 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2010/o/o2/a/solution

θ = (X^{T} X)^{- 1} X^{T} Y .

Write

A = (X^{T} X)^{- 1} X^{T}

. Since

A X = I

and

Cov (ϵ) = σ^{2} I

, we have

θ = θ + A ϵ

and

Cov (θ) = σ^{2} A A^{T} = σ^{2} (X^{T} X)^{- 1} .

This calculation needs the stated uncorrelated equal-variance errors, but not normality.

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