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

Treat the explanatory variables as fixed and put $S_{xx}={\sum }_i{x}_i^2$, $S_{zz}={\sum }_i{z}_i^2$. To identify both slopes, assume $S_{xx}>0$ and $S_{zz}>0$. The given centring and orthogonality conditions make the three columns $1,x,z$ pairwise orthogonal.

The log-likelihood function in the normal linear model, up to a constant independent of the coefficients, isThus maximum likelihood estimation minimizes the residual sum of squares. Setting its partial derivatives to zero givesUsing $\sum x_i=\sum z_i=\sum x_i{z}_i=0$ decouples these equations:The Hessian matrix of the residual sum of squares is $2\mathrm{diag}(n,S_{xx},S_{zz})$, which is positive definite. These are therefore the unique minimizers. If either squared sum is zero, the associated explanatory variable vanishes identically and its coefficient cannot be identified.

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