Balancing across sub-groups

[Tunga Kantarci]

Consider that there is some sample data in which one group is subjected to a policy reform while the other not. So there is a treatment and control group. Consider that treatment shows no causal effect on some outcome based on a DiD regression. Consider then conducting heterogeneity analysis with respect to labor market status to see if there is a treatment effect in sub-samples, in particular in two labor market groups. Say that these groups are employed and unemployed. Suppose there is a treatment effect in the unemployed group but not in the employed group. The critic for this heterogeneity analysis is that the found treatment effect can be confounded due to some unobserved or observed characteristic. Say that unemployed people are more often women and that women are more likely to respond to the policy reform. To address this critic, one would reweigh the two labor market groups with respect to gender, so that the two groups are balanced with respect to the share of men and women, and then conduct the DiD analysis within each group. Since share of men and women are now similar in the two groups, if there is still a treatment effect in the unemployed group, the effect is not confounded by gender. The first question is, how to generate weights to balance the two labor market groups with respect to gender, ahead of the DiD regressions within each group? The second question is that, to make the exercise a bit more complicated, how to generate weights to balance the two labor market groups with respect to gender, if I also balance (with respect to a set of covariates, using the ebalance package) the treatment and control groups within each labor market group (to satisfy the common trend assumption)?