Why do I still get results with LPM while logit and probit omit regressors?

[Antonio Cabrita]

Hello everybody,

Í'm comparing results from a linear probability model with logit and probit.

Now, I got confused by the fact that for some specifications logit and probit drop variables because they "predict success perfectly" while for the same model, LPM still gives coefficients for these regressors.

Why is it that? How can LPM still come up with results here? And are those LPM results useful?

Thanks a lot for your help!
Antonio

[Weiwen Ng]

This is not surprising. The outcome you observe is only 1 or 0. The linear probability model fundamentally treats the outcome as if it were continuous. In a logit or probit model, if a combination of independent variables perfectly predicts failure or success, e.g. you only observed 1s for females with graduate degrees, then a maximum likelihood estimate doesn't exist for that combination.

The linear probability model gets around this by the linearity assumption. It will still give out coefficients.

In general, linear probability models produce results that are more wrong than logit and probit. But you know what they say, all models are wrong, but some are useful. One of the ways linear probability models can go wrong is that they can produce probabilities outside the [0,1] interval. That suggests that when you have a lot of extreme probabilities, you really want to use a logistic or probit model instead. If you have the issue of perfect prediction, though, that makes me wonder if you have a lot of extreme probabilities. And perhaps you should investigate the user-written Firth logit command (link goes to a paper by our own Richard Williams), which is a penalized maximum likelihood method (whatever that is) that can produce consistent estimates even under complete separation or quasi-complete separation (which is what you have).

Some reading on linear vs logistic probability models here.

[Antonio Cabrita]

Thanks a lot Weiwen!

I was not aware of the firthlogit command, but it indeed seems to be viable option in my case! Thanks for the hint!

[Joseph Coveney]

Also

search penlogit

and

help bayes

for alternatives.