Multiple Linear Regression - 2 Independent Variables, one with the same values

Question

If I am running a multiple regression analysis with two independent variables (say, x1 and x2) and all the values of x1 are the same number, what issues does this present? I'm thinking in terms of Gauss-Markov assumptions.

Answer 1

Including a constant is equivalent to including a variable which is always equal to one:

Imagine you're interested in the regression:

$$ y_i = a + b x_i + \epsilon_i$$

An equivalent regression is to regress on two variables where $x_1 =1$ for all observations:

$$ y_i = a x_{1,i} + b x_{2, i} + \epsilon_i $$ Including a variable that is always equal to one is how you implement a constant in linear regression. For example, your data matrix X might be something like:

$$ X = \left[ \begin{array}{cc} 1 & 1.234 \\ 1 & .9374 \\ 1 & .7523 \\ 1 & 2.435 \\ 1 & 1.148 \end{array} \right] $$

What goes wrong if you include two constants...

If instead you had:

$$ X = \left[ \begin{array}{cc} 1 & 3 & 1.234 \\ 1 & 3 & .9374 \\ 1 & 3 &.7523 \\ 1 & 3 &2.435 \\ 1 & 3 &1.148 \end{array} \right] $$

This violates the Gauss-Markov assumption that the data matrix $X$ is full column rank. Here, $3x_1 = x_2$, they are linearly dependent, and the rank of $X$ is only 2.