5.2 Suppose that a researcher, using wage data on 250 randomly selected male wor

Question

5.2 Suppose that a researcher, using wage data on 250 randomly selected male workers and 280 female workers, estimates the OLS regression Wage = 12.52 + 2.12 × Male, R2-0.06, SER-42. (0.23) (0.36) where Wage is measured in dollars per hour and Male is a binary variable that is equal to 1 if the person is a male and 0 if the person is a female. Define the wage gender gap as the difference in mean earnings between men and women a. What is the estimated gender gap? b. Is the estimated gender gap significantly different from 0? (Compute the p-value for testing the null hypothesis that there is no gender gap.) c. Construct a 95% confidence interval for the gender gap. d. In the sample, what is the mean wage of women? Of men? e. Another researcher uses these same data but regresses Wages on Female, a variable that is equal to 1 if the person is female and 0 if the person a male. What are the regression estimates calculated from this regression? Wage _ × Female, R2 = SER =--' +

Explanation / Answer

a)

Since wage = 12.52 + 2.21*male

Where male is a dummy variable, 1 = male and 0 =female

Estimated gender gap: wagemale – wagefemale = (12.52+2.12)-12.52 = 2.12

b)

H0 : male = 0

H1 : male 0

t= male / s.e(male) = 2.12/0.36 = 5.889

Using software, p-value was calculated to 4.12455E-09. Since p-value < 0.05, we reject the null hypothesis and conclude that the gender gap is significantly different from zero. (Degrees of freedom= n -2 =530-2 = 528)

c)

mean wage of men(male=1 in the model) = 12.52+2.12 = 14.64

mean wage of women(male=0 in the model) = 12.52

d)

when female=1 and male=0, the model is given by:

wage = 0 + 1* female

Since in the previous model, male=1 and female=0, the model is given by:

wage = 0 + 1* male

The relationship between the regression coefficients.

0 = 0 + 1

0 + 1 = 0

0 = 14.64

1 = 12.52 – 14.64 = -2.12

Since the coefficient estimates are related, OLS residual is the same under the two regression equations. Therefore, SER and R2 are unchanged.

S.E(1) = [S.e(wage­male)2 + S.e(wage­female)2 ] = S.E(1)

S.E(0) = S.e(wage­male)2 = [S.e(1)2 - S.e(0)2 ] = [0.362 -0.232 ] = 0.28

Therefore the model is given by:

Wage = 14.64 – 2.12*female, R2 = 0.06 , SER=4.2

               (0.28)    (0.36)

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