Bounds on direct and indirect effects under treatment/mediator endogeneity and outcome attrition

Abstract

Causal mediation analysis aims at disentangling a treatment effect into an indirect mechanism operating through an intermediate outcome or mediator, as well as the direct effect of the treatment on the outcome of interest. However, the evaluation of direct and indirect effects is frequently complicated by non-ignorable selection into the treatment and/or mediator, even after controlling for observables, as well as sample selection/outcome attrition. We propose a method for bounding direct and indirect effects in the presence of such complications using a method that is based on a sequence of linear programming problems. Considering inverse probability weighting by propensity scores, we compute the weights that would yield identification in the absence of complications and perturb them by an entropy parameter reflecting a specific amount of propensity score misspecification to set-identify the effects of interest. We apply our method to data from the National Longitudinal Survey of Youth 1979 to derive bounds on the explained and unexplained components of a gender wage gap decomposition that is likely prone to non-ignorable mediator selection and outcome attrition.

Keywords:

• Bounds
• causal channels
• causal mechanisms
• direct effects
• indirect effects
• mediation analysis
• sample selection

JEL CLASSIFICATION:

• C21

Acknowledgments

Authors gratefully acknowledge the helpful comments of two anonymous referees and the editor Esfandiar Maasoumi.

Declaration of interest statement

The authors report there are no competing interests to declare.

Notes

1 The studies mentioned and our own investigate the sensitivity of direct and indirect effects to prespecified deviations from the identifying assumptions. Alternatively, one may derive worst case bounds, which are based on the possibly most extreme forms of violations of specific assumptions, which typically implies a rather wide range of admissible effect values. See for instance Kaufman et al. (2005), Cai et al. (2008), Sjölander (2009), and Flores and Flores-Lagunes (2010).

2 Note that Assumption A2 is not required for the identification of $E[Y(d,M(d))]$ for $d\in \{0,1\},$ but for $E[Y(d,M(1-d))].$

3 It is in principle possible to compute sharp bounds even for $\delta (1)$ and $\delta (0)$ at the cost of solving a more complex optimization problem. This issue is discussed in a greater detail in Appendix B.

4 Another approach for setting the entropy parameter is to consider the change in estimated probabilities induced by a change of the link function, e.g. by picking the probit instead of logit function. Formally, ${ϵ}_{i,\mathit{probit}}^{A3,1}=\frac{|{\widehat{p}}_{i,\mathit{probit}}^{A3}-{\widehat{p}}_i^{A3}|}{\sqrt{{\widehat{p}}_i^{A3}(1-{\widehat{p}}_i^{A3})}},$ where ${\widehat{p}}_{i,\mathit{probit}}^{A3}$ and ${\widehat{p}}_i^{A3}$ correspond to the estimated probabilites under a probit and logit model, respectively. One may thus pick the entropy parameter as average ${ϵ}_{\mathit{probit}}^{A3,1}={\sum }_{i=1}^n\frac{D_i·S_i·{ϵ}_{i,\mathit{probit}}^{A3}}{{\sum }_{i=1}^n{D}_i·S_i}.$

5 The NLSY79 data consists of three independent samples: a cross-sectional sample (6,111 subjects, or 48%) representing the non-institutionalized civilian youth; a supplemental sample (42%) oversampling civilian Hispanic, black, and economically disadvantaged nonblack/non-Hispanic young people; and a military sample (10%) comprised of youth serving in the military as of September 30, 1978 (Bureau of Labor Statistics, U.S. Department of Labor, 2001).

6 For instance, we excluded 502 persons reporting to work 1,000 hours or more in the past calendar year, but whose average hourly wages in the past calendar year were either missing or equal to zero. Furthermore, we dropped 54 working individuals with average hourly wages of less than $1 in the past calendar year. We also excluded 608 observations with missing values in the mediators.

7 We note that in the decomposition literature, it is frequently the male wages that are considered as reference (or ‘fair’) wages. This suggests considering $\theta (0)$ and $\delta (1)$ as unexplained and explained components, respectively. See Sloczynski (2013) for an in-depth discussion of reference group choice in the potential outcome framework.

8 We applied subsampling to the upper and lower bounds separately similar to Lafférs and Nedela (2017) or Demuynck (2015). A computationally more expensive stepdown procedure of Romano and Shaikh (2010) may be used to control for the asymptotic coverage of the whole identified set. Neither the subsample size nor the number of subsamples affected our results in an important way. For completeness, we describe the procedure in detail in Appendix C.

Additional information

Funding

This work was supported by the Slovak Research and Development Agency under the Grant APVV-17-0329 and under the Grant VEGA-1/0692/20.