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Identification and Estimation of Conditional Average Partial Causal Effects via Instrumental Variable

Yuta Kawakami, Manabu Kuroki, Jin Tian

TL;DR

This paper tackles heterogeneity in causal effects for a continuous treatment under an instrumental-variable model by introducing conditional average partial causal effects (CAPCE), defined as $\mathbb{E}[\partial_x Y_x|{\boldsymbol w}]$. It proves CAPCE is identifiable under a weaker separability assumption than required for $\mathbb{E}[Y_x|{\boldsymbol w}]$, enabling estimation in broader IV models. The authors develop three estimators—sieve CAPCE, parametric CAPCE, and RKHS CAPCE—with consistency and convergence guarantees, and demonstrate superior performance to traditional IV methods on both synthetic and real data when standard separability fails. This work provides a practical toolkit for uncovering how treatment effects vary across covariates in contexts where experimental control is limited, with clear implications for policy evaluation and causal inference in economics and related fields.

Abstract

There has been considerable recent interest in estimating heterogeneous causal effects. In this paper, we study conditional average partial causal effects (CAPCE) to reveal the heterogeneity of causal effects with continuous treatment. We provide conditions for identifying CAPCE in an instrumental variable setting. Notably, CAPCE is identifiable under a weaker assumption than required by a commonly used measure for estimating heterogeneous causal effects of continuous treatment. We develop three families of CAPCE estimators: sieve, parametric, and reproducing kernel Hilbert space (RKHS)-based, and analyze their statistical properties. We illustrate the proposed CAPCE estimators on synthetic and real-world data.

Identification and Estimation of Conditional Average Partial Causal Effects via Instrumental Variable

TL;DR

This paper tackles heterogeneity in causal effects for a continuous treatment under an instrumental-variable model by introducing conditional average partial causal effects (CAPCE), defined as . It proves CAPCE is identifiable under a weaker separability assumption than required for , enabling estimation in broader IV models. The authors develop three estimators—sieve CAPCE, parametric CAPCE, and RKHS CAPCE—with consistency and convergence guarantees, and demonstrate superior performance to traditional IV methods on both synthetic and real data when standard separability fails. This work provides a practical toolkit for uncovering how treatment effects vary across covariates in contexts where experimental control is limited, with clear implications for policy evaluation and causal inference in economics and related fields.

Abstract

There has been considerable recent interest in estimating heterogeneous causal effects. In this paper, we study conditional average partial causal effects (CAPCE) to reveal the heterogeneity of causal effects with continuous treatment. We provide conditions for identifying CAPCE in an instrumental variable setting. Notably, CAPCE is identifiable under a weaker assumption than required by a commonly used measure for estimating heterogeneous causal effects of continuous treatment. We develop three families of CAPCE estimators: sieve, parametric, and reproducing kernel Hilbert space (RKHS)-based, and analyze their statistical properties. We illustrate the proposed CAPCE estimators on synthetic and real-world data.
Paper Structure (26 sections, 12 theorems, 90 equations, 17 figures, 15 tables)

This paper contains 26 sections, 12 theorems, 90 equations, 17 figures, 15 tables.

Table of Contents

  1. Introduction
  2. Notation and Background
  3. Identification of CAPCE
  4. Estimation of CAPCE
  5. Sieve CAPCE Estimator
  6. Parametric CAPCE estimator
  7. RKHS CAPCE estimator
  8. Experiments
  9. Application in a Real-World Dataset
  10. Conclusion
  11. Proofs of Theorem 3.1 and RKHS CAPCE estimator
  12. Proof of Theorem \ref{['TEO2']}
  13. Identification theorem under IV model in Fig \ref{['DAG1d']}
  14. Derivation of RKHS CAPCE estimator
  15. Consistency of Sieve CAPCE Estimator
  16. ...and 11 more sections

Key Result

Theorem 3.1

Under SCM ${\cal M}_{IV}$ and Assumptions AS1 and AS2, CAPCE $\mathbb{E}[\partial_x Y_{x}|{\boldsymbol w}]$ is identifiable from distributions $\mathbb{P}(X, {\boldsymbol W}|Z)$ and $\mathbb{P}( Y |Z)$ via the integral equation: where $\mu(z)=\mathbb{E}[Y|Z=z_0]-\mathbb{E}[Y|Z=z], k(z,x,{\boldsymbol w})=\mathfrak{p}(X\leq x,{\boldsymbol W}={\boldsymbol w}|Z=z)-\mathfrak{p}(X\leq x,{\boldsymbol W}

Figures (17)

  • Figure 1: A causal graph representing the IV model.
  • Figure 3: A causal graph representing the IV setting with covariates when there is an edge ${\boldsymbol W} \rightarrow Z$.
  • Figure : (a) Parametric setting (A) (Means, $95 \%$ CI)
  • Figure : (a) True CAPCE
  • Figure : (a) True CAPCE
  • ...and 12 more figures

Theorems & Definitions (25)

  • Definition 1: CAPCE
  • Theorem 3.1: Identification of CAPCE
  • Theorem 4.1: Consistency
  • Theorem 4.2: Rate of Convergence
  • Theorem 4.3: Consistency
  • Theorem 4.4: Rate of Convergence
  • Theorem 4.5: Consistency
  • proof
  • proof
  • proof
  • ...and 15 more