7.9 Let d be a dummy (binary) variable and let z be a quantitative variable. Consider the model his is PR R REWOR y=B+8d+ Bz +8d+z+u version of a model with an interaction between a dummy variable and a quantitative variable. [An example is in equation (7.17).] Since it changes nothing important, set the error to zero, u = 0. Then, when d = 0 we can write the relationship between y and z as the function fo(z) = B+ Bz. Write the same relationship when d =1, where you should use fi(z) on the left-hand side to denote the linear function of z. (ii) Assuming that 8,0 (which means the two lines are not parallel), show that the value of z* such that fo(z) = f(z) is z* = -8/8. This is the point at which the two lines intersect [as in Figure 7.2(b)]. Argue that z* is positive if and only if &, and 8, have opposite signs. log(wage) (iii) Using the data in TWOYEAR.RAW, the following equation can be estimated: 2.289.357 female + 50 totcoll+.030 female - totcoll (0.011) (.015) (.003) n = 6,763, R = 202, (.005) where all coefficients and standard errors have been rounded to three decimal places. Using this equation, find the value of totcoll such that the predicted values of log(wage) are the same for men and women. (iv) Based on the equation in part (iii), can women realistically get enough years of college so that their earnings catch up to those of men? Explain.