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.
17,213 characters · 5 sections · 7 citation commands
Event Studies with Feedback
\onehalfspacing
Event studies in the panel data setting aim to quantify how treatment effects evolve over time and vary across units. When outcomes are persistent and covariates adjust endogenously to past outcomes and treatment exposure, observed dynamic responses may not correspond to a single causal mechanism. Instead, they may combine direct effects with indirect effects operating through endogenous covariate adjustments. This distinction is empirically relevant when treatment triggers equilibrium adjustment. For example, a minimum wage policy may have a direct effect on wages but may also lead to changes in firm-level input demand, which may in turn affect individual wages. Separating these effects permits an assessment of how much of the dynamic response reflects direct versus indirect effects through equilibrium-induced covariate adjustment.
This paper develops a dynamic event study framework that separates these two effects. The direct structural effect captures how outcomes respond to treatment holding the covariate path fixed, while the indirect adjustment effect operates through covariates that evolve endogenously over time. We consider a panel event study setting with units $i = 1,\ldots,N$ observed over periods $t = 0,\ldots,T$. For exposition, we present a simplified version of the model. The outcome evolves in calendar time $t$ according to
where $Y_{it}\in\mathbb{R}$ is the scalar outcome, $X_{it}\in\mathbb{R}^K$ is a vector of time-varying covariates, and $\alpha_i$ captures unit-specific heterogeneity. Treatment is characterized by an event-time $t_{0i}\in\{1,\dots,T\}$. Let $\mathcal{J}\subset\mathbb Z$ denote a finite set of event-time indices for which dynamic treatment effects are specified. For each $j\in\mathcal{J}$, the event-time indicator $D_{it}^j = \mathbbm{1}\{t - t_{0i} = j\}$ is a deterministic function of $t_{0i}$. The coefficients $\delta_{ij}$ represent unit-specific dynamic treatment effects at event time $j$ and capture the direct structural response to treatment.
Following botosaru2025time, we impose a parsimonious dynamic structure on these heterogeneous treatment effects by assuming that they follow an autoregressive process in event time:
where $\rho_\delta$ is a common persistence parameter and $\varepsilon_{ij}$ are idiosyncratic innovations. Let $\lambda_i = (\alpha_i, \delta_{i0})'$ denote the vector of unobserved heterogeneity, and denote its conditional distribution by $$H(\lambda_i \mid \mathcal I_i^0), \quad \mathcal I_i^0 = \left\{Y_{i0},X_{i0},t_{0i}\right\}.$$ The framework readily extends to higher-order dynamics, continuous treatment intensities, and event-time leads for testing anticipation effects. Gaussianity in (ref) and (ref) is imposed for likelihood-based estimation, while the identification results do not require a correct Gaussian specification.
The parameter $\beta$ captures the indirect adjustment channel, mapping changes in covariates into changes in outcomes. In many applications, these covariates are policy-reactive and adjust dynamically in response to treatment and past outcomes. As a result, treatment may affect outcomes not only directly, but also indirectly through sequentially exogenous covariate adjustments. Standard event study approaches typically rule out this feedback by imposing strict exogeneity of covariates. While botosaru2025time relax this requirement to allow for sequential exogeneity, they treat the covariate process as given and do not model or identify the adjustment mechanism itself.
This paper extends their framework by explicitly modeling and identifying the covariate feedback process. We allow covariates to be predetermined and impose a homogeneous feedback restriction, requiring the covariate adjustment rule to be common across units up to observable conditioning variables. This restriction separates the two channels: the distribution $H(\lambda_i \mid \mathcal I_i^0)$ governs unit-level heterogeneity in the direct structural treatment effect, while $\beta$ and the identified feedback process capture the indirect adjustment effects. The framework therefore permits the inclusion of policy-reactive covariates and delivers an explicit decomposition of event study dynamics beyond reduced-form comparisons, facilitating policy analysis in settings with sequentially exogenous covariate adjustment.
We denote the history of variables up to time $t$ as $Y_i^t = (Y_{i1}, \dots, Y_{it})'$ and $X_i^t = (X_{i1}', \dots, X_{it}')'$. Let $\theta = (\rho_Y, \rho_\delta, \beta, \sigma^2_U, \sigma^2_\varepsilon)'$ denote the vector of common structural parameters, and $\mathcal I_i^{t} = \left\{Y_i^{t},X_i^{t},\mathcal I_i^0\right\}$ denote the information set up to time $t$.
To achieve point identification of the feedback process in short panels, we impose the following restriction on the feedback process, following bonhomme2025dynamics.
Under Assumption (ref), covariates $X_{it}$ can depend on $\lambda_i$ through past outcomes in complex ways. Assumption (ref) breaks this dependence by requiring the conditional covariate adjustment rule (ref) to be the same across individuals, regardless of their $\lambda_i$, conditional on the observable history $\mathcal I_i^{t-1}$. This restriction delivers a clean separation between the direct structural effect and the indirect adjustment effect. Conditional on $(X_i^T,\mathcal I_i^0)$, the likelihood kernel for $Y_i^T$ given $\lambda_i$ coincides with the representation studied in botosaru2025time, while all information about the indirect adjustment effect is captured by $g(X_i^T \mid Y_i^T, \mathcal I_i^0)$. This factorization allows us to treat the feedback mechanism as a separately identified component and to apply the identification arguments of botosaru2025time to the conditional distribution of $Y_i^T$ given $(X_i^T,\mathcal I_i^0)$ to identify the direct structural effect. The positivity condition on $g(X_i^T \mid Y_i^T, \mathcal I_i^0)$ ensures that this decomposition is well-defined.
We now establish that the model delivers point identification of both the direct and indirect effects. Specifically, we identify: (i) the common parameters $\theta$, (ii) the distribution of unobserved heterogeneity $H(\lambda_i \mid \mathcal I_i^0)$, which characterizes the direct structural effect, and (iii) the feedback process $f_t(X_{it} \mid \mathcal I_i^{t-1})$, which characterizes the indirect adjustment effect.
The main contribution of our framework is that it enables the decomposition of dynamic treatment effects into direct and indirect effects, which allows counterfactual analysis incorporating both channels. The identified objects $\theta$, $H(\lambda_i \mid \mathcal I_i^0)$, and $\{f_t(X_{it} \mid \mathcal I_i^{t-1})\}_{t=1}^T$ characterize both the heterogeneous treatment effects and the dynamic adjustment of covariates following treatment.
The factorization in the proof of Theorem (ref) implies that in the first step, the estimation of $\theta$ and $H(\lambda_i \mid \mathcal I_i^0)$ can proceed as in botosaru2025time. In a second step, the homogeneous feedback process for covariates can be estimated by modeling the transition densities $f_t(X_{it}\mid \mathcal I_i^{t-1})$, using, e.g., parametric Markov models or sieve methods. Under our assumptions, the parameters governing the direct channel $(\theta,H)$ and the feedback mechanism enter the likelihood in separable blocks, so estimation of $(\theta,H)$ need not rely on a particular parametric specification of the feedback model, provided the feedback factor $g$ is estimated consistently on the relevant support. The feedback model can therefore be selected to balance flexibility and parsimony depending on the intended decomposition and counterfactual exercises.
Given an alternative treatment timing $t_{0i}^\ast$ and/or alternative initial conditions $\left\{Y_{i0}^\ast,X_{i0}^\ast\right\}$, one can construct joint counterfactual paths $\left\{Y_i^{T,\ast},X_i^{T,\ast}\right\}$ as described in Algorithm (ref). Starting from the counterfactual initial conditions $\mathcal I_i^{0,\ast}$, latent heterogeneity $\lambda_i$ is drawn from $H(\lambda_i\mid\mathcal I_i^{0,\ast})$ and counterfactual treatment effects are generated according to (ref). The system is then iterated forward in calendar time, drawing covariates from the estimated feedback process and constructing outcomes using (ref), which combines the direct effect through $\{\delta_{ij}^\ast\}$ and the indirect effect through $X_{it}^\ast$ and $\beta$.
The resulting counterfactual paths decompose dynamic event study responses into a direct structural component, driven by latent heterogeneity and treatment effect dynamics, and an indirect component operating through sequentially exogenous covariate adjustments. This decomposition is empirically relevant because it distinguishes between treatment effects that arise from heterogeneous structural responses versus those that operate through equilibrium adjustments. For example, in a minimum wage study, the direct effect captures how firm productivity responds to the wage change, while the indirect effect captures how firm adjustments in hours or employment composition feed back into productivity. The framework provides an approach to quantifying each effect, enabling researchers to assess their relative importance in driving event study dynamics and facilitating counterfactual analysis that propagates both effects.
First, the homogeneous feedback framework is not tied to the linear model in (ref). In principle, one can replace the outcome equation by a nonlinear panel model $Y_{it} \sim f_\theta(\cdot \mid \mathcal I_i^{t-1}, X_{it}, \lambda_i),$ where $\theta$ is a finite-dimensional parameter and $\lambda_i$ collects unobserved unit-specific heterogeneity. Examples include dynamic logit or probit models, Poisson or negative binomial count models, and Tobit models for censored outcomes. Under Assumptions (ref)--(ref), once we factor out the feedback term, the conditional distribution of $Y_i^T\mid X_i^T, \mathcal I_i^0$ depends on $(\theta, H)$ only through an average over $\lambda_i$. Identification of $(\theta,H)$ can then proceed by applying the relevant nonlinear identification results to this conditional distribution, while the feedback process $g(X_i^T \mid Y_i^T, \mathcal I_i^0)$ remains a separately identified component.
The homogeneous feedback assumption in (ref) can be relaxed to allow for observed group-specific covariate adjustment rules. That is, let $G_i$ denote an observed group indicator, such as industry, region, or demographic category, and assume that for each $t$, $f_t(X_{it} \mid \mathcal I_i^{t-1}, G_i, \lambda_i) = f_t(X_{it} \mid \mathcal I_i^{t-1}, G_i).$ That is, conditional on observable history and group membership, the covariate adjustment process does not depend on unobserved heterogeneity, but may vary across groups. Under this modification, the likelihood factorization applies group by group. In particular, for each $g$, $\theta$ and $H(\lambda_i \mid \mathcal I_i^0, G_i = g)$ are identified as before.