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PLEASE DO PART 5,6,7,8 !!!!
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PLEASE DO PART 5,6,7,8 !!!!
PLEASE DO PART 5,6,7,8 !!!!
Exercise 2. Recall the (population) covariance of two random variables X, Y is given by Cov(X,Y)= E((X - E(X)) (Y - E(Y))), or equivalently, Cov(X,Y) = E(XY) - E(X)E(Y). When X = Y, we simply have the variance of the random variable, say Cov(x,x) = Var(X). If we don't know the covariance or variance of the two random variables, we can estimate it given a sample of X1,Y1, X2, Y2,..., Xn, Y, by the sample covariance (or sample variance when X = Y) which is given by Sxy = ΣΧ - X) (Υ. - Y), n 1 i=1 where X = 12-1X,Y= 12-1Y; are the sample means of X and Y. (i) Suppose c is a constant, show that Cov(c, X) = 0 for any random variable X. (ii) Let a, b be two constants, X, Y, Z are three random variables, show that Cov(Z, aX+Y) = aCov(Z,X) + b Cov(Z,Y). 1 (iii) Suppose a random variable u has mean zero, i.e., E(u) = 0. Prove that Cov(X, u) = E(Xu). (iv) Suppose that E (u|X) = 0. Show that Cov(X, u) = 0. (Hint: You may use the law of iterated expectation and the result from Part (iii)). (v) Now consider a simple linear regression model Y = Bo + BiX+u. Assume the first assumption of least square holds, that is, E (u X) = 0. Show that Cov(X,Y) B1 = Var(X) (Hint: First calculate the Cov(Y,X)—you may use the results in previous step and the useful formulas in part (i) and (ii)). (vi) Based the result in Part (v), use the sample covariance of X and Y and sample vari- ance of X to construct an estimator of B1. (vii) Now suppose we have omitted some variables and consequently we have Cov(X, u) 70. Show that Cov(X, u) B1 = B1+ Var(X) where B1 = Cov(X,Y) In fact, you have proved the omitted variable bias formula intro- (x) duced in class. (viii) Assume there is another variable Z, satisfying Cov(X,Z) # 0, Cov(Z,u) = 0. Show that Cov(Y,Z) B1 = Cov(X,Z) Based on the result above, use the sample covariances about X, Y and Z to construct an estimator of B1. (Note that we don't assume Cov(X, u) = 0 or E (UX) = 0 here. Such Z variable is called the instrumental variable, which helps construct a consistent estimator when the assumption E (UX) = 0 is violated thus OLS fails to work.)
Exercise 2. Recall the (population) covariance of two random variables X, Y is given by Cov(X,Y)= E((X - E(X)) (Y - E(Y))), or equivalently, Cov(X,Y) = E(XY) - E(X)E(Y). When X = Y, we simply have the variance of the random variable, say Cov(x,x) = Var(X). If we don't know the covariance or variance of the two random variables, we can estimate it given a sample of X1,Y1, X2, Y2,..., Xn, Y, by the sample covariance (or sample variance when X = Y) which is given by Sxy = ΣΧ - X) (Υ. - Y), n 1 i=1 where X = 12-1X,Y= 12-1Y; are the sample means of X and Y. (i) Suppose c is a constant, show that Cov(c, X) = 0 for any random variable X. (ii) Let a, b be two constants, X, Y, Z are three random variables, show that Cov(Z, aX+Y) = aCov(Z,X) + b Cov(Z,Y). 1 (iii) Suppose a random variable u has mean zero, i.e., E(u) = 0. Prove that Cov(X, u) = E(Xu). (iv) Suppose that E (u|X) = 0. Show that Cov(X, u) = 0. (Hint: You may use the law of iterated expectation and the result from Part (iii)). (v) Now consider a simple linear regression model Y = Bo + BiX+u. Assume the first assumption of least square holds, that is, E (u X) = 0. Show that Cov(X,Y) B1 = Var(X) (Hint: First calculate the Cov(Y,X)—you may use the results in previous step and the useful formulas in part (i) and (ii)). (vi) Based the result in Part (v), use the sample covariance of X and Y and sample vari- ance of X to construct an estimator of B1. (vii) Now suppose we have omitted some variables and consequently we have Cov(X, u) 70. Show that Cov(X, u) B1 = B1+ Var(X) where B1 = Cov(X,Y) In fact, you have proved the omitted variable bias formula intro- (x) duced in class. (viii) Assume there is another variable Z, satisfying Cov(X,Z) # 0, Cov(Z,u) = 0. Show that Cov(Y,Z) B1 = Cov(X,Z) Based on the result above, use the sample covariances about X, Y and Z to construct an estimator of B1. (Note that we don't assume Cov(X, u) = 0 or E (UX) = 0 here. Such Z variable is called the instrumental variable, which helps construct a consistent estimator when the assumption E (UX) = 0 is violated thus OLS fails to work.)