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

The least-squares estimators choose the intercept and slope to minimize the sum of squared regression residuals:After observation, replace $Y_i$ by $y_i$ to obtain the numerical estimates. Their definition does not require a normal distribution for the errors. A unique slope requires some variation in the fixed predictors; under ${\sum }_i{x}_i=0$, this is $S_{xx}={\sum }_i{x}_i^2>0$, assumed below. If all $x_i=0$, the slope is unidentifiable.