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Past exam of the mathematics course of the University of Oxford

2017

Prelims

Mathematics III

{c}

Substitute the <linear regression model> into the estimators and use the centring and <orthogonality> conditions:
$$
\widehat\beta_0=\beta_0+\frac1n\sum_i\epsilon_i,\qquad
\widehat\beta_1=\beta_1+\frac{\sum_i x_i\epsilon_i}{S_{xx}}.
$$
A linear combination of <independent> <normal random variables> is normal. Their <means> are $\beta_0$ and $\beta_1$, while <variance additivity for independent random variables> gives
$$
\operatorname{Var}(\widehat\beta_0)=\frac{n\sigma^2}{n^2}=\frac{\sigma^2}{n},\qquad
\operatorname{Var}(\widehat\beta_1)=\frac{\sigma^2\sum_i x_i^2}{S_{xx}^2}=\frac{\sigma^2}{S_{xx}}.
$$
Consequently
$$
\boxed{\widehat\beta_0\sim N(\beta_0,\sigma^2/n),\qquad
\widehat\beta_1\sim N(\beta_1,\sigma^2/S_{xx}).}
$$

Solution

Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 8 / a / ii / Solution

Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 8 / a / ii / Solution

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