= Solution
{c}
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, is
$$
\ell(\beta_0,\beta_1,\beta_2)=-\frac1{2\sigma^2}\sum_i(Y_i-\beta_0-\beta_1x_i-\beta_2z_i)^2.
$$
Thus <maximum likelihood estimation> minimizes the <residual sum of squares>. Setting its <partial derivatives> to zero gives
$$
\sum_i(Y_i-\beta_0-\beta_1x_i-\beta_2z_i)=0,\quad
\sum_i x_i(Y_i-\beta_0-\beta_1x_i-\beta_2z_i)=0,\quad
\sum_i z_i(Y_i-\beta_0-\beta_1x_i-\beta_2z_i)=0.
$$
Using $\sum x_i=\sum z_i=\sum x_iz_i=0$ decouples these equations:
$$
\boxed{\widehat\beta_0=\overline Y,\qquad
\widehat\beta_1=\frac{\sum_i x_iY_i}{S_{xx}},\qquad
\widehat\beta_2=\frac{\sum_i z_iY_i}{S_{zz}}.}
$$
The <Hessian matrix> of the <residual sum of squares> is $2\operatorname{diag}(n,S_{xx},S_{zz})$, 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.
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