Chapter 6.6, Problem 14E

Let $\bar{x}=\frac{1}{n}\left(x_1+\cdots +x_n\right)$ and $\bar{y}=\frac{1}{n}\left(y_1+\cdots +y_n\right)$. Show that the least-squares line for the data (x₁, y₁), …, (x_n, y_n) must pass through ( $\bar{x}$, $\bar{y}$). That is, show that $\bar{x}$ and $\bar{y}$ satisfy the linear equation $\bar{y}={\hat{\beta }}_0+{\hat{\beta }}_1\bar{x}$. [Hint: Derive this equation from the vector equation y = X $\hat{\beta }$ + ε. Denote the first column of X by 1. Use the fact that the residual vector ε is orthogonal to the column space of X and hence is orthogonal to 1.]