 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ib/paper-2/18h/e/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

X

makes

X^{T} : R^{n} \to R^{p}

surjective. Given

x \in R^{p}

, choose

u

with

x = X^{T} u

. The fitted values for the unrestricted and restricted regressions are

X β = P Y, X β = P_{0} Y .

Therefore

x^{T} β = u^{T} P Y, x^{T} β = u^{T} P_{0} Y .

Using

cov (Y) = σ^{2} I_{n}

and idempotence of the projections,

var (x^{T} β) = σ^{2} u^{T} P u, var (x^{T} β) = σ^{2} u^{T} P_{0} u .

Part (d) now gives

var (x^{T} β) \leq var (x^{T} β) .

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