Example: Adding a Binary Variable to the Model

Consider the following regression equation:

Where $X_1$ is a person's age and $\hat{Y}$ is their predicted income in 1000 euros.

Now suppose that, besides a person's age, you would also like to incorporate into the model whether or not the person has a Dutch nationality. This variable can take on two values: either you are Dutch or you aren't.

To incorporate this variable into the model, a dummy variable $X_2$ can be introduced, which takes on a value of $1$ if the person in question is Dutch and a value of $0$ if the person has another nationality.

Suppose the new model is described by the following regression equation:

Here $b_2=5$. So if you have a Dutch nationality, the model predicts you will earn $5000$ euros more than a person of the same age, but with a different nationality.

On the basis of this dummy variable, it is possible to construct two models: one for people with a Dutch nationality and one for people with another nationality.

• The predicted income of someone with a Dutch nationality is:
  • $\hat{Y_1}=9X_1-12+5\cdot1=9X-7$
• The predicted income of someone with a different nationality is:
  • $Y_2=9X_1-12+5\cdot0=9X-12$

Notice that both equations have the same regression coefficient but different intercepts. The distance between the two regression lines is equal to the coefficient of the dummy variable, $b_2=5$.

Example: Multiple Dummy Variables

Consider the following regression equation:

Where $X$ is a person's age and $\hat{Y}$ is their predicted income in 1000 euros.

Now suppose that, instead of treating age like a quantitative variable, you want to treat age like a categorical variable by grouping people into $4$ age groups: Child, Teen, Adult, and Elder.

Since there are $4$ age groups (levels), you need $k-1=4-1=3$ dummy variables:

• The variable $X_1$ is one if the person is a child and zero otherwise.
• The variable $X_2$ is one if the person is a teen and zero otherwise.
• The variable $X_3$ is one if the person is an adult and zero otherwise
• For an elderly person, $X_1, X_2$, and $X_3$ are all zero.

On the basis of these three dummy variables, you can construct four different regression models, one for each age group.