Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 9 / a / i

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- Solution i

Solution

Minimize the residual sum of squares

Q (b_{0}, b_{1}, b_{2}) = \sum_{i} (Y_{i} - b_{0} - b_{1} x_{i 1} - b_{2} x_{i 2})^{2} .

Setting its three partial derivatives to zero gives the normal equations

i \sum (Y_{i} - \hat{β}_{0} - \hat{β}_{1} x_{i 1} - \hat{β}_{2} x_{i 2}) = 0, i \sum x_{ij} (Y_{i} - \hat{β}_{0} - \hat{β}_{1} x_{i 1} - \hat{β}_{2} x_{i 2}) = 0 (j = 1, 2) .

These equations characterize the linear least squares minimizers, since

Q

is a convex quadratic. A unique coefficient vector additionally requires full column rank of the design matrix.

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