We now introduce a second indicator variable Gi for gender which is 1 if female and 0 if male. The regressions equation is now ln(Wi)=1 +2Ei +3Xi +4Xi2 +4Bi +5Gi +ei estimate this equation by least squares and report your results.

(a) Test the null that 5 = 0. Should the alternative be one- or two-sided?

(b) How does the presence of this new variable impact the significance of the black coefficient 4? Test the joint null that 4 = 5 = 0. Does this joint test use one- or two-sided alternatives.

(c) Report the results for the White test applied to this regression. Outline what is involved in the calculations. What is this a test of and is it a one-sided or two-sided test.

(d) Suppose we reject in (c), then what are the implications for the inferences in (a) and (b)? What can be done to avoid these difficulties?

(e) The justification for a joint test is that it is more powerful. Discuss how we might sometimes encounter a situation where one of two individual test rejects the null but the joint test does not