Central Limit Theorem follows from the Crameer-Wold Device. This result states the fol- lowing. Let {b, } be a sequence of random k x 1 vectors. If A b,, converges to a normal random variable for every k x 1 constant non-zero vector A, then b,, converges to a multi- variate normal random variable.Consider the Linear Regression Model 1. For any X = x, let Y = x8 + U, where B E R*. 2. X is exogenous. 3. The probability model is { f(u; ) is a distribution on R : Ey[U] = 0, Var, [U] = 02, 0 > 0}. 4. Sampling model: {YT, is an independent sample, sequentially generated using Y, = x;8+ U. i = 1, ...,n, where the U, are a IID(0, (?). 1.1 Theoretical Part Let X be the n x k matrix of regressors, and let Y be the n x 1 vector of the dependent variable. 1. Compute Bus by minimizing (Y - XB) (Y - XB), where B ERk. 2. Prove that BLS - B, V(B, 0) ER* x R. 3. Prove that vn (BLS - B) is asymptotically distributed as multivariate normal with zero mean, and report the asymptotic covariance matrix. Hint: The multivariate version of theConsider the regression model with no intercept Yi = BXite, i= 1, 2, ..., n. B is an unknown parameter. X1, ..., Xnare given constants.6, i = 1, .., n are iid random variables withN (0, o'). Assume o' is unknown. a.) Find MLE for B and o'. b. ) Find distribution for B(hat). How can use this to a confidence interval for B. c.) Find a prediction interval for Y (hat) when = x. d.) Find an unbiased estimator for of edit. In part c.) Find a prediction for Y(hat) when X=x. edit. part b.) "how can we use this to find find a confidence interval."Amanda's utility function is Leontief and given by Wa(Xarya) = min(xa. yal while Brenda's utility is linear and given by us(xb. )b) = XD + VD Their endowments are wa = (2,3) and wo = (4, 1). Show this economy in an Edgeworth Box Show the Individually Rational allocations and show in Edgeworth Box Find the Pareto Efficient allocations for this economy and show in the Edgeworth Box. Show the core of the economy. Find the Walrasian Equilibrium of the economy and show in your Edgeworth box. Prove directly that the First Welfare Theorem holds Does the Second Welfare Theorem hold for allocation ((3,3)a, (3,1),}? If so how?Consider a pure exchange economy with two individuals (Adam and Bob) and two goods (x and y). Their utility functions are given by UA(TA, yA) = JAYA and UB(TB, yB) 0.5 0.5 = "ByB Initially, Adam has 100 units of good x and Bob has 100 units of good y (a) Find the general equilibrium price ratio px/py and the general equilibrium allocation (their demand bundles in the equilibrium). Illustrate this general equilibrium allocation using Edgeworth box. (b) Find all Pareto efficient allocations in this economy, and illustrate all these Pareto efficient allocations in the Edgeworth box. (c) The first theorem of welfare economics states that the general equilibrium allocation is Pareto efficient. Is the equilibrium allocation you found in (a) efficient and why? Show transcribed image text4. Consider a two person {1 and 2], two good {A and B] exchange economy. Social welfare forthe economy i5: W=U1+U2. a. Draw social indifference curves for the economy in question. What is the rate at which society is willing to trade one person's welfare for another? b. Suppose that person 1's utility function is U1=A+2El and person 2's utility function is U2=3A+EL Suppose that each person is endowed with 15 units of goods A and B {making a total of 3!] units in the economy]. Determine the optimal allocation in the economy, underthe condition that both individuals receive at least as much utility as their endowed consumption bundle. NOTE The marginal utitr'es for person I are: Muzl, MUB=2 The marginal utilities for person .2 are: MUAzi Muazl. c. Suppose social welfare is now characterized as: Wzmin[U1,U2], re-do part B. Describe how the optimal allocations change and why. 1.2 Simulation Part Let k = 3 where X = [1, X1, X2] with regression parameters given by (8o, B1, 82), respectively. Suppose we wish to test the hypothesis that one parameter is equal to the reciprocal of the other: Ho : g" ( B1, B2) = B1 - 1/B2 = 0. (1) An alternative form to the above hypothesis which is algebraically equivalent under the null hy- pothesis is Ho : 95 (81, B2) = B1B2 - 1 = 0. (2) 1. Calculate the Wald statistic for each of the above forms of restriction that defines the null hypothesis. Are these statistics identical ? 2. In the simulation exercise we will compare the relative performance of the two Wald statis- tics. Consider the following DGPs in Ho with X, and X2 obtained using