Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 7 / b / i

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- Solution i

Solution

Differentiate the ordinary least squares objective in a. Its normal equations for linear least squares are

\sum_{i} (Y_{i} - α - β x_{i}) = 0, \sum_{i} x_{i} (Y_{i} - α - β x_{i}) = 0.

Using

\sum_{i} x_{i} = 0

and

S_{xx} = \sum_{i} x_{i}^{2} > 0

gives

α = \overline{Y}, β = \frac{\sum _{i} x _{i} Y _{i}}{S _{xx}} .

The Hessian matrix of the objective in

(a, b)

is

2 diag (n, S_{xx})

, which is positive-definite, so this is its unique global minimum. For the observed estimates, replace

Y_{i}

by

y_{i}

.

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Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 7 / c / iii / Solution

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