help solve these complex problems plz

Problem 1. (36 points.) Consider the linear regression model Y = XB+ with pre-determined (i.e., non-stochastic) regressors X. All of the usual as- sumptions hold except, instead of Var(E) = 02In, we have Var(E) = E(EE') = La, where E, is a generic symmetric positive definite matrix. In other words, we are assuming that the errors are not necessarily homoskedastic and un- correlated. (1) (4 points) The traditional least-squares estimator is BLS = (XX)-X'Y. Define, now, a new estimator BGLS = (X'S,]X) 'X'S,ly. Call it "generalized least-squares" (GLS) estimator. Show that E(BGLS) = B, i.e., BGLS is unbiased. (2) (6 points) Show that the variance of Bys under the new assumption is (XX)X'EX(X'X ) -1. (3) (6 points) Show that the variance of BGLS under the new assumption is (X_'X) -1. (4) (8 points) Show that the variance of BGLS is not larger than the vari- ance of the traditional least-squares estimator BLs. In other words, show that (XX)-1XEX(X'X)-1 - (XE-1X)-120.Hint: This is the same as showing (X'S1X)-(XX)(X'EX) '(X'X) 2 0. Write the expression in terms of a suitable "quadratic form" using a suitable "idempotent and symmetric matrix" and the proof is almost complete. The proof is very similar to the one showing that BLS is BLUE in the standard regression model. Assume, now, that the regression Y = XB +8 is in a time-series context. The error terms et are independent of r and modeled as an MA(1) process. Specifically, write Et = Ut - Out-1, where u, is a white noise process with mean zero and variance o?.4. )For the following total revenue and total cost functions of a firm: TR=220-0.5Q^2, TC= 1/3Q^3- 8.5Q^2+50Q+90 4a) Determine the level of output at which the firm maximizes its total profit. 4b) Determine the maximum profit that the firm could earn. 5) A firms total-revenue and total-cost functions are, TR=4Q , TC= 0.04Q^3-0.9Q^2+10Q+5 5a) Determine the best level of output. 5b) Determine the total profit of the firms at its best level of output. 6.) Given the following cost function, determine the level of (nonzero) output at which the cost function is minimized, and the level of the costs. AC= 200-Q^2 7.) Given the following cost function, determine the level of (nonzero) output at which the cost function is minimized, and the level of the costs. MC=200-48Q+3Q^2 9) The Warren & Smith Company manufactures commercial zippers of two kinds, kind X and kind Y. Its production department estimates that the average-cost function of the firm is: AC=X*2+2Y^2-2XY-2X-6Y+20 9a) The manager of the firm would like to know the level of output of zipper X and zipper Y at which the average cost of the firm is minimized, and the level of this minimum average cost. 9b) The firm expects an order that will require it to produce a total output of 6 units of both kinds of zippers (each unit may be a large number of zippers), and so the manager would also like to know how many of each type of zipper the firm must produce to minimize its average cost, and what its minimum average cost would be if it receives the order. The manager gives this assignment to two researchers who use different methods to obtaintheir answers. EViews: Given the Coffee price and quantity data, develop each of the items 2 - 4 requested above