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I want to conduct a simple propensity score estimation where the treatment $D_i$ is a binary variable ($D_i=1$ individual $i$ participates in the labor market program, zero otherwise). I estimate the propensity score using a simple probit model using various explanatory variables (including gender).

Then I want to calculate the average treatment effect (ATE) (and the average treatment effect on the treated, ATET) using e.g. caliper matching with replacement or a simple 1:1 matching with replacement.

The question: how can I check whether the ATEs of e.g. male and female individuals differ, i.e. whether $$(\text{ATE|i is male} - \text{ATE|i is female}) = 0$$

How can I make inference (whether the difference between the ATEs is different from zero)? Would you recommend bootstrap? If this was the case how would I do it?

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