Multivariate fractional logit with binary endogenous regressor using control function approach

[Clemence Kieny]

Dear Statalisters,

This is my first post, so please let me know in case I should expose my problem differently.

I am trying to fit a multivariate fractional logit model with a binary endogenous regressor.

1/ Without taking into account the endogeneity issue, my code looks like this:

xi: fmlogit t_1_share t_2_share t_3_share t_4_share t_5_share, etavar($x $controls)
                        
                        estimates store time_est
        
                        forvalues i = 1/5 {
                                    estimates restore time_est
                                    margins, dydx($x) predict(pr outcome(t_`i'_share))  post        
                        }

Where:

• t_1_share to t_5_share= fractional outcome variables (all between 0 and 1, summing up to 1)
• $x=endogenous explanatory variable (binary)

2/ I however need to take into account the fact that $x is endogenous.

From reading Wooldridge (JHR, 2015, http://jhr.uwpress.org/content/50/2/420.abstract) and previous posts (https://www.statalist.org/forums/for...quadratic-term , https://www.statalist.org/forums/for...ch-with-probit, https://www.statalist.org/forums/for...ndent-variable ,
https://www.statalist.org/forums/for...nous-regressor), it is my understanding that I should use a 2-step control function approach.

I am however not sure whether this would still work even if my EEV is binary and how to adapt the method to a fmlogit setting.

Would something like the following code make sense?

reg $x $z $controls
predict vh, resid
xi: fmlogit t_1_share t_2_share t_3_share t_4_share t_5_share, etavar($x $controls vh)
                        
                        estimates store time_est
        
                        forvalues i = 1/5 {
                                    estimates restore time_est
                                    margins, dydx($x) predict(pr outcome(t_`i'_share))  post        
                        }

Where $z is a binary instrument that I use in other parts of my paper to assess the impact of my EEV $x on continuous outcome variables.

Any insight would be useful. Thank you very much in advance for your help.

Clémence

[Fei Wang]

As the cited threads pointed out, the CF approach described in #1 is valid when the EEV is roughly continuous. For you case, a better alternative could be jointly estimating the two stages using, for example, cmp (from SSC). One equation (second stage) is the fractional logit (you may try Stata's official command fracreg), and the other equation (first stage) is a logit or probit of the EEV on the exclusive instrument and other exogenous covariates.

[Clemence Kieny]

Dear Fei,

Thank you so much for your answer and sorry for my late response due to the winter vacation.

I will try the CMP approach. However, after reading some more (e.g. Section C., p. 427 of Wooldrighe (JHR, 2015)), I see that in the case of a binary EEV I may still use a CF approach with a 1st stage probit, predicting the generalized residuals and including them in the second stage in a similar fashion as in the standard CF approach. Do you think this could be appropriate in my context and is this the correct way to do it?

I have read some threads suggesting to use predict gr, score (e.g., https://www.statalist.org/forums/for...ith-panel-data; https://www.statalist.org/forums/for...atory-variable). However, I looked at the Stata help for post estimation commands for probit which indicates that predict, score returns the "first derivative of the log likelihood with respect to xjβ". I fail to understand how this is the same as the generalized residuals I am trying to compute.

Thank you so much in advance for your help,

Best,

Clémence

[Clemence Kieny]

Hello again,
Would anyone have an insight on whether the generalized residuals approach would be appropriate and whether I can indeed use predict gr, score in order to obtain them?
Thanks a lot in advance,
Clémence