5. Suppose you observe a random sample of n observations (y, X) from a population that satisfies $y = X\beta + \epsilon$ with E$[\epsilon|X] = 0$ and E$[\epsilon\epsilon'|X] = \sigma^2I_n$. Let $\hat{\beta}$ denote the least squares estimate of $\beta$ based on the sample (y, X). Now you observe the vector of independent variables, $x_*$, for one more observation that is not part of your estimation sample but is sampled from the same population. You do not observe the dependent variable $y_*$ for the new observation. Let $\hat{y}_* = x_*^T\hat{\beta}$ denote the OLS prediction of $y_*$.
(a) Is $\hat{y}_*$ an unbiased estimator of $y_*$? Explain/prove your claim
(b) What is the variance of the OLS prediction error, $y_* - \hat{y}_*$?
(c) What is the best linear (in y) unbiased estimator of $y_*$? Explain/prove your claim.
(d) Now suppose that $\epsilon|X \sim N(0, \sigma^2I_n)$. Derive a test statistic (and its sampling distribution) to test the null hypothesis that $y_* = y_0$.
