Identification and Auto-Debiased Machine Learning for Outcome-Conditioned Average Structural Derivatives

Abstract

This study proposes a new class of heterogeneous causal quantities, referred to as outcome-conditioned average structural derivatives (OASDs), in a general nonseparable model. An OASD is the average partial effect of a marginal change in a continuous treatment on individuals located on different parts of an outcome distribution, irrespective of individuals’ characteristics. We show that OASDs extend the unconditional quantile partial effects (UQPE) proposed by Firpo, Fortin, and Lemieux to that conditional on a set of outcome values by effectively integrating the UQPE. Exploiting such relationship brings about two merits. First, unlike UQPE that is generally not $\sqrt{n}$-estimable, OASD is shown to be $\sqrt{n}$-estimable. Second, our estimator achieves semiparametric efficiency bound which is a new result in the literature. We propose a novel, automatic, debiased machine-learning estimator for an OASD, and present asymptotic statistical guarantees for it. The estimator is proven to be $\sqrt{n}$-consistent, asymptotically normal, and semi-parametrically efficient. We also prove the validity of the bootstrap procedure for uniform inference for the OASD process. We apply the method to Imbens, Rubin, and Sacerdote’s lottery data.

KEYWORDS:

• Debiased machine learning
• Doubly/locally robust score
• Heterogeneity
• Local average structural derivative
• Semiparametric efficiency bound
• Unconditional quantile partial effect

Supplementary Materials

The online supplementary material comprises an appendix, the code and 18 figures. The appendix contains technical proofs, simulation results, and empirical studies. The code is applied to both simulation and empirical studies. The 18 accompanying figures depict the results of the simulations.

Acknowledgments

The three authors contribute to the article equally. We are grateful to the Editor Ivan Canay, the Associate Editor, and two anonymous referees for their very useful and insightful comments. We also thank Chunrong Ai and Zheng Zhang for kindly sharing their dataset. All of the remaining errors are ours.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 For example, we have ${\eta }_1=1$ for $\mathcal{l}=1$; ${\eta }_1=4/3$ and ${\eta }_2=-1/6$ for $\mathcal{l}=2$; ${\eta }_1=3/2,{\eta }_2=-3/10$ and ${\eta }_3=1/30$ for $\mathcal{l}=3$.

2 We can also use post-lasso estimator to replace γ.

3 Our OASDs are defined on $({\tau }_1,{\tau }_2)\in (0,1)\times (0,1)$ with ${\tau }_1<{\tau }_2$. Let $c_0,c_1>0$ and $c_2<1$ be three constants. The metric space $\mathcal{U}$ can be defined as $\mathcal{U}=\left\{({\tau }_1,{\tau }_2):c_1\le {\tau }_1+c_0\le {\tau }_2\le c_2\right\}$, which is a bounded upper triangular.

Additional information

Funding

This research is supported by the National Natural Science Foundation of China (Grant Nos. 71873080, 72273076, 72103126, 72333002), the National Social Science Fund of China Major Project (Grant No. 23&ZD074) and the Fundamental Research Funds for the Central Universities (Grant No. 2023110077).