 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2012/o/o2/a/ii/solution

Independence and common error variance give

Var (ϵ) = σ^{2} I_{n}

. For the fixed matrix

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

, the covariance transformation rule gives

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

Thus the covariance is determined by the error variance and the inverse Gram matrix of the design columns.

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