fractional probit model added to -cmp-

[David Roodman]

The latest version of cmp, now on SSC, adds the fractional probit model of Papke and Wooldridge (1996) as a model type. This is same model also implemented in isolation in Stata 14's -fracreg- command.

As usual, the purpose of cmp is not mimic other commands, but showing how it can do so is informative. In Stata 14, these give the same results:

webuse 401k
fracreg probit prate mrate ltotemp age i.sole
margins, dydx(mrate)

cmp setup
cmp (prate = mrate ltotemp age i.sole), ind($cmp_frac) qui
margins, dydx(mrate) predict(pr)

You can also do bivariate fractional probits, IV fractional probits, etc.

Install with "ssc install cmp, replace". Comments welcome.

--David

[Dandan Yu]

Hello, Dr. Roodman,

Thanks for your effort on 'cmp' command. It's very powerful and useful.

I have a question about using fractional probit model in a multi-equation SUR system, consisting of, for example, a probit and a fractional probit.
Since I haven't found an error term in the modeling of fractional probit, I couldn't well understand the meaning of estimated 'rho' in the output.
Is this the correlation coefficient between the conditional mean of the fractional response and the error term in the latent probit variable?
Or how should I describe this term accurately?

Thanks so much!
Dandan Yu

[sim sh]

hi, did u figure this out?

[David Roodman]

It's an interesting question. Fractional regression models are fit with quasi-maximum likelihood. This means that you maximize a representation of the data-generating process, a likelihood, for estimation purposes even if that representation is not necessarily based on the true model, and even based on any fully coherent model at all.

The same thing happens when you run a probit regression with robust standard errors. The likelihood that is maximized to fit the probit model assumes that the error has the same distribution in every observation. But the "robust" option implies otherwise, in particular that there is heteroskedasticity. The "robust" part is ignored during estimation of the parameters. After estimation, the robust correction is factored into the standard errors.

In cmp, the fractional regression model is fit by using a generalization of the probit model, just as with Stata's "fracreg probit" command. If the fractional outcome is 0 or 1, the computed likelihood is the same as for the probit model. If the fractional outcome is in between, then the computed log likelihood is a weighted average of the log likelihoods for the 0 and 1 outcomes. So in this sense there is an error term. It is the error term of the probit model that is being used for estimation purposes. That is what the rho estimates from cmp refer to.

I suggest you compare the methods and formulas for probit and fracreg with the probit model.

[sim sh]

ok , so rho estimates should come insignificant. is it normal for value of sig2 and rho 2 to be blank in z and p value. basically 4 things are showing at bottom : lnsig_2, atanhrho_12 , sig_2 and rho_12 , how to interpret them?

[David Roodman]

If you are getting missing or dropped values for parameters, that probably means the model is having difficult converging. In general there is no mathematical guarantee of convergence, unless the model is exactly correct and data are infinite.

sig_2 = exp(lnsig_2). the estimated variance of the error for equation 2. rho_12 = tanh(atanhrho_12). the estimated correlation between the errors in equation 1 and equation 2.