Estimation of the Local Conditional Tail Average Treatment Effect

Abstract

The conditional tail average treatment effect (CTATE) is defined as a difference between the conditional tail expectations of potential outcomes, which can capture heterogeneity and deliver aggregated local information on treatment effects over different quantile levels and is closely related to the notion of second-order stochastic dominance and the Lorenz curve. These properties render it a valuable tool for policy evaluation. In this article, we study estimation of the CTATE locally for a group of compliers (local CTATE or LCTATE) under the two-sided noncompliance framework. We consider a semiparametric treatment effect framework under endogeneity for the LCTATE estimation using a newly introduced class of consistent loss functions jointly for the CTE and quantile. We establish the asymptotic theory of our proposed LCTATE estimator and provide an efficient algorithm for its implementation. We then apply the method to evaluate the effects of participating in programs under the Job Training Partnership Act in the United States.

Keywords:

• Causal inference
• Conditional tail expectation
• Endogeneity
• Semiparametric estimation
• Treatment effects

Acknowledgments

We are grateful to the Editor, Ivan Canay, the Associate Editor and two anonymous reviewers for valuable comments and suggestions on previous versions of the article. We thank seminar participants in 2019 macroeconometric modeling workshop (Academia Sinica), 2020 Annual Meeting of Taiwan Econometric Society, 2021 Delhi Winter School-the Econometric Society, The 5th International Conference on Econometrics and Statistics (EcoSta 2022), 2022 Asian Meeting of the Econometric Society, CRETA Seminar, National Chengchi University and National Taiwan University for helpful comments.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

1 Let L(x, y) denote a loss function for obtaining a statistical functional of a random variable. In our case, $L=F_{\tau }^{\mathit{\text{sp}}}$ and $x=(q,e)$. Let $\mathcal{F}$ denote a class of distribution functions and F be an element in $\mathcal{F}$. Let $\lambda :\mathcal{F}\mapsto \mathbb{S}$ denote a statistical functional which maps $F\in \mathcal{F}$ to a set $\mathbb{S}$, which in our case amounts to the set ${\mathbb{R}}^2$. L(x, y) is consistent for the statistical functional $\lambda (F)$ if $E_F\left[L(\lambda (F),Y)\right]\le E_F\left[L(x,Y)\right]$ for all $F\in \mathcal{F}$ and all $x\in \mathbb{S}$ and a random variable Y following the distribution F. If a loss function is consistent and $E_F\left[L(\lambda (F),Y)\right]=E_F\left[L(x,Y)\right]$ implies $x=\lambda (F)$, the loss function is said to be strictly consistent.

Additional information

Funding

Le-Yu Chen acknowledges financial support from Academia Sinica. Yu-Min Yen is supported by National Science and Technology Council, Taiwan, Grant ID: MOST 110-2410-H-004-172-MY2.