hi all

im running some regressions and want to examine the difference between high income areas and low income areas.
I have created a dummy variable; income=1 if high income, income=0 if otherwise.

i was wondering what the difference is between running the following sets of regressions and the reasons behind choosing either of them.
(my y, dependent, variable is score, my x,independent, variable is sat)

The first, using the 'if' command:
a) reg score sat if income==1
b) reg score sat if income==0



The second, using an interaction term, incsat =income*sat
c) reg score sat incsat

I believe the Coefficient of sat in regression a will give the marginal effect of sat in high income areas while that in regression b will be the marginal effect for low income areas.
I also think that in regression c the coefficient of sat will be the marginal effect for low income areas, while the coefficient of incsat will be the difference in marginal effects, thus adding the two coefficients will give me the marginal effect in high income areas.


unless im mistaken in my metrics (please correct me if I am), I cannot see any differences/ reasons in favour of one method.
If anyone can explain id really appreciate it because i need to pick a method asap haha!

thank you