From Mark Schaffer:
Question: Dave Giles, in his econometrics blog, has spent a few blog entries attacking the linear probability model.
The first of these is the more convincing (at least for me): he cites Horace & Oaxaca (2006) who show that the LPM will usually generate biased and inconsistent estimates. Biasedness doesn’t bother me so much but inconsistency does, especially as it apparently carries over to estimates of the marginal effects.
Dave’s conclusion is that one should use probit or logit unless there are really good reasons not to (e.g., endogenous dummies or with panel data).
You’ve been staunch defenders of estimating the LPM using OLS, so I’d be very interested to see your views on this.
There are three arguments here: (1) The LPM does not estimate the structural parameters of a non-linear model (Horace and Oaxaca, 2006); (2) the LPM does not give consistent estimates of the marginal effects (Giles blog 1) and (3) the LPM does not lend itself towards dealing with measurement error in the dependent variable (Giles blog 2). The structural parameters of a binary choice model, just like the probit index coefficients, are not of particular interest to us. We care about the marginal effects. The LPM will do a pretty good job estimating those. If the CEF is linear, as it is for a saturated model, regression gives the CEF – even for LPM. If the CEF is non-linear, regression approximates the CEF. Usually it does it pretty well. Obviously, the LPM won’t give the true marginal effects from the right nonlinear model. But then, the same is true for the “wrong” nonlinear model! The fact that we have a probit, a logit, and the LPM is just a statement to the fact that we don’t know what the “right” model is. Hence, there is a lot to be said for sticking to a linear regression function as compared to a fairly arbitrary choice of a non-linear one! Nonlinearity per se is a red herring. As for measurement error, we would welcome seeing more applied work taking this seriously. Of course, plain vanilla probit is not the answer.