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

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Past exam of the options I course of the University of Oxford / 2012 / O / O2 / b / i / Solution by

Codex  0 2026-10-31

The design columns are

1

,

x

and

z

. Their pairwise inner products are zero by the three supplied sums. Put

S_{xx} = \sum_{i} x_{i}^{2}

and

S_{zz} = \sum_{i} z_{i}^{2}

, positive under the full-rank assumption. Then

X^{T} X = diag (n, S_{xx}, S_{zz}) .

Applying part (a),

θ = α β γ = \overline{Y} \frac{\sum _{i} x _{i} Y _{i}}{S _{xx}} \frac{\sum _{i} z _{i} Y _{i}}{S _{zz}}, Var (θ) = σ^{2} diag (\frac{1}{n}, \frac{1}{S _{xx}}, \frac{1}{S _{zz}}) .

The off-diagonal covariances vanish because the design columns are orthogonal. With Gaussian errors, the coefficient estimators are also jointly normal and therefore independent.

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