Show that beta is an unbiased estimator of beta and obtain

Question

1. Consider the linear regression model

y = X_β + u,

where y is an n x 1 vector of observations on the dependent variable, X is an n x k matrix of observations on k non-stochastic explanatory variables, β = (β₁,...................,β_k) is a k x 1 vector of unknown coefficients, u is an n x 1 vector of unobservable random disturbances such that E(u) = 0 and E(uu') = σ²I_n, σ² is an unknown positive constant, and I_n is the it n x n identity matrix. Let β' = (X'/X)^-1X'y be the ordinary least squares (OLS) estimator of β.

(a) What assumption about X guarantees the existence of β'? Give an example where this assumption fails.

(b) Define the OLS residuals as u^ = y XB^. Show that X'u^ = 0.

(c) Show that β^ is an unbiased estimator of β and obtain its variance-covariance matrix. How can the standard errors of the elements of β^ be estimated?

(d) Let β = Cy be an unbiased estimator of β, where C is a non-stochastic k x it matrix. Is there a matrix C for which le  is a more efficient estimator than β? Explain.

(e) How do your answers to (c) and (d) change if E(uu') = σ²Ω, where Ω is a symmetric and positive definite it n x n matrix (with Ω ≠ I_n)? Explain in detail.