Discussion
Consider the multiple regression set-up \(\{Y, X\beta, \sigma^{2}I\}\) with \(p\)-regressors. The ordinary least squares estimator (OLSE) \(\hat{\beta}\) is the vector which minimizes \(E(Y - X\beta)(Y - X\beta)'\). Which of the following statements is not true under this set-up ?
Answer: C
The statement that is not true is \(\hat{\beta}\) is unique if and only if rank(X) \(\lt p\).
The normal equations \(X'X\hat{\beta} = X'Y\) always have a solution, and the fitted vector \(X\hat{\beta}\) is always unique, being the orthogonal projection of Y onto the column space of X (A). The solution \(\hat{\beta}\) is unique exactly when \(X'X\) is invertible, that is when rank(X) = \(p\) (B). When rank(X) \(\lt p\) there are infinitely many solutions, which is the opposite of C.
Even then, a linear function \(K'\hat{\beta}\) is the same for every solution precisely when \(K'\beta\) is estimable, which is D.
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