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Identification in Multiple Treatment Models under Discrete Variation

Vishal Kamat, Samuel Norris, Matthew Pecenco

TL;DR

The paper tackles causal inference with multiple treatments and discrete instruments by embedding selection into a multidimensional threshold-crossing model and expressing effects as marginal treatment response (MTR) functionals. It develops a two-step bound procedure: first, fix the selection primitives (up to a finite-dimensional parameter) and bound the MTRs via linear/convex programs; second, union these bounds over admissible selection-primitives to form the identified set, accommodating both parametric and nonparametric MTRs. The authors provide estimation and inference procedures, including a two-step algorithm and a recycling-based test, and apply the method to re-evaluate the Head Start program in KW (Kline and Walters, 2016), assessing robustness to assumptions about selection and rival options. The approach yields informative bounds on policy-relevant effects and welfare measures (e.g., PRTE and MVPF) under weaker identification assumptions, highlighting when empirical conclusions are robust or sensitive to model specification.

Abstract

We develop a marginal treatment effect based method to learn about causal effects in multiple treatment models with discrete instruments. We allow selection into treatment to be governed by a general class of threshold crossing models that permit multidimensional unobserved heterogeneity. An inherent complication is that the primitives characterizing the selection model are not generally point-identified. Allowing these primitives to be point-identified up to a finite-dimensional parameter, we show how a two-step computational program can be used to obtain sharp bounds for a number of treatment effect parameters when the marginal treatment response functions are allowed to satisfy only nonparametric shape restrictions or are additionally parameterized. We demonstrate the benefits of our method by revisiting Kline and Walters' (2016) empirical analysis of the Head Start program. Our approach relaxes their point-identifying assumptions on the selection model and marginal treatment response functions, allowing us to assess the robustness of their conclusions.

Identification in Multiple Treatment Models under Discrete Variation

TL;DR

The paper tackles causal inference with multiple treatments and discrete instruments by embedding selection into a multidimensional threshold-crossing model and expressing effects as marginal treatment response (MTR) functionals. It develops a two-step bound procedure: first, fix the selection primitives (up to a finite-dimensional parameter) and bound the MTRs via linear/convex programs; second, union these bounds over admissible selection-primitives to form the identified set, accommodating both parametric and nonparametric MTRs. The authors provide estimation and inference procedures, including a two-step algorithm and a recycling-based test, and apply the method to re-evaluate the Head Start program in KW (Kline and Walters, 2016), assessing robustness to assumptions about selection and rival options. The approach yields informative bounds on policy-relevant effects and welfare measures (e.g., PRTE and MVPF) under weaker identification assumptions, highlighting when empirical conclusions are robust or sensitive to model specification.

Abstract

We develop a marginal treatment effect based method to learn about causal effects in multiple treatment models with discrete instruments. We allow selection into treatment to be governed by a general class of threshold crossing models that permit multidimensional unobserved heterogeneity. An inherent complication is that the primitives characterizing the selection model are not generally point-identified. Allowing these primitives to be point-identified up to a finite-dimensional parameter, we show how a two-step computational program can be used to obtain sharp bounds for a number of treatment effect parameters when the marginal treatment response functions are allowed to satisfy only nonparametric shape restrictions or are additionally parameterized. We demonstrate the benefits of our method by revisiting Kline and Walters' (2016) empirical analysis of the Head Start program. Our approach relaxes their point-identifying assumptions on the selection model and marginal treatment response functions, allowing us to assess the robustness of their conclusions.
Paper Structure (35 sections, 8 theorems, 107 equations, 2 figures, 5 tables)

This paper contains 35 sections, 8 theorems, 107 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Setup
  3. Observed and Potential Variables
  4. Selection Model
  5. Parameters of Interest
  6. Identified Set
  7. Identification
  8. Two-Step Approach
  9. Identified Set under Parametric MTRs
  10. Identified Set under Nonparametric MTRs
  11. Parametrization of Selection Model Primitives
  12. Estimation and Statistical Inference
  13. Estimation
  14. Statistical Inference
  15. Estimating the Returns to Head Start
  16. ...and 20 more sections

Key Result

Proposition 1

For $h \in \mathbf{H}$, let $\mathbf{M}(h)$ satisfy Assumption ass:PM. If $\mathbf{M}^*(h)$ is empty then so is $\theta(\mathbf{M}^*(h),h)$, and if it is not empty, then $\text{closure}(\theta(\mathbf{M}^*(h),h)) = [\theta_L(h),\theta_U(h)]$, where

Figures (2)

  • Figure 1: Partition satisfying Definition \ref{['def:P']} in Example \ref{['ex:double']} with $\mathcal{Z} = \{z,z'\}$ and $\mathcal{X} = \{x\}$, and where the parameter of interest is the $\text{ATE}$ between two treatments. In Panel (a), note that $\mathcal{U}^{\text{sm}}_{0,z|x}(g) = \mathcal{U}'_3 \cup \mathcal{U}'_4$, $\mathcal{U}^{\text{sm}}_{1,z|x}(g) = \mathcal{U}'_1 \cup \mathcal{U}'_2$, $\mathcal{U}^{\text{sm}}_{0,z'|x}(g) = \mathcal{U}'_2 \cup \mathcal{U}'_4$, and $\mathcal{U}^{\text{sm}}_{1,z'|x}(g) = \mathcal{U}'_1 \cup \mathcal{U}'_3$.
  • Figure 2: $p$-values for null hypothesis that $\text{MVPF}_{\text{adj}}$ is at most 1 for different values of $\kappa_{\text{ben}}$

Theorems & Definitions (17)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Proposition 1
  • Definition P
  • Proposition 2
  • Example 5: continues=ex:arum
  • Proposition 3
  • Example 6: continues=ex:sequential
  • ...and 7 more