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

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.

💡 Research Summary

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The paper tackles a pervasive problem in observational social‑science research: treatments are often continuous, non‑randomly assigned, and generate heterogeneous effects that differ across sub‑populations even at the same intensity. Traditional causal methods for continuous treatments focus on estimating a single average dose‑response function (ADRF) under the assumptions of weak unconfoundedness and positivity. In many real‑world settings these assumptions are violated because the distribution of treatment intensity varies dramatically with covariates, leading to limited common support and biased ADRF estimates.

To address this, the authors introduce the Clustered Dose‑Response Function (Cl‑DRF) framework. Instead of a population‑wide ADRF, Cl‑DRF estimates a separate dose‑response curve for each latent subgroup (cluster). The key methodological innovations are:

1. Relaxed Identification Assumptions – Weak unconfoundedness and positivity are required only within each cluster, not globally. By restricting attention to more homogeneous sub‑populations, the plausibility of these assumptions improves dramatically.

2. Joint Estimation of Clusters and DRFs – The number of clusters K is treated as an unknown parameter. An EM‑like iterative algorithm alternates between (a) updating cluster memberships, (b) estimating cluster‑specific generalized propensity scores (GPS), and (c) fitting cluster‑specific outcome regressions. This mirrors classic cluster‑wise regression but incorporates the continuous‑treatment setting through GPS modeling.

3. Parametric GPS Modeling – For computational tractability the authors adopt a parametric (linear or polynomial) model for the GPS within each cluster. Although less flexible than fully non‑parametric alternatives, this choice enables simultaneous estimation of the clustering structure and the DRFs.

Theoretical results show that, under the within‑cluster positivity condition, the Cl‑DRF estimator is consistent and √N‑convergent. The authors also derive the bias‑variance trade‑off associated with misspecifying K and propose information‑criterion (AIC/BIC) based selection as a practical solution.

A series of Monte‑Carlo simulations illustrates the advantages of Cl‑DRF. Four designs are examined: (i) clusters with distinct treatment ranges, (ii) nonlinear dose‑response shapes, (iii) heterogeneous slopes, and (iv) severe common‑support violations. In all cases, conventional GPS‑based or recent non‑parametric weighted estimators produce biased ADRFs, sometimes failing to identify any causal effect. By contrast, Cl‑DRF recovers the true cluster‑specific curves with negligible bias, even when some clusters have virtually no overlap in treatment intensity with others.

The empirical application estimates the impact of European Union Cohesion Funds on regional economic growth. Using a panel of EU NUTS‑2 regions (2000‑2020) with detailed covariates (initial GDP, industry composition, human capital, institutional quality, etc.), the authors let the data determine four clusters:

• Cluster 1 – High‑income, high‑absorption regions – The estimated DRF is approximately linear and positive across the observed funding range; marginal returns do not diminish, suggesting that additional funds continue to boost growth.

• Cluster 2 – Medium‑income, diversified economies – Positive returns at low to moderate funding levels, but a modest flattening as intensity rises.

• Cluster 3 – Low‑income, industry‑dependent regions – Gains are present but taper off more quickly; the DRF shows a clear concave shape.

• Cluster 4 – Low‑absorption capacity regions – The DRF is flat or slightly negative; beyond a small threshold, extra funding yields no measurable growth benefit.

These findings overturn the naïve interpretation that a single ADRF would suggest diminishing returns after a modest funding level. Instead, the heterogeneity uncovered by Cl‑DRF reveals that policy‑relevant marginal effects differ dramatically across regions. The authors argue that optimal allocation should (i) concentrate additional resources in high‑absorption clusters where returns are sustained, and (ii) complement funding in low‑absorption clusters with capacity‑building measures (e.g., administrative reforms, project management training) before further financial injections.

The paper acknowledges limitations: the reliance on parametric GPS may miss complex treatment assignment mechanisms, and the EM‑style algorithm can be computationally intensive when K is large. Future work is suggested on integrating non‑parametric or machine‑learning based propensity models and on scalable Bayesian clustering approaches.

Overall, the study makes a substantive methodological contribution by providing a feasible, theoretically grounded tool for estimating heterogeneous dose‑response relationships in settings with limited common support, and it demonstrates the practical relevance of this tool through a policy‑critical analysis of EU cohesion funding.