Discussion
Consider the linear model \(\underline{Y}_{n-1} = X\underline{\theta}_{p-1} + \xi\) with \(E(\xi) = \underline{0}\) \(Cov(\xi) = \sigma^{2}I_{n}\). Then which of the following statements is always true ?
Answer: D
The always-true statement is that l'θ is estimable only for l in the column space of X'.
A linear function \(\underline{l}'\underline{\theta}\) is estimable if some linear function \(\underline{a}'\underline{Y}\) has expectation \(\underline{l}'\underline{\theta}\) for all \(\theta\). Since \(E(\underline{a}'\underline{Y}) = \underline{a}'X\underline{\theta}\), this needs \(\underline{l} = X'\underline{a}\), that is, \(\underline{l}\) must lie in the column space of X' (the row space of X).
If X does not have full column rank, \(\underline{\theta}\) itself is not estimable, it has no BLUE, and some \(\underline{l}'\underline{\theta}\) are not estimable. So the other three statements hold only in the full rank case, not always.
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