Fractional response model with sample selection

[Guest]

Hello all!
I am planning to run a fractional regression model to determine the share of expenditure in total expenditure. Since, the share lies between 0 and 1, i understand that fractional logit is the appropriate model. However, since since all the individuals are not involved in expenditure activity, there is selection bias too. And my data is panel in nature. I know, I have to combine fractional logit model in panel data with selection bias. Unfortunately, I am unsure of what to do in Stata. It would be great if anyone could let me know if the steps below are right or not:
1) run a xtprobit regression to account for participation
2) get inverse mills ratio from participation equation
3) Add Inverse mills ratio as one of the explanatory variables in the glm command
I know that getting IMR and adding it as one of the explanatory variables in case of linear model works, following Wooldridge(1995). But what about the non-linear model case. Does it work the same way?

[Jeff Wooldridge]

Hi Guest:

I've been meaning to reply when I got a free moment. First, are you sure you have a "selection" problem? If a share is zero it can still be modeled using a fractional response if you think a logit mean function is a good approximation. or, you can use a two-part model.

If you want to study it as a "selection" problem, I refer you to two relatively recent papers, which I've attached. You effectively have to combine them because my 2014 paper discusses the fractional case but in the Semykina-Wooldridge paper we only explicitly cover the binary case. Here's the important point: the exact same method described in the S-W case works for the fractional case, for the same reasons I discuss in the 2014 paper. The tricky thing is getting canned software to do the estimation. You can use -heckprobit- and the pooled correlated random effects probit method, but you'd need to edit the ado file so that the check on the binary nature of the dependent variable is turned off. I once successfully did this with -biprobit- because the issue is essentially the same: the estimators are consistent with fractional response, but Stata doesn't allow fractional responses in these commands.

Your proposed solution is something I suggest as an approximation in my 2014 paper, but, strictly speaking, it is not consistent. It is perfectly fine as a test for selection, as I discuss in my paper.

But, again, let me remind you: a zero is not the same as a missing variable. The methods in Papke and Wooldridge (2008, Journal of Econometrics) can also be used.

JW

[Carlo Lazzaro]

Jeff:
thanks for sharing.

[Richard Williams]

You may want xtefracreg. Unfortunately that program doesn't exist, at least not yet.

Jeff, do you think it might be a relatively simple matter to tweak eprobit and xteprobit so they worked with fractional variables? For those who don't have Stata 16 yet, most of the Extended Regression Models are described at

https://www.stata.com/manuals/erm.pdf

[Guest]

Thank You Prof. Wooldridge for replying