I'm a little suspicious of Brown's criteria. Consider the ethnicity, "Analytic Philosophers" (a name they use for themselves): they trace their intellectual ancestry back to Socrates. They possess a shared history of thought and the traditions of discourse (including artifical languages, customs of annotation, etc.) developed in that history. The community has more than a casual concern with Athens as the cradle of philosophy (whether they assert or deny that status) and, of course, there's a concern with the continuation of their community's influence.

(Similar made-up ethnicities could probably be made for Computer Programmers, Elvis Fans, Goths and Role-players, among others.)

I'll admit to having pulled a fast one with "intellectual ancestry", but by the rest of the criteria, "Analytic Philosopher", seems like a more respectable ethnic identity than, "Pakeha", despite our intuitions to the contrary. And the name change to "New Zealander" doesn't seem to do us any favours in making the identity more robust.

The conclusion here is what you knew already: ethnicity is a made-up notion. If you don't happen to identify as any particular ethnicity, just ask yourself if a self-identifying "member" of any ethnic groups would say that you were one of them. Tick the boxes that apply.

I'll probably tick "New Zealand European" and "Other" (Pakeha). They seem about as different as English and Scottish, and both would claim me.