Modeling heterogeneous treatment effects in the presence of endogeneity

Abstract

An inappropriate handling of cross-sectional heterogeneity renders estimates of causal effects inaccurate and uninformative. The present paper discusses how the direct modeling of cross-sectional differences via semiparametric models represents a useful bridge between a statistical approach, where the conditional distribution of the dependent variable returns any value of the outcome given any value of the explanatory variables, and an econometric analysis, where functions and parameters have direct policy implications. The explicit modeling of heterogeneity across different groups improves the quality of the estimates, mitigates their dependence upon the chosen instrumental variable, diminishes the self-selection problem, and fosters the acquisition of useful information for the entire sample.

Keywords:

• Generalized structured models
• heterogeneous effects
• marginal treatment effects
• semiparametric IV estimation
• varying coefficients

JEL CLASSIFICATION:

• C14
• C26
• C51

Acknowledgments

We are very grateful to the anonymous referee who helped us a lot and has improved this paper quite significantly. Our thanks goes also to the editors. Financial support from the Swiss National Science Foundation, project 200021-192345 is acknowledged by the second author.

Notes

1 In the limit case, $\mathrm{d}.\mathrm{a}.[{\epsilon }_{\mathrm{i}}{|\mathrm{D}}_{\mathrm{i}}]$ equals the entire probability density function of ${\epsilon }_i|D_i.$ Note that knowledge about the unconditional distribution of ${\epsilon }_i$ would not necessarily imply $\mathbb{E}[{\epsilon }_i|D_i,X_i]\ne 0.$

2 It is also possible to integrate over the elements of X_i and Z_i not contained in Q_i.

3 When only full interactions exist, you have $g_c+g_{\mathit{VXW}}$ and $E_c+E_{XW}.$ In a linear model you may imagine g_{V XW} as a coefficient times (VXW), accordingly centered, i.e., a newly defined covariate that is not a linear combination of the others.

4 However, he also specified $\tilde{g}$ as a simple linear function, so that the first step had only consequences for interpretation, not for dimension reduction.

5 Instead of ${\omega }_i\stackrel{\mathit{iid}}{\sim }N(0,1)$ one may simulate other types of noise, as for example in the moment matching bootstrap. In case of discrete D_i, it is possible to draw $D_i^{*}$ from the estimated distribution.

6 LS has a strongly asymmetric distribution on [0,10], with its mode at 8. The transformation makes it almost symmetric. For the sake of interpretation it is rescaled to [0,10] afterwards.

7 We are aware of the doubts concerning the exogenous nature of this IV and we do not specifically argue in favor of it. We use it for illustrative purposes given its broad popularity.