Somebody explain this to me: when you estimate a probit, but the DGP actually follows some other distribution, say logit, the estimates are inconsistent. But when we estimate an LPM, the estimates are supposedly consistent even if the DGP was actually say probit. How does that work?
LPM and consistency

The question you have to ask is consistent *with respect to what*? If the estimand of interest is the MSEbest linear approximation of the regression function, then OLS is consistent regardless of the underlying distribution (provided the estimand is welldefined under that distribution). If the estimand is the probitlinear coefficients, then of course OLS won’t recover those.

The question you have to ask is consistent *with respect to what*? If the estimand of interest is the MSEbest linear approximation of the regression function, then OLS is consistent regardless of the underlying distribution (provided the estimand is welldefined under that distribution). If the estimand is the probitlinear coefficients, then of course OLS won’t recover those.
Yeah, that was my hunch.
Now if your dependent variable is binary (say, company went public, or didn't go public), then LPM is the wrong specification regardless. So what does it mean then to have consistent estimates? Consistent to the best linear approximation when the model is inherently nonlinear. That's nonsensical, isn't it?

Remember we care about ME. OLS most certainly will do a good job recovering those.
The question you have to ask is consistent *with respect to what*? If the estimand of interest is the MSEbest linear approximation of the regression function, then OLS is consistent regardless of the underlying distribution (provided the estimand is welldefined under that distribution). If the estimand is the probitlinear coefficients, then of course OLS won’t recover those.
Yeah, that was my hunch.
Now if your dependent variable is binary (say, company went public, or didn't go public), then LPM is the wrong specification regardless. So what does it mean then to have consistent estimates? Consistent to the best linear approximation when the model is inherently nonlinear. That's nonsensical, isn't it? 
Remember we care about ME. OLS most certainly will do a good job recovering those.
LPM recovers a constant as an estimate of the ME, based on a linear approximation to a nonlinear model (which obviously should have a nonconstant ME). Sounds pretty useless to me.

Remember we care about ME. OLS most certainly will do a good job recovering those.
LPM recovers a constant as an estimate of the ME, based on a linear approximation to a nonlinear model (which obviously should have a nonconstant ME). Sounds pretty useless to me.
This usually works well for estimating average treatment effects. And it can easily accommodate an IV setup where you need to instrument for treatment. Also handles fixed effects.
So it's not useless. Just depends on what the question and underlying model are.

So it's not useless. Just depends on what the question and underlying model are.
Usually none. It's a reg monkey tool.
I'm glad that most econometricians and applied people, even those doing mostly structural work, aren't as closedminded as you.
Now get back to your outstanding hardcore modelbased logit estimation, the world is waiting for your insights.

Remember we care about ME. OLS most certainly will do a good job recovering those.
LPM recovers a constant as an estimate of the ME, based on a linear approximation to a nonlinear model (which obviously should have a nonconstant ME). Sounds pretty useless to me.
This usually works well for estimating average treatment effects. And it can easily accommodate an IV setup where you need to instrument for treatment. Also handles fixed effects.
So it's not useless. Just depends on what the question and underlying model are.Yes, you can estimate average treatment effect for a model that is known to be wrong.
You can accomodate IVs and fixed effects for a model that is known to be wrong.
But to economists, it is more important to be able to run IVs, fixed effects, etc than to use a model that is correct. Because those get you the pubs.

Remember we care about ME. OLS most certainly will do a good job recovering those.
LPM recovers a constant as an estimate of the ME, based on a linear approximation to a nonlinear model (which obviously should have a nonconstant ME). Sounds pretty useless to me.
This usually works well for estimating average treatment effects. And it can easily accommodate an IV setup where you need to instrument for treatment. Also handles fixed effects.
So it's not useless. Just depends on what the question and underlying model are.Yes, you can estimate average treatment effect for a model that is known to be wrong.
You can accomodate IVs and fixed effects for a model that is known to be wrong.
But to economists, it is more important to be able to run IVs, fixed effects, etc than to use a model that is correct. Because those get you the pubs.Wow, we found a guy here who knows the correct model.

Wow, we found a guy here who knows the correct model.
I am with you on LPM in the plain vanilla case, but IV and fixed effects are a very different story. If you run a LPM, you know you are estimating the best linear approximation to the true conditional expectation function, and coefficients can be interpretet as approximations to marginal effects. These "best" approximations may not be "good", but at least you know with you are doing.
With IV for example there is no such result. IVLPM does not estimate the best linear approximation to anything. It is totally unclear what the parameters mean that you are getting out of such a procedure.

Show me a realworld example where the LPM coefficients and the probit marginal effects are meaningfully different. I’ve been running it both ways for 20 years and haven’t had it happen yet.
As an aside, the binary dep car just implies some nonlinear RHS; tbere’s no theorem that says it’s a linear index embedded in the standard normal CDF. That is a modeling choice, nothing more.