Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The CATT SATT on the MATT: semiparametric inference for sample treatment effects on the treated

Andrew Yiu

Abstract

We study variants of the average treatment effect on the treated with population parameters replaced by their sample counterparts. For each estimand, we derive the limiting distribution with respect to a semiparametric efficient estimator of the population effect and provide guidance on variance estimation. Included in our analysis is the well-known sample average treatment effect on the treated, for which we obtain some unexpected results. Unlike the ordinary sample average treatment effect, we find that the asymptotic variance for the sample average treatment effect on the treated is point-identified and consistently estimable, but it potentially exceeds that of the population estimand. To address this shortcoming, we propose a modification that yields a new estimand, the mixed average treatment effect on the treated, which is always estimated more precisely than both the population and sample effects. We also introduce a second new estimand that arises from an alternative interpretation of the treatment effect on the treated with which all individuals are weighted by the propensity score.

The CATT SATT on the MATT: semiparametric inference for sample treatment effects on the treated

Abstract

We study variants of the average treatment effect on the treated with population parameters replaced by their sample counterparts. For each estimand, we derive the limiting distribution with respect to a semiparametric efficient estimator of the population effect and provide guidance on variance estimation. Included in our analysis is the well-known sample average treatment effect on the treated, for which we obtain some unexpected results. Unlike the ordinary sample average treatment effect, we find that the asymptotic variance for the sample average treatment effect on the treated is point-identified and consistently estimable, but it potentially exceeds that of the population estimand. To address this shortcoming, we propose a modification that yields a new estimand, the mixed average treatment effect on the treated, which is always estimated more precisely than both the population and sample effects. We also introduce a second new estimand that arises from an alternative interpretation of the treatment effect on the treated with which all individuals are weighted by the propensity score.
Paper Structure (20 sections, 14 theorems, 101 equations, 1 figure, 1 table)

This paper contains 20 sections, 14 theorems, 101 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Set-up and point estimation
  3. Estimands and inference
  4. The literal and figurative interpretations
  5. Figurative estimands
  6. Literal estimands
  7. Proofs of results in the main text
  8. Proof of Proposition 1
  9. Proof of Proposition 2
  10. Proof of Proposition 3
  11. Proof of Theorem 1
  12. Proof of Proposition 4
  13. Proof of Proposition 5
  14. Proof of Theorem 2
  15. Proof of Proposition 6
  16. ...and 5 more sections

Key Result

Proposition 1

Under Assumption ass::patt_eff, the estimator $\hat{\psi}$ in (eqn::eff_est) admits the asymptotically linear expansion $n^{1/2}(\hat{\psi} - \psi_{patt}) = n^{1/2}\mathbb{P}_{n}(\dot{\psi}) + o_{\mathbb{P}}(1)$, and $\Vert\dot{\psi}\Vert_{\mathbb{P}} < \infty$.

Figures (1)

  • Figure 1: Hasse diagram describing the partial ordering on the estimands; figurative estimands; literal estimands; $\psi_{1} \leftarrow \psi_{2}$ means that $\psi_{1} \preceq \psi_{2}$.

Theorems & Definitions (28)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Remark 2
  • Definition 1
  • Proposition 3
  • Definition 2
  • Theorem 1
  • Proposition 4
  • Proposition 5
  • ...and 18 more