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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?

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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.

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