beta differs from overall subgroup

Question

My plain regression looks as follows:

tendaystockreturn = alpha + beta * dummy variable that takes a value of 1 in case a director purchases shares

Now I want to test whether a subgroup of directors will have a higher beta. In other words, I want to know whether the beta of a subgroup differs from the beta of the overall group.

The overall director dummy variable has 52 out of 267 obseverations where it takes a value of 1. The subgroup director dummy variable has 17 out of 267 observation where it takes a value of 1.

How should I put this in the regression? Should I make an interaction of both dummy variables or not? Or should I run both regression separately and then compare?

Answer 1

There's not enough information in your question to clearly see what you're doing (you should edit to explain more clearly what your data - including the response variable - and model consist of), but it's still possible to give some important advice.

If you compare a subgroup with the overall (which includes the subgroup) you will induce dependence that you can easily avoid.

Logically, comparing the subgroup with everything but the subgroup yields exactly the same information as comparing overall, but makes for a simpler, more standard analysis.

(Consider: if A is the same as B, then A is the same as A&B combined.)

So you should instead compare the subgroup with the remainder.