 Solution

Assume $S_{xx}={\sum }_i{x}_i^2>0$, which is necessary to identify the slope in linear regression through the origin. The joint normal likelihood has logarithm, apart from a constant,Complete the square in $\beta$. With $\hat{\beta }={\sum }_i{x}_i{Y}_i/S_{xx}$ and $\mathrm{RSS}={\sum }_i(Y_i-\hat{\beta }{x}_i{)}^2$,The maximum likelihood estimation therefore first minimizes at $\hat{\beta }$; differentiating in ${\sigma }^2$ then maximizes atThis is a genuine finite maximum when $\mathrm{RSS}>0$. If all $x_i=0$, the slope is unidentifiable. If $\mathrm{RSS}=0$, the likelihood is unbounded as ${\sigma }^2\downarrow 0$, so no positive-variance maximum exists. For nonzero design and $n\ge 2$, zero residual has probability zero under the stated positive-variance model.