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Marginal treatment effects in the absence of instrumental variables

Zhewen Pan, Zhengxin Wang, Junsen Zhang, Yahong Zhou

Abstract

We propose a method for defining, identifying, and estimating the marginal treatment effect (MTE) without imposing the instrumental variable (IV) assumptions of independence, exclusion, and separability (or monotonicity). Under a new definition of the MTE based on reduced-form treatment error that is statistically independent of the covariates, we find that the relationship between the MTE and standard treatment parameters holds in the absence of IVs. We provide a set of sufficient conditions ensuring the identification of the defined MTE in an environment of essential heterogeneity. The key conditions include a linear restriction on potential outcome regression functions, a nonlinear restriction on the propensity score, and a conditional mean independence restriction that will lead to additive separability. We prove this identification using the notion of semiparametric identification based on functional form. And we provide an empirical application for the Head Start program to illustrate the usefulness of the proposed method in analyzing heterogenous causal effects when IVs are elusive.

Marginal treatment effects in the absence of instrumental variables

Abstract

We propose a method for defining, identifying, and estimating the marginal treatment effect (MTE) without imposing the instrumental variable (IV) assumptions of independence, exclusion, and separability (or monotonicity). Under a new definition of the MTE based on reduced-form treatment error that is statistically independent of the covariates, we find that the relationship between the MTE and standard treatment parameters holds in the absence of IVs. We provide a set of sufficient conditions ensuring the identification of the defined MTE in an environment of essential heterogeneity. The key conditions include a linear restriction on potential outcome regression functions, a nonlinear restriction on the propensity score, and a conditional mean independence restriction that will lead to additive separability. We prove this identification using the notion of semiparametric identification based on functional form. And we provide an empirical application for the Head Start program to illustrate the usefulness of the proposed method in analyzing heterogenous causal effects when IVs are elusive.
Paper Structure (16 sections, 14 theorems, 235 equations, 12 figures, 3 tables)

This paper contains 16 sections, 14 theorems, 235 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Identification
  4. Estimation
  5. Empirical application
  6. Conclusion
  7. Additional results of the empirical application
  8. Proof of Theorem \ref{['theorem:main']}
  9. Further discussion on the nonlinearity assumption
  10. Limited valued outcomes
  11. Sufficient conditions and proofs for Theorem \ref{['theorem:AN1']}
  12. Sufficient conditions
  13. Proof of Theorem \ref{['theorem:AN1']}
  14. Auxiliary lemmas
  15. Implementation guidance
  16. ...and 1 more sections

Key Result

Theorem 1

If Assumptions CMI, L, NL, and S hold, then $\beta _{d}$ and $g_{d}\left( p\right)$ at all $p$ in the support of the propensity score $P$ are identified for $d=0,1$.

Figures (12)

  • Figure 1: An informal test for Assumption NL1
  • Figure 2: Common support
  • Figure 3: MTE curves under normal specification
  • Figure 4: MTE curves under semiparametric specification
  • Figure A.1: Semiparametric estimates of marginal structural functions
  • ...and 7 more figures

Theorems & Definitions (31)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Theorem 1
  • Theorem 2
  • Theorem C.1
  • proof
  • ...and 21 more