Question 1.

1a. White male: E[WAGE|EDUC=16] = β 1 + β2EDUC = –9.482 + 2.474*16 = 30.102

White female: E[WAGE|EDUC=16] = β 1 + β2EDUC + β4FEMALE = –9.482 + 2.474*16 –  4.224 = 25.878

Black male: E[WAGE|EDUC=16] = β 1  + β2EDUC + β3BLACK = –9.482 + 2.474*16 – 2.065 = 28.037

Black  female:  E[WAGE|EDUC=16]  =  β 1    +  β2EDUC  +  β3BLACK+  β4FEMALE+

β5(BLACK × FEMALE) = –9.482 + 2.474*16 – 2.065 –  4.224 + 0.533 = 24.346

The difference in the conditional expected hourly wage between while males and white females is equal to 30.102 – 25.878 = 4.224, which is equal (in absolute value) to the coefficient of the FEMALE dummy variable.

The difference in the conditional expected hourly wage between while males and black males is equal to 30.102 – 28.037 = 2.065, which is equal (in absolute value) to the coefficient of the BLACK dummy variable.

The difference in the conditional expected hourly wage between black males and black females is equal to 28.037 – 24.346 = 3.691, which is equal to 4.224 - 0.533, that is, to the combined effect of the coefficients of the FEMALE dummy variable and the interaction term BLACK ×

FEMALE.

1b. The coefficient β5 of the interaction term BLACK × FEMALE shows how the effect on the hourly wage of being a female (denoted by the coefficient β4) is changed when the female is also black. Hence the total effect on the hourly wage of being a female is equal to β4 + β5 among blacks.

Alternatively, β5  shows how the effect on the hourly wage of being black (denoted by the coefficient β3) is changed when a black person is also a female. Hence the total effect on the hourly wage of being black is equal to β3 + β5 among females.

1c. One would jointly test the variables denoting being black and a female by performing a joint test of significance of the coefficients of the variables BLACK, FEMALE and BLACK × FEMALE, that is of the coefficients β3, β4  and β5 . The test statistic is going to have a F distribution and will be equal to

     ሺ ோ  െ  ⃞ ሻ/3

 ோ /ሺ െ 3ሻ  ,

where  ோ  denotes the sum of squared residuals from the restricted model, that is the model in which β3, β4  and β5  are set to zero, ⃞  denotes the sum of squared residuals from the unrestricted model, that is the model in which β3, β4  and β5  are included, and N denotes the number of observations. The number 3 is used because we are testing 3 restrictions.

1d.  The  fact that both BLACK  and BLACK  × FEMALE have  statistically  insignificant coefficients does not necessarily imply that being black has no effect on the hourly wage. If the two variables are strongly collinear (i.e., strongly correlated), testing one coefficient in an individual test implies that one controls for the other variable, which implies that the variable tested will have little variability and thus is likely to be statistically insignificant. In other words, the common underlying concept of the two variables which creates their strong correlation (in this case race), is captured by the variable that is not tested, and thus the tested variable has little additional explanatory power.

On the other hand, when both variables are jointly tested, which is the correct way to test for the total effect of being black, then we are testing an equation in which race plays a role against an equation is which it does not. In our context, it could be the case that race affects wages, and thus the joint test could reject the null.

To put things differently, in a joint test of the coefficients of BLACK and BLACK × FEMALE the common concept (race) underlying the two possibly highly correlated regressors is not being kept constant by not testing (and thus controlling for) a variable that also captures the concept of race.

The more general point is that if one is interested in testing for the effect of a particular concept (in our case race), one should jointly test all the terms in which it appears, that is, including all interaction terms.

Question 2.

2.1. Holding all other factors fixed, we have

ΔE[log(WAGE)] = β 1 + β2ΔEDUC + β3(ΔEDUC×PAREDUC) = (β2 + β3PAREDUC)ΔEDUC and dividing both sides by ΔEDUC gives the result

Alternatively, one can just take the derivative of the regression equation with respect to EDUC.

The sign of β3  is not obvious; although, β3  > 0 if we think that the more highly educated the child’s parents, the more it gets out of another year of education.

2.2. Using the values for parental equation equal to PAREDUC = 32 and PAREDUC = 24 to interpret the coefficient on EDUC × PAREDUC, we obtain that the difference in the estimated hourly wage is .00078(32 – 24) = .0062, or about .62 percentage points.