With k-1 dummy variables, why do we need a constant?

During lesson 5, Jeremy mentioned that when setting up dummy variables for tabular data, we include a constant for k-1 dummy variables. Meanwhile, if you have k dummy variables, you don’t need a constant. Why do we have to have a constant for k-1 dummy variables?


Let’s consider an example where we have a table of values with dependent variable income (y) and independent variables age (x1) and marital status (x2). age takes numerical values and marital status can be single, married, or divorced.

Let us introduce k = 3 dummy variables for marital status - x21 = 1 if single, x22 = 1 if married, and x23 = 1 if divorced. Our regression function will look like:

y = a1*x1 + a21*x21 + a22*x22 + a23*x23

If we choose to use k - 1 = 2 dummy variables, we can eliminate x23 from above equation to rewrite it as:

y = a1*x1 + (a21 - a23)*x21 + (a22 - a23)*x22 + a23
or, y = a1*x1 + a21'*x21 + a22'*x22 + a23

Now a23 coefficient becomes the bias which we’ll have to estimate.