Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 9 / a / ii / Solution

The centered regressors make the first normal equation read $n{\hat{\beta }}_0={\sum }_i{Y}_i$. DefineThe remaining equations are $A{\hat{\beta }}_1+B{\hat{\beta }}_2=u$ and $B{\hat{\beta }}_1+C{\hat{\beta }}_2=v$. Provided $D=AC-B^2>0$, matrix inversion givesBy the Cauchy-Schwarz inequality, $D\ge 0$. If $D=0$, the two regressor columns are linearly dependent, and their separate slopes are not uniquely identifiable. One still has well-defined linear least squares fitted values, but cannot use the displayed divisions by $D$.