The Effect of Including a Constant Column in OLS

Prerequisites: Understanding partitioned least squares and its solution.

Setup

Given the linear model:

We can partition the columns of $X$ into $X_1$ and $X_2$ to get:

Partitioned Least Squares Result

Recall that $\pmb b_2 = (\tilde X_2' \tilde X_2)^{-1} \tilde X_2'\tilde{\pmb y}$ and that each variable with a tilde is the result of being partialled out by $M_1 = I - X_1(X_1'X_1)^{-1}X_1'$.

Applying result to inclusion of a column of 1's

Consider the case where $X_1 = \pmb \iota$, then each column of $\tilde X_2$ is the form:

We also have $M_0 \pmb y$ so $\tilde{\pmb y} = \ddot{\pmb y}$ so all variables in the regression are effectively converted to deviations from their means in the regression. (See here for more on $M_0$)

Which brings us to the result including a coefficient has on the solution to regression:

Deriving the coefficient on $\pmb \iota$

Recall that $\pmb b_1 = (X_1'X_1)^{-1}X_1'(y - X_2 \pmb b_2)$.

Also, notice that $\pmb b_1$ is just a a scalar $b_1$ (because $X_1$ is just one column).

We have: