# Interaction terms in cross-sectional regression

I have a question regarding the use of interaction terms in a cross-sectional regression model.

Currently I am working on a study for which I have a sample consisting of roughly 500 observations, which can be divided under type 1 and type 2 (similar to e.g. male and female). My interest goes out to finding whether the outcome in the dependent variable is significantly different between groups. After performing a difference in means t-test I found out that there is a significant difference, but I want to further control whether this difference holds by adding in total 8 independent variables. I performed an OLS regression including all these variables and a dummy that noted 1 for type 1 and 0 for type 2, but I was told to include interaction term effects. This way my model would look something like:

$$a + b_1x_1 + b_2x_2 + b_3x_3 + \dotsm + b_9x_1x_2 + b_10x_1x_3 + \dotsm + e$$ (I basically multiplied every predictor variable with the dummy to find the additional interaction terms)

I am wondering:

1. if I add a dummy to a regression with multiple predictor variables into a regression, why can I not interpret the outcome for the dummy as already being controlled for these variables?

2. Is it allowed to use the above model? I found models specifying e.g. $$x_1 + x_2 + x_1x_2$$, but I could hardly find any good reads on adding multiple independent variable interaction terms.

3. to me it seems highly problematic to use these many interaction terms, as my intention is to find out what the effect is for the dummy variable (type 1), and whether it is significant. I am aware that adding interaction terms gives me no more chance to directly interpret the coefficient and the significance, but I could hardly imagine how I can still make any valid assumptions with this amount of interaction terms?

4. Just as a check of my knowledge, if the interaction term lets say dummy1 x education comes out significant over this entire interaction model, can I then correctly assume that the coefficient shows the difference in the effect the predictor education has on the dependent variable for all type 1 when compared to type 2?

I hope that I explained my current problem clearly, if in need for any clarification I will gladly do so.

It's hard to give a specific answer to your question as I don't now your research question. But I'll try to give a general interpretation which might lead the way.

It appears to me that you where told to do a "non-fully" saturated model. (So if you want to read up on models with several interaction terms "saturated model" is what I would goggle if I where you.) I.e. include interactions between your dummy variable $D$ and your other 8 independent variables so that the model becomes, \begin{eqnarray} y_i = \alpha &+& \phi D_i + \pi_1 x_{1i} + \pi_2 x_{2i} + \ldots + \pi_8 x_{8i} \\ &+& \beta_1 (D_i \cdot x_{1i}) + \beta_2 (D_i \cdot x_{2i}) + \ldots + \beta_8 (D_i \cdot x_{8i}) + \varepsilon_i \end{eqnarray} where $\phi$ is your coefficient of interest. Why you would want to do this is hard to know without knowing your research question but perhaps you got a motivation from the person who told you to do so?

I would interpret the coefficients in this fashion; $\phi$ is the effect of $D=1$ in contrast to being zero holding the other variables constant. Say that, as in your example, $x_1$ is years of education. Then $\beta_1$ is difference between the two groups $D=0,1$ in the effect on $y$ of an additional year of schooling. So if e.g. $\beta<0$ (disregarding significance), well then returns from one year of extra education is lower for those who are $D=1$. Note that including all these interaction terms completely changes the interpretation of $\phi$ as this estimate now corresponds to someone who has $x_1=x_2=\ldots=x_8=0$. So yes, in general I would say that it appears problematic to include all these interaction terms.

If I add a dummy to a regression with multiple predictor variables into a regression, why can I not interpret the outcome for the dummy as already being controlled for these variables?

I think you are confusing "interaction" with "control". An interaction means that the relation between one predictor variable and the outcome variable is different at different levels of the other predictor variable -- or, in your case, that e.g. education has a different effect for type 1 and type 2.

Is it allowed to use the above model? I found models specifying e.g. x1+x2+x1x2 , but I could hardly find any good reads on adding multiple independent variable interaction terms.

Provided that you have a big enough sample and meet the other assumptions of regression, yes, it is.

to me it seems highly problematic to use these many interaction terms, as my intention is to find out what the effect is for the dummy variable (type 1), and whether it is significant. I am aware that adding interaction terms gives me no more chance to directly interpret the coefficient and the significance, but I could hardly imagine how I can still make any valid assumptions with this amount of interaction terms?

If there are interactions, then not including them would lead to invalid conclusions.

Just as a check of my knowledge, if the interaction term lets say dummy1 x education comes out significant over this entire interaction model, can I then correctly assume that the coefficient shows the difference in the effect the predictor education has on the dependent variable for all type 1 when compared to type 2?

I'm not sure what you mean by "entire interaction model" but I think you have the right idea (although I can't be certain).