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

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- Solution ii

Solution

The centered design gives

β = β + S_{xx}^{- 1} \sum_{i} x_{i} ϵ_{i}

. By independence of the errors, their off-diagonal covariances vanish. Thus variance additivity for independent random variables gives

Var (β) = \frac{\sum _{i} x _{i}^{2} σ ^{2}}{S _{xx}^{2}} = \frac{σ ^{2}}{S _{xx}} .

A larger spread of predictor values decreases the slope variance for fixed error variance.

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