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Stock and Watson, Exercise 2.3

Stock and Watson, Exercise 2.3. (a) Compute E(Y). We first need to find the density of y, p(y). Using formulas discussed in class (i.e., recovering the marginal density from the joint), we find Pr(Y=0) = .045 + .005 = .05 Pr(Y=1) = .709 + .241 = .95

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Stock and Watson, Exercise 2.3

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  1. Stock and Watson, Exercise 2.3 Econometrics 472

  2. Econometrics 472

  3. (a) Compute E(Y). We first need to find the density of y, p(y). Using formulas discussed in class (i.e., recovering the marginal density from the joint), we find Pr(Y=0) = .045 + .005 = .05 Pr(Y=1) = .709 + .241 = .95 These two numbers define the density function for the employment random variable. Econometrics 472

  4. Now, given the density function, it follows that E(Y) = 0 Pr(Y=0) + 1 Pr(Y=1) = Pr(Y=1) = .95 Econometrics 472

  5. (b) Show that the unemployment rate (the fraction of the labor force that is unemployed) is 1-E(Y). • Let n denote the total population of the labor force, and write n = nu + ne, where nu is the number unemployed and ne is the number employed. Econometrics 472

  6. The fraction of the labor force unemployed is (nu/n) = (n-ne)/n = 1-(ne/n). Note that (since n refers to the population) E(Y) = (1/n) ∑yi =(ne)/n. Therefore, the fraction of the labor force unemployed is 1-E(Y). Econometrics 472

  7. (c) Calculate E(Y|X=1) and E(Y|X=0). • First, we need to find the conditional density function p(Y|X=1). Econometrics 472

  8. Note that Pr(Y=1|X=1) = Pr(Y=1,X=1)/Pr(X=1) = .241/.246 = .98 Similarly, Pr(Y=0|X=1) = Pr(Y=0,X=1)/Pr(X=1) = .005/.246 = .02. Econometrics 472

  9. Therefore, E(Y|X=1) = 1Pr(Y=1|X=1) + 0 Pr(Y=0|X=1) = Pr(Y=1|X=1) = .98. By similar arguments, one can calculate E(Y|X=0) = Pr(Y=1|X=0) = .709/.754 = .94. Econometrics 472

  10. (d) Derive the unemployment rate for college grads and non-college grads. • We appeal to the result in part (b): The unemployment rate for college grads is 1-E(Y|X=1) = 1-.98 = .02. (The conditioning on X=1 restricts us to the subpopulation of college grads). • Similarly, the rate for non-college grads is 1-E(Y|X=0) = 1-.94 = .06. Econometrics 472

  11. (e) Suppose that someone reports being unemployed (Y=0). What is that probability that she is a college grad? A non-college grad? • In other words, we seek to find Pr(X=1|Y=0) and Pr(X=0|Y=0). Econometrics 472

  12. This is straight-forward: Pr(X=1|Y=0) = Pr(X=1,Y=0)/Pr(Y=0) = .005/.05 = .1. Pr(X=0|Y=0) = Pr(X=0,Y=0)/Pr(Y=0) =.045./05 =.9. In other words, if a person is observed to be unemployed, there is a 90 percent that she is not a college graduate. Econometrics 472

  13. (f) Are educational achievement and employment status independent? Just by looking at the answers to the previous questions, we know that the answer is no – we showed that p(Y|X) does not equal p(Y). One can also establish this by showing that E(XY) does not equal E(X)E(Y). Econometrics 472

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