Estimation of counterfactual distributions with a continuous endogenous treatment

Abstract

In this article, I propose a method to estimate the counterfactual distribution of an outcome variable when the treatment is endogenous, continuous, and its effect is heterogeneous. The types of counterfactuals considered are those in which the change in treatment intensity can be correlated with the individual effects or when some of the structural functions are changed by some other group’s counterparts. I characterize the outcome and the treatment with a triangular system of equations in which the unobservables are related by a copula that captures the endogeneity of the treatment, which is nonparametrically identified by inverting the quantile processes that determine the outcome and the treatment. Both processes are estimated using existing quantile regression methods, and I propose a parametric and a nonparametric estimator of the copula. To illustrate these methods, I estimate several counterfactual distributions of the birth weight of children, had their mothers smoked differently during pregnancy.

Keywords:

• Copula
• counterfactual distribution
• endogeneity
• policy analysis
• quantile regression
• unconditional distributional effects

JEL classification::

• C31
• C36

Acknowledgments

Departamento de Economía, Universidad de Cantabria, Avenida de los Castros, s/n, 39005 Santander, Spain. I would like to thank the editor, Esfandiar Maasoumi, the associate editor, two anonymous referees, Manuel Arellano, Stéphane Bonhomme, Domenico Depalo, Bryan Graham, Giuseppe Ilardi, Michael Jansson, James Powell, Demian Pouzo, Enrique Sentana, and seminar participants at CEMFI and University of California, Berkeley for their helpful comments and discussion. All remaining errors are my own. I can be reached via email at [email protected]

Notes

1. Other related works include Firpo, Fortin, and Lemieux (2009), who proposed an estimator based on reweighting of the influence function to estimate distributional effects under exogeneity, or Frölich and Melly (2013), who proposed a nonparametric estimator of the unconditional quantile treatment effect for the subpopulation of compliers with an endogenous, binary treatment.

2. Because these variables are unidimensional, the amount of heterogeneity of the model is restricted. In particular, it rules out models that are not monotonic on the unobservables, such as random coefficients models. Unfortunately, these models are not nonparametrically identified (Hahn and Ridder, 2011; Hoderlein, Holzmann, and Meister, 2017; Kasy, 2011; Masten, 2018) and they would further complicate the present analysis. Deriving set-identification results with multidimensional unobserved heterogeneity is beyond the scope of this article.

3. This system uses the Skorohod representation, which states that a random variable φ_i can be written in terms of its quantile function: φ_i = Q(U_i), where $U_i\sim U(0,1)$.

4. By definition, a copula is the multivariate distribution of $(U_1,...,U_m)$ such that their marginal distributions are uniformly distributed on the unit interval. Sklar (1959) showed that any multivariate distribution of the continuously distributed variables $X_1,...,X_m$, with respective marginal cdfs $F_1\left(x_1\right),...,F_m\left(x_m\right)$, there exists a unique cdf C, such that $\mathbb{P}(X_1\le x_1,...,X_m\le x_m)=C(F_1\left(x_1\right),...,F_m\left(x_m\right))$. The conditional copula is defined as $C(F_1\left(x_1\right),...,F_{\bar{m}-1}\left(x_{\bar{m}-1}\right),F_{\bar{m}+1}\left(x_{\bar{m}+1}\right),...,F_m\left(x_m\right)|F_{\bar{m}}\left(x_{\bar{m}}\right))=\frac{\partial }{\partial F_{\bar{m}}\left(x_{\bar{m}}\right)}C(F_1\left(x_1\right),...,F_m\left(x_m\right))$. Lastly, the copula density is defined as $c\left(F_1\left(x_1\right)\right),...,F_m\left(F_m\left(x_m\right)\right)=\frac{{\partial }^m}{\partial F_1\left(x_1\right)\cdot ...\cdot \partial F_m\left(x_m\right)}C(F_1\left(x_1\right),...,F_m\left(x_m\right))$.

5. The relation of the copula and the MTE can be immediately generalized to the case of multidimensional unobserved heterogeneity. For example, assume that it depends on ${\left\{U_m\right\}}_{m=1}^M$. Define the multidimensional copula by $C_{\mathrm{UV}|X_2}(U_1,...,U_M,V|x_2)$. Then, the MTE would be calculated as in Eq. (3(3) $$\begin{array}{l}\mathbb{E}[\frac{\partial g_1(X_1,X_2,U_d)}{\partial X_1}|X_1=x_1,X_2=x_2,V=v,D=d]={\int }_0^1{\mathrm{\nabla }}_1{g}_d(x_1,x_2,u)c_{U|V{X}_2}^d(u|v,x_2)\mathrm{du}\end{array}$$(3) ) by integrating the derivative of g_d with respect to the multidimensional copula, holding V constant.

6. This is a particular case of the Bernstein copula.

7. Note that in many circumstances, the ability of the policy maker to implement such counterfactuals may be limited, making them informative of such potential effects theoretically.

8. Chesher (2003) and Imbens and Newey (2009), who studied the nonparametric identification of nonseparable models using a control function approach. Other papers proposed semiparametric methods which do not suffer from the curse of dimensionality, such as Jun (2009), or Lee (2007) who assumed the model to be separable. Alternatively, Ma and Koenker (2006) proposed a parametric model of Chesher (2003). Another recent paper is D’Haultføeuille and Février (2015), which established the identification based on similar conditions to those in Torgovitsky (2015). On the other hand, (Hoderlein and Mammen 2007) discussed the identification without monotonicity.

9. This constant serves to avoid the estimation of extreme quantiles. See, for example, Chernozhukov and Hansen (2005).

10. Additionally, the linear specification requires regressors to take either positive or negative values, but not both, as that would make the process non monotonic. See Koenker (2005) for further details.

11. I discuss the properties of the estimator when this assumption is relaxed in Appendix C.

12. With some slight abuse of notation, the metric spaces used for the convergence omit the spaces of group indicators. They should be implicitly included throughout the article.

13. The finite sample performance is shown in a Monte Carlo exercise in Appendix E.

14. See Fang and Santos (2019) for further details.

15. The asymptotic distribution of the counterfactual unconditional distribution and quantile function is obtained analogously to that of the first kind of counterfactual, shown in Theorem 1. Its proof is omitted for the sake of brevity.

16. Even for such estimator it would not be possible to establish the asymptotic normality as in Theorem 1, since the estimator of the conditional copula converges at a rate slower than $\sqrt{n}$. See Appendix B.7 for further details.

17. Note that the weight for each individual is the same in every step.

18. The bootstrap validity for the second kind of counterfactual estimators is analogously proved and therefore omitted.

19. Note that the bootstrapped covariance matrix requires additional conditions to be valid. See Kato (2011) for further details.

20. See, for example, Woutersen and Ham (2013).

21. For a more detailed description of the datasets, see Evans and Ringel (1999).

22. The age variable is introduced in the regressions as actual age minus 18.

23. Note that the extreme quantiles take values outside this range. Because these are less precisely estimated, I do not comment them in the text.

24. As stated in the text, this counterfactual is not Hadamard differentiable, so the bootstrap does not consistently approximate the asymptotic distribution of the estimator. Regardless, because the alternative method reported in Remark 5 is infeasible, I report these inconsistent estimates.

25. I only report the estimates with these two copulas because the Frank and Clayton copulas either yield very similar results or fit the actual data much worse. The results with them are available upon request.

26. Some of this notation is the standard in the literature of empirical processes. See VanDerVaart2000.

27. See, for example, Theorem 1 in Pratt (1960).

28. Notice that it is not possible to apply the extended continuous mapping theorem to conclude that if ${\hat{U}}_{d,i}\stackrel{P}{\to }{U}_{d,i}$ and ${\hat{V}}_i\stackrel{P}{\to }{V}_i$, then $\mathbf{1}({\hat{U}}_{d,i}\le u)\stackrel{P}{\to }\mathbf{1}(U_{d,i}\le u)$ or $\mathbf{1}({\hat{V}}_i<v)\stackrel{P}{\to }\mathbf{1}(V_i<v)$ uniformly in (u, v). Hence, a different argument is required for the proof.

Additional information

Funding

This work is part of the I+D+i project Ref. TED2021-131763A-I00 financed by MCIN/AEI/10.13039/501100011033 and by the European Union NextGenerationEU/PRTR. I gratefully acknowledge financial support from the Spanish Ministry of Universities and the European Union-NextGenerationEU (RMZ-18).