I've blogged previously about specification testing in the context of Logit and Probit models. For instance, see here and here.
Testing for homoskedasticity in these models is especially important, for reasons that are outlined in those earlier posts. I won't repeat all of the details here, but I'll just note that heteroskedasticity renders the MLE of the parameters inconsistent. (This stands in contrast to the situation in, say, the linear regression model where the MLE of the parameters is inefficient, but still consistent in this case.)
If you're an EViews user, you can find my code for implementing a range of specification tests for Logit and Probit models here. These include the LM test for homoskedasticity that was proposed by Davidson and MacKinnon (1984).
More than once, I've been asked the following question:
"When estimating a Logit or Probit model, we set the scale parameter (variance) of the error term to the value one, because it's not actually identifiable. So, in what sense can we have heteroskedasticity in such models?"This is a good question, and I thought that a short post would be justified. Let's take a look: