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Heterogeneous Responses to Continuous Treatments: A Cluster-Based Causal Framework

Augusto Cerqua, Roberta Di Stefano, Raffaele Mattera

Abstract

When treatments are non-randomly assigned, continuous, and yield heterogeneous effects at the same intensity, causal identification becomes particularly challenging. In such contexts, existing approaches often fail to provide policy-relevant estimates of the relationship between treatment intensity and outcomes, especially in the presence of limited common support. To fill this gap, we introduce the Clustered Dose-Response Function (Cl-DRF), a novel estimator designed to uncover the continuous causal relationship between treatment intensity and the dependent variable across distinct subgroups. Our approach leverages both theoretical and data-driven sources of heterogeneity, relying on relaxed versions of the conditional independence and positivity assumptions that are plausible across various observational settings. We apply the Cl-DRF estimator to estimate subgroup-specific dose-response relationships between European Cohesion Funds and economic growth. In contrast to much of the literature, higher funding increases growth in more developed regions without diminishing returns, while limited absorptive capacity prevents other regions from fully benefiting.

Heterogeneous Responses to Continuous Treatments: A Cluster-Based Causal Framework

Abstract

When treatments are non-randomly assigned, continuous, and yield heterogeneous effects at the same intensity, causal identification becomes particularly challenging. In such contexts, existing approaches often fail to provide policy-relevant estimates of the relationship between treatment intensity and outcomes, especially in the presence of limited common support. To fill this gap, we introduce the Clustered Dose-Response Function (Cl-DRF), a novel estimator designed to uncover the continuous causal relationship between treatment intensity and the dependent variable across distinct subgroups. Our approach leverages both theoretical and data-driven sources of heterogeneity, relying on relaxed versions of the conditional independence and positivity assumptions that are plausible across various observational settings. We apply the Cl-DRF estimator to estimate subgroup-specific dose-response relationships between European Cohesion Funds and economic growth. In contrast to much of the literature, higher funding increases growth in more developed regions without diminishing returns, while limited absorptive capacity prevents other regions from fully benefiting.
Paper Structure (28 sections, 3 theorems, 39 equations, 20 figures, 2 tables)

This paper contains 28 sections, 3 theorems, 39 equations, 20 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A motivating example
  3. Related works
  4. Approaches for estimating ADRF
  5. Approaches for estimating heterogeneous treatment effects
  6. The Clustered Dose-Response-Function Estimator
  7. The causal framework for continuous treatments
  8. The Clustered Dose-Response Function approach
  9. The Cl-DRF estimation algorithm
  10. Choosing the number of clusters
  11. Revisiting the motivating example
  12. Simulation results
  13. Estimating the impact of the EU Cohesion Policy through the Cl-DRF estimator
  14. Background
  15. Data
  16. ...and 13 more sections

Key Result

Lemma 1

Fix $c\in\{1,\dots,C\}$ and $t\in\mathcal{T}_c$. Under Assumption A1 and Assumption B1--B3, the cluster-specific dose--response function is identified and admits the representation

Figures (20)

  • Figure 1: Relationship between treatment and outcome for all units and at the cluster level (left-hand side) and histogram of distribution of treatment within clusters (right-hand side). Simulated data with $n = 800$ units divided into $C = 4$ equally sized clusters are used. Treatment is assigned based on cluster-specific functions of two pre-treatment covariates, and outcomes are generated using distinct treatment–outcome relationships within each cluster.
  • Figure 2: ADRF without cluster structure by using the hirano2004propensity approach and following huling2023independence. True average relationship (dashed black line) represents the average of the four simulated relationships as in equation \ref{['simulexample']}; HI2004 (red line) is the estimated ADRF by using the hirano2004propensity approach; HGC 2024 (purple line) is the ADRF estimated by using the huling2023independence approach.
  • Figure 3: DRFs estimated using the hirano2004propensity approach within each cluster, under the assumption that the cluster structure is known, compared to the true simulated relationships. The colored solid lines represent the four simulated relationships for all values of treatment as in equation \ref{['simulexample']}; the dashed black lines are the estimated DRFs by using the hirano2004propensity approach.
  • Figure 4: Average dose-response function. Example based on 40 simulated unit-dose responses.
  • Figure 5: Selection of the number of clusters. IC values \ref{['eq:bic']} for different values of $C$, in the set $C = \{1, 2,3,4,5,6,7\}$. We choose $C$ with the Elbow criterion. We detect an elbow for $C=4$.
  • ...and 15 more figures

Theorems & Definitions (10)

  • Definition 1: Cluster-specific DRF
  • Remark 1
  • Lemma 1: Identification of the cluster-specific dose--response function
  • Definition 2: Cluster-specific generalized propensity score
  • Lemma 2: GPS-based representation within clusters
  • Proposition 1: Closed-form updates in the Cl-DRF algorithm
  • Remark 2: Absence of the GPS update step
  • proof
  • proof : Proof of Lemma \ref{['cor:gps_representation']}
  • proof : Proof of Proposition \ref{['prop:updates']}