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I have a question concerning the interaction with dummy variables.

Consider the following OLS regression

$Y_{it} = b_1 x_{it} + b_2 (dum_{it} * x_{it}) + e_{it}$ .

In some studies (not all) that I have seen, they include the dummy variable as well. Thus, the equation becomes

$Y_{it} = b_1 x_{it} + (b_2 dum_{it} * x_{it}) + b_3 dum_{it} + e_{it}$ .

What are exactly the pros and cons of including / excluding the dummy variable? Or, is there a need to include the variable?

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Standard practice is to include the dummy variable as well. Using your equations, we could say that x represents age, dum represents sex (0 = female, 1 = male) and Y represents income. In the first equation, we would get an estimate of the effect of age (b1) on income and an estimate of the difference in effect of age on income. In the second equation, we will also get an estimate of the effect of sex on income. A major reason to include the effect of sex on income, even if we might not be at all interested in that effect, is that if there is a difference, we will get a more accurate estimate of the regression coefficient of the interaction.

If we do not include the main effect of sex (as in your first equation), the specific effect of sex on income will distort the estimate of the effect of the interaction.

I haven't heard of any advantages of leaving out the main effect unless you are certain that there is no effect (in which case it shouldn't matter if you leave it out) and you have a small sample size so that the number of parameters you can estimate is a serious limitation.

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In most cases, it is not adequate to include an interaction without all the variables in the modell, because those interactions become difficult to interpret. With dummy variables, it is a bit different. In your first model, you allow the linear modell to have two different slopes for the two groups that differ in the dummy. However, you do not allow them to have different intercepts. This may be usefull in special cases but if there is no special reason, allow the model to compute different intercepts. If the different intercept term is significant, you are happy to have seen this. It makes your models description of reality better. If it is not significant, nothing is lost. So the only reason I could think of the first version is, if you have a very small n and strong reasons to believe, that the to groups share a common intercept. The only downside is, that you will have to compute more coefficients, which can become an issue if you have a small n and many dummys.

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