Suppose you observe a random sample of n observations (y,x) from a population that satisfies y=xβ+ε with E[ε |x|]=0 and E[εε^T|x|]=σ^2I_n. Let hat{eta} denote the least squares estimate of β based on the sample (y,x). Now you observe the vector of independent variables, x_{**}^T, 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_{**}^That{eta} 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 ε|x∼N(0,σ^2I_n). Derive a test statistic (and its sampling distribution) to test the null hypothesis that y_{**}=y_0.
5. Suppose you observe a random sample of n observations (y,X) from a population that satisfies y=X+e with E[e|X]=0 and E[ee'|X]=2I_n. Let hat{eta} denote the least squares estimate of β based on the sample (y,X). Now you observe the vector of independent variables, ,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 y. = ' denote the OLS prediction of y.
(a) Is y. an unbiased estimator of y.? Explain/prove your claim.
(b) What is the variance of the OLS prediction error, y* - y.?
(c) What is the best linear (in y) unbiased estimator of y.? Explain/prove your claim.
(d) Now suppose that X ~ N(0,σI_n). Derive a test statistic (and its sampling distribution) to test the null hypothesis that y. = 10.