Nonparametric identification and estimation of heterogeneous causal effects under conditional independence

Abstract

In this article, I propose a nonparametric strategy to identify the distribution of heterogeneous causal effects. A set of identification restrictions proposed in this article differs from existing approaches in three ways. First, it extends the random coefficient model by allowing potentially nonlinear interactions between distributional parameters and the set of covariates. Second, the causal effect distributions identified in this article give an alternative to those under the rank invariance assumption. Third, identified distribution lies within the sharp bound of distributions of the treatment effect. I develop a consistent nonparametric estimator exploiting the identifying restriction by extending the conventional statistical deconvolution method to the Rubin causal framework. Results from a Monte Carlo experiment and an application to wage loss of displaced workers suggest that the method yields robust estimates under various scenarios.

Keywords:

• Conditional independence
• heterogeneous causal effects
• nonparametric estimation
• statistical deconvolution

JEL CLASSIFICATION:

• C14
• C21
• C51

Acknowledgments

I would like to thank Professor Esfandiar Maasoumi, the editor, an anonymous associate editor, and two anonymous referees for their invaluable comments. The views expressed in this article are the views of the author alone and do not necessarily reflect the views of the author’s employer or any other entity with which the author may be associated.

Notes

1 For example, Heckman and Robb (1985), Heckman et al. (1997), and Heckman et al. (2006) discuss how the understanding of heterogeneity in causal effects can alter policy implications in the context of empirical policy evaluation with observational data.

2 For a comprehensive review of the partial identification approach to distributional treatment effects, see Abbring and Heckman (2007).

3 It is important to note that the propensity to selection is bounded away from zero so that no observation is strictly excluded from either group. If the propensity score tends to 0 or 1, the standard inferential theory with $\sqrt{n}$-consistency and asymptotic normality fails even for the average treatment effect (D’Amour et al., 2021; Khan and Nekipelov, 2013; Khan and Tamer, 2010; Ma and Wang, 2020; Rothe, 2017). Such a case is ruled out under the identification scheme proposed in this article.

4 Zimmer and Trivedi (2006), for example, propose a method based on parameterization of the degree of dependence between potential outcomes in terms of their rank correlation. While it provides an alternative point-identification scheme to the rank invariance, a parametric approach often has difficulties in justifying any particular design to model the dependence between potential outcomes unless it is based on a structural assumptions.

5 The case may be interpreted as a linear shift model where the distribution of ε governs the degree of dispersion in Δ while X determines the conditional mean of Δ.

6 In a formal notation, the difference between Condition 3 and rank invariance is as follows. Suppose that $(Y_1,Y_0)$ and $(Y_1^{\prime },Y_0^{\prime })$ are two random draws from the joint distribution of potential outcomes conditional on the same X. With Condition 3, there exists positive probability of “reversed rank” in the sense that $\text{Prob}((Y_1^{\prime }-Y_1)(Y_0^{\prime }-Y_0)<0)>0.$ On the other hand, the rank invariance does not allow such possibility as $\text{Prob}((Y_1^{\prime }-Y_1)(Y_0^{\prime }-Y_0)>0)=1.$

7 Most of the widely-used parametric distributions fall into the category of either super-smooth or ordinary-smooth distributions. For example, normal, mixture normal, and Cauchy distributions have super-smooth characteristic functions while gamma and exponential distributions have ordinary-smooth characteristic functions.

8 Some studies on nonparametric estimation for panel regression models discuss technical conditions to achieve uniform convergence of the deconvolution estimator for all conditional distributions. See Horowitz and Markatou (1996), Neumann (2007), Bonhomme and Robin (2010), Evdokimov (2010), and Canay (2011).

9 Trimming is a practical approach to avoid irregularities in inverse propensity score weighting estimators discussed in Khan and Tamer (2010). D’Amour et al. (2021) argue that it is useful in particular when the dimension of covariate is high though the implication of the resulting treatment effect estimator is limited to bounded support.

10 For example, Fan (1991b) requires ${\phi }_{\xi }$ to be twice differentiable over $\mathbb{R}$ which is overly restrictive than what is necessary to achieve consistency of the distributional effect estimator of Δ. On the other hand, Johannes (2009), suggests a uniform kernel ${\phi }_{\xi }(\omega /a_n)=\mathbf{1}(|\omega /a_n|\le 1)$ with a_n tends to infinity as n increases.

11 Results are similar with alternative tuning parameters including uniform kernel and triangular kernel.

12 As discussed by Bassett and Koenker (1982), the difference between empirical distributions does not necessarily result in a monotonically increasing function. Therefore, I follow the suggestion by Chernozhukov et al. (2010) to sort quantile effect estimates in increasing order with respect to u.

13 See proof of Proposition A.1 in Appendix C for details.

14 See Belloni et al. (2015) and Hansen (2015) for the list of known basis functions and their bounding sequences.