 Solution

ID: past-exam-of-the-probability-statistics-and-options-ii-course-of-the-university-of-oxford/2011/o/o5/b/solution

Write

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

. Since

A X = I

, the estimator is

θ = θ + A ϵ

. Therefore

E θ = θ, Var (θ) = A (σ^{2} I) A^{T} = σ^{2} (X^{T} X)^{- 1} .

Its expectation proves unbiasedness. It is also jointly normal, being an affine transformation of the normal error vector.

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