Thread by @jmwooldridge

I've become a believer in always reporting "robust" standard errors. This may seem obvious, but there are nuances. And I'm not talking indiscriminate clustering -- I'll comment on that at some point. Let's start with random sampling from a cross section.

#metricstotheface

We know to use heteroskedasticity-robust standard errors for linear regression automatically. For two reasons:

1. Var(y|x) might depend on x.
2. E(y|x) is wrong and we are only estimating the LP.

Both cause heteroskedasticity.

#metricstotheface

We've discussed quasi-MLE, such as fractional logit and Poisson regression. If E(y|x) is correct, we want standard errors robust to general variance misspecification.

#metricstotheface

Interesting fact: Because flogit and Poisson use the canonical link function, the same robust variance matrix falls out whether the mean is assumed correct or not. (Just like the linear case.) This isn't the case for fprobit, for example.

Now it gets more subtle. If y is binary and I estimate logit, probit, cloglog, should I use "robust" standard errors?

If I do, I'm admitting my model for P(y=1|x) is wrong. It has virtually nothing to do with "heteroskedasticity." It means I think P(y=1|x) is wrong.

I think admitting our models are wrong is healthy, but you must know you're doing it.

probit y x1 ... xk, robust

is not magically consistently estimating coefficients in the presence of heteroskedasticity.

It is still useful to say: like the LPM, my probit model is only an approximation. But it might well approximate APEs -- and could very well be better than linear. I'm going to show robust standard errors in all cases.

Same comments hold for something like Tobit. Any misspecification of the underlying assumptions causes inconsistent estimates of the betas (and sigma). Yet, the estimated Tobit mean function might be a good approximation.

Using vce(robust) with tobit is not producing consistent estimators of the betas in the presence of heteroskedasticity in the latent variable model. And you wouldn't want it to. You want the best approximation to, say, E(y|x) given the model you're using.