Show that the least-squares estimate of the conditional distribution of \(Y\) given \(Z\) is

\[
N_{n}\left(Q Y+P \hat{\mu}, s^{2} P Vight)
\]

for some scalar \(s^{2}\). Show that the least-squares estimate is singular and is supported on the \(k\)-dimensional coset \(Q Y+\mathcal{K}\). Explain why self-consistency requires \(K Q=\) \(K\).