Fractional probit with a binary endogenous explanatory variable

[Varsha Vaishnav]

Hello,

I have a dependent variable in the interval [0,1] and a binary endogenous regressor. How can I proceed to estimation in such a case in stata?

Thanks

[Jeff Wooldridge]

There are at least two possibilities. In my 2014 Journal of Econometrics paper, I showed that the same log likelihood used for the case with y1 binary and y2 binary applies to y1 fractional, y2 binary. In other words, you can, in principle, use biprobit. The practical problem is that Stata applies a data check on y1 (and y2, I imagine) and replaces any nonbinary variable with a binary variable. You don't want that.

In biprobit.ado, you will find a command _fv_check_depvar `dep1', which I believe is checking whether y1 is binary. I don't feel confident enough to edit this myself -- although I did successfully do it many years ago, and then lost the ado file. I would created a new ado called fracbiprobit.ado, or something like that, and remove the data check on y1. Then, use the vce(robust) option and you'll be all set. You need to compute average partial (marginal) effects as when y1 is binary.

A second possibility is what is either called the control function or two-stage residual includes (2SRI). In my 2014 paper, I suggested using the generalized residual for y2 from a first-stage probit and then inserting those, in a flexible way, into the second stage fractional probit. Terza and coauthors suggested something slightly different when y1 is binary.

The CF approach is simple but it is at best an approximation. You should bootstrap to get proper standard errors for the APEs. Below, the generalized residual interacts with everything and one can even justify a triple interaction. Note that L > K so we have at least one IV for y2.

probit y2 z1 ... zL
predict gr2, score
fracreg probit y1 1.y2 z1 ... zK gr2 1.y2#c.gr2 c.z1#c.gr2 ... c.zK#c.gr2 1.y2#c.gr2#c.z1 ... 1.y2#c.zK#c.gr2
margins, dydx(y2)

[Varsha Vaishnav]

Sir, the inclusion of the interactions is required?

[Jeff Wooldridge]

No, not required. In the simplest case, just add gr2. The interactions are intended to mimic the endogenous switching model in the linear case. They can matter.

[Varsha Vaishnav]

Many thanks for your quick responses, Sir. Means a lot.

[George Ford]

I tried to recode biprobit along Jeff's recommendations. I could get an estimate, but they were questionable.

Probit won't estimate with a share, so I recoded using fracreg. Didn't like it for some reason.

[Varsha Vaishnav]

Hello Sir,

Sir, I have a query regarding the standard errors. The survey data I'm utilizing adopts a two-stage stratified sampling design, so I have used clustered standard errors for other regressions. Is it okay if I use clustered in the case of fractional response as well instead of bootstrapping?