Let d be a dummy (binary) variable and let z be a quantitative variable. Consider the model
y = (0 + (0d + (1z + (1d (z + u;
this is a general version of a model with an interaction between a dummy variable and a quantitative variable.
(i) 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 f0(z) = (0 +(1z. Write the same relationship when d = 1, where you should use/,(z) on the left-hand side to denote the linear function of z.
(ii) Assuming that (1 ( 0 (which means the two lines are not parallel), show that the value of z* such that f0 (z*) = f1 (z*) is z* = - (0/(1. This is the point at which the two lines intersect. Argue that z* is positive if and only if (0 and (1 have opposite signs.
(iii) Using the data in TWOYEAR.RAW, the following equation can be estimated:

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.

log(wage) = 2.289-.357 female + .50 totcoll + .030 female-totcoll (0.011) 015 .003) n 6,763, R2 .202. 005)