Let d be a dummy (binary) variable and let z be a quantitative variable. Consider the model y 5 b0 1 d0d 1 b1z 1 d1d # z 1 u;

this is a general version of a model with an interaction between a dummy variable and a quantitative variable. [An example is in equation (7.17). ]

(i) Since it changes nothing important, set the error to zero, u 5 0. Then, when d 5 0 we can write the relationship between y and z as the function f0 1z2 5 b0 1 b1z. Write the same relationship when d 5 1, where you should use f1 1z2 on the left-hand side to denote the linear function of z.

(ii) Assuming that d1 2 0 (which means the two lines are not parallel), show that the value of z*

such that f0 1z*2 5 f1 1z*2 is z* 5 2d0/d1. 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 d0 and d1 have opposite signs.

(iii) Using the data in TWOYEAR, the following equation can be estimated:

log1wage2 5 2.289 2 .357 female 1 .50 totcoll 1 .030 female # totcoll 10.0112 1.0152 1.0032 1.0052 n 5 6,763, R2 5 .202, 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.