast, because the pre-recession economy was above the typical level of financial health (the mean D2D was high), worsening financial conditions pushed firms closer to (and then below) the long-run mean, which is the region where responsiveness is highest, according to our estimates. Thus, the two approaches yield very different implications regarding the states of micro-level financial conditions when monetary policy is most effective: the linear estimates imply monetary policy is most effective at times when firms tend to be financially the strongest (responsiveness from the linear estimates also tends to be high when dispersion in D2D is high), whereas the nonlinear estimates imply monetary policy is most effective when firms tend to be closer to the long-run average level of D2D (responsiveness from the nonlinear estimates also tends to be high when dispersion in D2D is low). In sum, allowing for a flexible approach to estimating the impact of macro shocks on micro variables can be critical for properly assessing the macro-level implications of micro-level heterogeneity.
State-dependent local projections are widely used to study heterogeneous responses to aggregate shocks, yet their causal interpretation in nonlinear environments has remained unclear. This paper provides a unified resolution. It shows that state-dependent LPs recover causal impulse responses under a sufficient condition on the conditional mean that is satisfied in environments with state-dependent exposure to a linear shock and constant propagation dynamics. This structure is naturally satisfied in linearized (first-order) solutions of heterogeneous-agent macroeconomic models and in a large class of macro-finance models.
The analysis also reveals that common empirical practice is generally not innocuous. Even in environments where a causal interpretation is valid, standard linear interactions that are misspecified fail to recover the impulse response function and instead identify non-causal projection objects. Recovering the causal estimand therefore generally requires estimating state dependence nonparametrically. We provide a nonparametric sieve approach that achieves this in LPs and delivers asymptotically valid inference both pointwise and for the entire response function.
These results reframe how LPs should be used in practice. In nonlinear settings where shocks affect both exposure and propagation, state-dependent LPs generally lack a causal interpretation. In contrast, in micro--macro environments where heterogeneity operates through exposure to an aggregate shock that enters linearly and propagation is governed by stable structural primitives, state-dependent LPs can recover economically meaningful causal effects when implemented nonparametrically. This distinction reconciles the use of state-dependent LPs in empirical work with the structure of modern heterogeneous-agent and macro--finance models.
More broadly, the paper highlights a general lesson: identifying heterogeneous causal effects is fundamentally more demanding than estimating average effects, and requires moving beyond parametric approximations. By clarifying both the scope and the limitations of state-dependent LPs, the framework developed here provides a foundation for credible empirical analysis of heterogeneous transmission mechanisms in macroeconomics and macro-finance.