How can I do a Chow test with the robust variance estimates, that is, after estimating with regress, vce(robust)?

First, see the FAQ How can I compute a Chow test statistic?. The point of that FAQ is that you can do Chow tests using Stata’s test command and, in fact, Chow tests are what the test command reports.

Well, that’s not exactly right. test uses the estimated variance–covariance matrix of the estimators, and test performs Wald tests,

       W  =  (Rb-r)'(RVR')-1 (Rb-r)

where V is the estimated variance–covariance matrix of the estimators.

For linear regression with the conventionally estimated V, the Wald test is the Chow test and vice versa.

You might say that you are performing a Chow test, but I say that you are performing a Wald test. That distinction is important, because the Wald test generalizes to different variance estimates of V, whereas the Chow test does not. After regress, vce(robust), for instance, test uses the V matrix estimated by the robust method because that is what regress, vce(robust) left behind.

Thus the short answer is that you estimate your model using regress, vce(robust) and then use Stata’s test command. You then call the result a Wald test.

If you are bothered that a Wald test produces F rather than chi-squared statistics, also see the FAQ Why does test sometimes produce chi-squared and other times F statistics?