Chapter 11.12, Problem 71E

Consider the general linear model

$Y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\cdots +{\beta }_k{x}_k+\epsilon$,

where E(ε) = 0 and V(ε) = σ². Notice that ${\hat{\beta }}_i=a^{\prime }\hat{\beta }$, where the vector a is defined by

$a_j=\{\begin{array}{ll}1,\hfill & \text{if}j=i,\hfill \\ 0\hfill & \text{if}j\ne i.\hfill \end{array}$

Use this to verify that E( ${\hat{\beta }}_i$) = β_i and V( ${\hat{\beta }}_i$) = c_iiσ², where c_ii is the element in row i and column i of (X/X)⁻¹.