1) Suppose that in the population, the dependent variable Y is related to the explanatory variable X and the unobserved variable e, Y = atBX +E where a and B are the unknown population intercept and the slope coefficient respectively. You have a random sample of size n, {(Xi, Yi) : i = 1, ...,n}, such that realizations of Xi and Y; for i = 1, ...,n are not all the same. You seek estimators for a and B. i) What problem do ordinary least squares estimators a and B solve? State this problem mathematically. ii) Derive the first order conditions that a and 3 satisfy. iii) The OLS estimator B is given by, B = Li(Xi - mx) Yi Et (Xi - mx)? Show that S can be rewritten, B = B+ Lil(X, -mx)ci Er(Xi -mx)2 Hint: Remember CT(Xi - mx) =0 and Et(Xi - mx)? = Et(Xi -mx)Xi. iv) Assume that E(c|X) = 0. Since {(Xi, Yi) : i = 1, ...,n} is a random sample, this implies that E(EX1, X2, . .., Xn) = 0 for i = 1, 2, ... n. Use your answer from (iii) to compute E(BIX1, ..., Xn) under this assumption. v) Use the law of iterated expectations to show that B is an unbiased estimator for B when the assumption E(EX ) = 0 holds. vi) Assume further that Var(EX) = o'. Since {(X, Yi) : i = 1, ...,n} is a random sample, this implies that Var(e|X1, X2, ..., Xn) = o' for i = 1, ...,n. Compute Var(BIX1, . .., Xn) under these assumptions