 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/9/c/i/solution

Here both regressor columns are already centered, so

Z = X V

and

X = Z V^{T}

β = (β_{1}, β_{2})^{T}

Xβ = Z V^{T} β

. Thus

β = V^{T} β, β_{1} = V_{11} β_{1} + V_{21} β_{2}, β_{2} = V_{12} β_{1} + V_{22} β_{2} .

The intercept stays

β_{0}

, because centering made the transformation a rotation of the regressor columns without an added translation.

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