Consider the sample regression of equation (13.58), where x1i is a dummy variable and x2i is a continuous variable:

Imagine that x1i identifies women. Section 13.2 explains why we don’t want to add a dummy variable identifying men, say x3i, to this regression. However, how do we know whether or not the effect of x2i is different for men than for women? Is there a reason why we shouldn’t add an interaction term between x2i and x3i to this regression, so that it looks like

(a) Recall that x1i = 0 when x3i = 1 and x1i = 1 when x3i = 0. Prove that, for each observation, x2i x1i x2i x3i x2i = + .

(b) Consider the following auxiliary regression:
x a b x x b x x 2i x x 1i 2i x x 3i 2i i 2 1 2 3 = + + + error .
Prove that, if a = 0, bx x 2 1 = 1, and bx x 2 3 = 1, this regression would fit perfectly:
All errors would be equal to zero.

(c) Explain, intuitively, why the regression of chapter 11 would choose the values a = 0, bx x 2 1 = 1 , and bx x 2 3 = 1 if we were to actually calculate this regression.

(d) Recall our discussion in section 11.4. There, we demonstrate that the multivariate regression uses only the part of each explanatory variable that is not associated with any of the other explanatory variables. Based on the answer to part

c, explain why x2i has no parts that are not related to x1iâ•›x2i and x3iâ•›x2i.

(e) Based on the answer to part

d, explain why, intuitively, the regression with which this question begins cannot be calculated.
(fâ•›) Return to table 1.4. The regression there contains interaction terms for age and women and for age and men. It can be calculated because it omits a crucial variable. What is this variable?
(g) This analysis shows that there are two specifications that can be calculated.
Regression can estimate either the general effect of x2i and the difference between the effects of x2i when x1i = 0 and x1i = 1, as in section 13.5, or the absolute effects of x2i when x1i = 0 and x1i = 1, as in table 1.4.
However, it cannot estimate a general effect and separate absolute effects for each of the cases x1i = 0 and x1i = 1. Explain, intuitively, why.

Transcribed Image Text:

y=a+bx+bx2 +b3xx2; +e;.