 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii/8/a/ii/solution

Substitute the linear regression model into the estimators and use the centring and orthogonality conditions:

β_{0} = β_{0} + \frac{1}{n} \sum_{i} ϵ_{i}, β_{1} = β_{1} + \frac{\sum _{i} x _{i} ϵ _{i}}{S _{xx}} .

β_{0}

and

β_{1}

Var (β_{0}) = \frac{n σ ^{2}}{n ^{2}} = \frac{σ ^{2}}{n}, Var (β_{1}) = \frac{σ ^{2} \sum _{i} x _{i}^{2}}{S _{xx}^{2}} = \frac{σ ^{2}}{S _{xx}} .

Consequently

β_{0} \sim N (β_{0}, σ^{2} / n), β_{1} \sim N (β_{1}, σ^{2} / S_{xx}) .

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