Heckman procedure with endogenous binary treatment

[Afsanul Hasib]

Hi everyone,

I am trying to run a simple Heckman procedure where we have 1 endogenous treatment variable and 1 other covariate. My purpose is to try to compare the LATE from using an 2sls IV procedure and from the heckman procedure. However, since the treatment is endogenous, should I use the fitted values from the first stage in the Heckman procedure, or just run it with normally without using the fitted values?

Code:
Simple: heckman outcome treatment, select(selection = treatment x2) twostep
Using fitted from 2sls: heckman outcome treatment, select(selection = fitted_treatment x2) two step

[Jeff Wooldridge]

Plugging in the fitted values is generally inconsistent. You can instead follow Procedure 19.6.2 in my 2010 MIT Press book. You obtain the inverse Mills ratio from the first stage probit, and then add those in a second stage, instrumenting for the EEV at the same time.

[Jeff Wooldridge]

To make this convincing, you need an instrument for treatment and another exogenous variable that determines treatment. I'll take z1 to be the variable that primarily acts as IV for treatment. If you have other controls, include them in the selection and ivregress stages.

probit selection z1 z2
predict imr, score
ivregress outcome (treatment = z1) imr if select, robust

The t statistic on imr is valid for testing the null of no selection bias. It allows treatment to be endogenous. You should bootstrap all steps to get proper standard errors if you use the estimates that include the imr.