Fractional Response Model with a binary endogenous explanatory variable

[Kamola Abdurasulova]

Dear fellow researchers,



I have a fractional response model with a binary endogenous explanatory variable (EEV). What modelling approaches would you suggest? I am thinking of estimating the EEV in the first stage using probit, and adding the residuals to the second stage fractional logit regression...


Thank you for in advance.

[Jeff Wooldridge]

I show in my 2014 Journal of Econometrics paper that including the generalized residual from a first-stage probit into a fractional probit is a solution under certain assumptions. Alternatively, you can apply "biprobit" but where y1 is a fractional outcome. The practical problem is Stata's built-in data check. I once managed to override it by editing the ado file for biprobit, but, regrettably, I lost that. It's an easy change, and then you'd use vce(robust) to account for the fact that y1 is fractional.

The two-step procedure is easier but then you should bootstrap the standard errors.

probit y2 z1 z2
predict gr2, score
fracreg probit y1 y2 gr2 z1, vce(r)

You get a valid test of the null of exogeneity in the last step as the t statistic on gr2.

[Kamola Abdurasulova]

Thank you very much professor Jeff Wooldridge !

I estimated my model using the two-step approach as you suggested. In the second stage, gr2 is negative and statistically significant. Does it mean after controlling for endogeneity in y2, any unobserved factor that increases y2 is associated with a decrease in y1?

[Jeff Wooldridge]

You'll want to obtain the average partial effect of y2 to get the magnitude. Plus, for added flexibility, you should probably interact y2 with gr2, and y2 with z1 -- to mimic the linear switching regression model.

probit y2 z1 z2
predict gr2, score
fracreg probit y1 i.y2 c.gr2 i.y2#c.gr2 z1 c.z1#c.gr2, vce(r)
margins, dydx(y2)
test 1.y2 1.y2#c.gr2

[Kamola Abdurasulova]

Thank you very much professor Jeff Wooldridge ! I appreciate your help!