Test equality of coefficients in separate regressions when populations are not independent

Question

I have three regressions with the same IVs.

Equation 1: Y1 = X

Equation 2: Y1 = X

Equation 3: Y2 = X

X is a vector of IVs with B1-B8 coefficients. Equations 2 and 3 are tested on the same population (P1), and Equation 1 is a larger population (P2) which includes P1. Y1 and Y2 are different but similar measures for the same construct. Y2 is not available for the whole populations P2.

I would like to test for the equality of coefficient B1 across the three regressions. From my limited stats understanding, the Z test from Testing equality of coefficients from two different regressions (from Paternoster) is appropriate when the populations are independent. My populations are identical in Eq 2 and 3, but have different DVs. The populations P1 and P2 are definitely not independent, since P1 is a subset of P2.

Can someone provide guidance of whether there is an appropriate test of equality of coefficients B1 across the three populations? Perhaps the above-referenced Z-score will work?

Answer 1

A simple way would be to do the following:

EQ2 compared to EQ3: First I would ensure that the different DVs are indeed measuring the same construct, both by a face/content validity standpoint (does it make theoretical sense to you and others), and then from a correlational perspective (with two variables, you could run something like Cronbach's $\alpha$ but I would just correlate them). I would also look at the distributions of the variables, to see that they have a similar shape (hopefully near-normal if continuous?) and that their variance is similar (you may want to test that as well). This is important because interpreting the coefficient magnitude is going to be based on the scale of the DVs. If the relationship holds up, then use seemingly unrelated regression. This is very easy to do in Stata, simply store the results and then use the suest command. It should be straightforward in other programs as well. This will compare your coefficients, and as long as you can make a strong enough case that the measures are similar, I would say the comparison is valid.

EQ1 to EQ2: The case here is simpler than you make it out to be, since P1 is a subgroup within P2. The easiest way to do this would be to simply interact IV for B1 with a group identifier, and test the statistical and substantive significance of the coefficient. If it is significant, then you have a differential effect for the subgroup. There are other ways to go about this, but this should do the trick.