OLS Residual Orthogonality and Decomposition

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Linear Regression.

Your Task
Let $y \in \mathbb{R}^n$ and $X \in \mathbb{R}^{n \times p}$ with $\text{rank}(X) = p$. The Ordinary Least Squares (OLS) estimator is $\hat{\beta} = (X^T X)^{-1} X^T y$.
(1) Prove that the residual vector $e = y - X\hat{\beta}$ is strictly orthogonal to the column space of $X$, i.e., $X^T e = 0$.
(2) Show that if $X$ contains an intercept (constant column), the sum of residuals is exactly zero ($\sum_{i=1}^n e_i = 0$), and prove the total sum of squares decomposition $\text{TSS} = \text{ESS} + \text{RSS}$.

Constraints

• The design matrix $X \in \mathbb{R}^{n \times p}$ has full column rank $p < n$.
• Disturbance terms satisfy standard zero-mean conditional homoskedasticity unless stated otherwise.