Reverse regression continued. Suppose that the model in Exercise 5 is extended to y = x

Chapter 7, Problem 6

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Reverse regression continued. Suppose that the model in Exercise 5 is extended to y = x + d+, x = x +u. For convenience, we drop the constant term. Assume that x, and u are independent normally distributed with zero means. Suppose that d is a random variable that takes the values one and zero with probabilities and 1 in the population and is independent of all other variables in the model. To put this formulation in context, the preceding model (and variants of it) have appeared in the literature on discrimination. We view y as a wage variable, x as qualifications, and x as some imperfect measure such as education. The dummy variable d is membership (d = 1) or nonmembership (d = 0)in some protected class. The hypothesis of discrimination turns on < 0 versus = 0. a. What is the probability limit of c, the least squares estimator of , in the least squares regression of y on x and d? [Hints: The independence of x and d is important. Also, plim d d/n = Var[d] + E2[d] = (1 ) + 2 = . This minor modification does not affect the model substantively, but it greatly simplifies the TABLE 7.8 Ship Damage Incidents Period Constructed Ship Type 19601964 19651969 19701974 19751979 A 0 4 18 11 B 29 53 44 18 C 1 1 21 D 0 0 11 4 E 0 7 12 1 Source: Data from McCullagh and Nelder (1983, p. 137). algebra.] Now suppose that x and d are not independent. In particular, suppose that E [x | d = 1] = 1 and E [x | d = 0] = 0. Repeat the derivation with this assumption. b. Consider, instead, a regression of x on y and d. What is the probability limit of the coefficient on d in this regression? Assume that x and d are independent. c. Suppose that x and d are not independent, but is, in fact, less than zero. Assuming that both preceding equations still hold, what is estimated by (y | d = 1) (y | d = 0)? What does this quantity estimate if does equal zero?