# Interaction between dummies and a continuous variable

Lately I have had some trouble interpreting the interaction term I use in a regression.

I have two dummy variables and a continuous variable. For the explanation, BLACK takes the value 1 if the respondent is black and 0 otherwise. MAR takes the value 1 if married and 0 otherwise. I also include a continuous variable for the education of the respondent EDUC. I create the variable MARBLACK which takes the value one for married blacks.

When building my model, I estimate:

$$EARNINGS= \alpha + \beta_1 BLACK + \beta_2 MAR + \beta_3 MARBLACK + \beta_4 BLACK*EDUC + \beta_5 MAR*EDUC + \beta_6 MARBLACK*EDUC + \beta_7 EDUC$$

Here is my vision of it:

$\alpha + \beta_1 BLACK + \beta_2 MAR + \beta_3 MARBLACK + \beta_4 BLACK*EDUC + \beta_5 MAR*EDUC + \beta_6 MARBLACK*EDUC + \beta_7 EDUC$ gives the predicted value for married blacks

$\alpha + \beta_1 BLACK + \beta_4 BLACK*EDUC + \beta_7 EDUC$ gives the predicted value for non-married blacks

$\alpha + \beta_2 MAR + \beta_5 MAR*EDUC + \beta_7 EDUC$ gives the predicted value for married non-blacks

$\alpha + \beta_7 EDUC$ gives the predicted value for non-married non-blacks (my referent class).

So, if I want to calculate the premium of being married and black, I would keep $\beta_1 BLACK + \beta_2 MAR + \beta_3 MARBLACK + \beta_4 BLACK*EDUC + \beta_5 MAR*EDUC + \beta_6 MARBLACK*EDUC$ and not simply $\beta_3 MARBLACK + \beta_6 MARBLACK*EDUC$

My intuition is that being black and married, three effects accrue to the respondent: the fact of being black, the fact of being married, and the fact of belonging to both categories.

However, discussing with another researcher, she claims that it is only the MARBLACK coefficients that I should consider.

She explains it taking conditional expectations:

Let $E[EARNINGS|BLACK=1,MAR=1]= \alpha + \beta_1 BLACK + \beta_2 MAR + \beta_3 MARBLACK + \beta_4 BLACK*EDUC + \beta_5 MAR*EDUC + \beta_6 MARBLACK*EDUC + \beta_7 EDUC$

and $E[EARNINGS|BLACK=0,MAR=1]= \alpha + \beta_2 MAR + \beta_5 MAR*EDUC + \beta_6 + \beta_7 EDUC$

She then compute the effect of the change, i.e. $E[EARNINGS|BLACK=1,MAR=1]-E[EARNINGS|BLACK=0,MAR=1]= \beta_1 BLACK + \beta_3 MARBLACK + \beta_4 BLACK*EDUC + \beta_6 MARBLACK*EDUC$. She calls it EARNINGSbis.

She then goes on and do the following:

$E[EARNINGSbis|MAR=1]=\beta_1 BLACK + \beta_3 MARBLACK + \beta_4 BLACK*EDUC + \beta_6 MARBLACK*EDUC$

and $E[EARNINGSbis|MAR=0]=\beta_1 BLACK + \beta_4 BLACK*EDUC$

She again computes the effect of the change and she is left with $\beta_3 MARBLACK + \beta_6 MARBLACK*EDUC$

Could any one of you please clarify what my method and her method mean ? And especially where they deviate in terms of interpretation ?