What is variance of $b$, the OLS estimator of $β$, when $u\sim N(A,σ²I)$?

Question

When $u\sim N(0,σ²I)$ I understand how to determine the Var$(b)=σ²(X'X)^{-1}$ however when $u\sim N(A,σ²I)$ I do not understand how to find the variance. $A$ is $n \times 1$.

Answer 1

Here's the answer for those interested:

Var(b)=E[(b-E[b])(b-E[B])']

Var(b)=E[(β+(X'X)^(-1)X'u-β-(X'X)^(-1)X'(A))(β+(X'X)^(-1)X'u-β-(X'X)^(-1)X'(A))']

Var(b)=E[((X'X)^(-1)X'u-(X'X)^(-1)X'(A))((X'X)^(-1)X'u-(X'X)^(-1)X'(A))']

Var(b)=E[((X'X)^(-1)X'(u-(A))((X'X)^(-1)X'(u-(A))']

Var(b)=E[(X'X)^(-1)X'(u-(A))(u-(A))'X(X'X)^(-1)]

E[(u-(A))(u-(A))']= Var(u), which is σ²I from $u\sim N(A,σ²I)$

Var(b)=σ²I(X'X)^(-1)X'X(X'X)^(-1)

Var(b)=σ²(X'X)^(-1)