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 Incoming links: Linear regression model

Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 8 / a / ii / Solution 2026-10-25

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}} .

A linear combination of independent normal random variables is normal. Their means are

β_{0}

and

β_{1}

, while variance additivity for independent random variables gives

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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Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 9 / a / iii / Solution 2026-10-25

If the two regressors are strongly associated, the data may provide little support for varying one while holding the other fixed. Multicollinearity makes the separate slope estimates unstable, and exact linear dependence makes them unidentifiable. A conditional slope may also be an extrapolation beyond combinations actually observed.

Furthermore, a slope in a linear regression model is not automatically a causal effect. A causal interpretation needs appropriate design or assumptions excluding confounding; independence of the errors across observations and zero mean alone do not establish those assumptions. The coefficients describe the specified linear mean relation, but changing a regressor by intervention can have a different interpretation.

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