Mills' ration in panel data regressions (sample selection bias)

[Nick Baradar]

Dear All,

I am currently addressing sample selection bias in my panel data and am attempting to incorporate Mills' ratio. However, I'm encountering difficulties with Stata commands.

Initially, I estimated the probability of observation selection using the xtprobit command, including one exogenous variable. Then, I used the

predict phat,xb

command and subsequently generated Mills' ratio with the

gen mills = exp(-0.5 * phat^2) / (sqrt(2 * _pi) * normalden(phat))

command. Finally, I ran the xtreg command, incorporating the Mills' ratio but omitting the exogenous variable.

I would greatly appreciate advice on whether the set of independent variables used in the xtprobit command for estimating the probability of observation selection should be identical to the set used in the final xtreg command. While I understand the necessity of including one exogenous variable in the first stage of the xtprobit regression, I'm uncertain if other variables in these two regressions should be exactly the same. Additionally, if you notice any issues with the approach I'm pursuing, or you have any other additional advice (any recourse that discussed a very similar case on addressing sample selection bias with panel data using Stata), I would be immensly grateful.

Thank you very much in advance!

Best,
Nick

[Andrew Musau]

With panel data, the instruments employed might be correlated with the unobserved heterogeneity, thus rendering the fixed effects two-stage least squares (FE2SLS) estimator inconsistent. Semykina and Wooldridge proposed a correlated random effects estimator that addresses the issues with FE2SLS (see https://myweb.fsu.edu/asemykina/selectionPS.pdf for discussions of these issues and a description of the proposed estimation method). In Stata, this estimator is implemented by xtheckmanfe from SSC.

ssc install xtheckmanfe, replace
help xtheckmanfe

Reference:
Semykina, Anastasia, and Jeffrey M. Wooldridge. 2010. "Estimating panel data models in the presence of endogeneity and selection." Journal of Econometrics 157 (2):375-380.