9.6 Understanding Uncertainty

Under the relatively stricter assumptions of constant error variance, the variance of a slope coefficient is given by

\[ V(\hat{\beta_j}) = \frac{\sigma^2}{SST_j (1-R_j^2)} \]

Definition 9.1 A similar formulation is given in FOAS as definition 4.2.3,

\[ \hat{V}_{C}[\hat{\beta}] = \hat{\sigma}^2 \left( X^{T} X \right)^{-1} \rightsquigarrow \hat{\sigma}^{2}{\left(\mathbb{X}^{T}\mathbb{X}\right)}, \] where \(\hat{\sigma}^{2} = V[\hat{\epsilon}]\)

Explain why each term makes the variance higher or lower:

• \(\hat{\sigma}^2\) is the variance of the error \(\hat{\epsilon}\)
• \(SST_j\) is (unscaled) variance of \(X_j\)
• \(R_j^2\) is \(R^2\) for a regression of \(X_j\) on the other \(X\)’s