# Regression and interactions

I am trying to understand what is meant when I see something of the phrases like "full interactions between group a, b, and c, as well as variable x fully interacted with with each group" ( i just generalized the group names) for regressions.

If I am running a regression, and say the different groupings are white, black, urban, rural. does this mean the equation is, where black and rural are indicators for each group :

$$y = \beta_0 + \beta_1 *Black + \beta_2*rural + \beta_3*(rural*black) + (\gamma_0 + \gamma_1 *Black + \gamma_2*rural + gamma_3*(rural*black)) * x + error$$

am I correct in thinking this equation is essentially fitting 4 different lines in one regression? so each of the beta's represent an intercept for each group, and the each of the gammas are the differential slopes for each group?

The most general linear-in-parameters model would look like this:

$$y = \beta_0 + \beta_1 \cdot Black + \beta_2 \cdot Rural + \beta_3 \cdot Rural \times Black \\ + \beta_5 \cdot x + \beta_5 \cdot Black \times x + \beta_6 \cdot Rural \times x \\ + \beta_7 \cdot Rural \times Black \times x + error$$

Without additional context, I would interpret "full interactions between group a and b, as well as variable x fully interacted with each group" as the model above with $$\beta_7=0$$, which is equivalent to dropping $$Rural \times Black \times x$$ from the regression. I don't think there is a group $$c$$ in your model. I believe here "full" is a synonym for is sometimes called "factorial".

The expected value line for White urbanites is the same in both your and the general model: $$y = \beta_0 + \beta_5 \cdot x$$

For a rural Black observation, the expected value line is:

$$y = (\beta_0 + \beta_1 + \beta_2 + \beta_3) + (\beta_4 + \beta_5 + \beta_6 + \beta_7) \cdot x$$

In your model, $$\beta_7=0$$ would be set zero.

You can get the other 2 lines by plugging ones and zeros where appropriate.