Interactions Among Dummy Variables
Quantitative regressors in regression models often have an interaction among each other. In the same way qualitative regressors, or dummies, can also have interaction effects between each other and these interactions can be depicted in the regression model. For example,in a regression involving determination of wages, if 2 qualitative variables are considered, namely, gender and marital status,there could be an interaction between marital status and gender; in the sense that, there will be a difference in wages for female-married and female-unmarried. These interactions can be shown in the regression equation as illustrated by the example below.
Suppose we consider the following regression, where Gender and Race are the 2 qualitative regressors and Years of education is the quantitative regressor:
- Yi = β1 + β2D2 + β3D3 + αXi + Ui
where,
- Y = Hourly Wages (in $)
- X = Years of education
- D2 = 1 if female, 0 otherwise
- D3 = 1 if non-white and non-Hispanic, 0 otherwise
There may be an interaction that occurs between the 2 qualitative variables, D2 and D3. For example, a female non-white / non-Hispanic may earn lower wages than a male non-white / non-Hispanic. The effect of the interacting dummies on the mean of Y may not be simply additive as in the case of the above examples, but multiplicative also, such as:
- Yi = β1 + β2D2 + β3D3 + β4(D2D3) + αXi + Ui
From this equation, we obtain:
Mean wages for female, non-white and non-Hispanic workers:
- E(Yi|D2 = 1, D3 = 1, Xi) = (β1 + β2 + β3 + β4) + αXi
Here,
- β2 = differential effect of being a female
- β3 = differential effect of being a non-white and non-Hispanic
- β4 = differential effect of being a female, non-white and non-Hispanic
The model shows that the mean wages of female non-white and non-Hispanic workers differs by β4 from the mean wages of females or non-white / non-Hispanics. Suppose all the 3 differential dummy coefficients are negative, it means that, female non-white and non-Hispanic workers earn much lower mean wages than female OR non-white and non-Hispanic workers as compared to the base group, which in this case is male, white or Hispanic (omitted category).
Thus, an interaction dummy (product of two dummies) can alter the dependent variable from the value that it gets when the two dummies are considered individually.
Read more about this topic: Dummy Variable (statistics)
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