(c)
Figure xr2.28(c) Residuals from linear wage model
The residuals are plotted against education in Figure xr2.28(c). There is a pattern evident; as EDUC increases, the magnitude of the residuals also increases, suggesting that the error variance is larger for larger values of EDUC—a violation of assumption SR3. If the assumptions SR1-SR5 hold, there should not be any patterns evident in the residuals.
(b) The estimated model equations, including the one from part (b), are given in Table xr2-28
Table xr2-28
CEDUCNSSE
part (b)allCoeff
Std. err.−10.4000
(1.9624)2.3968
(0.1354)1200220062.3
part (c)maleCoeff
Std. err.−8.2849
(2.6738)2.3785
(0.1881)672144901.4
femaleCoeff
Std. err.−16.6028
(2.7837)2.6595
(0.1876)52869610.5
whiteCoeff
Std. err.−10.4747
(2.0806)2.4178
(0.1430)1095207901.2
blackCoeff
Std. err.−6.2541
(5.5539)1.9233
(0.3983)10511369.7
The white equation is obtained from those workers who are neither black nor Asian. From the results, we can see that an extra year of education increases the expected wage rate of a white worker more than it does for a black worker. And an extra year of education increases the expected wage rate of a female worker more than it does for a male worker.
Exercise 2.28 (continued)
(e) The estimated quadratic equation is
The marginal effect is . For a person with 12 years of education, the estimated marginal effect of an additional year of education on expected wage is 2(0.0891)(12) = 2.1392. That is, an additional year of education for a person with 12 years of education is expected to increase wage by $2.14. For a person with 16 years of education, the marginal effect of an additional year of education is 2(0.0891)(16) = 2.8523. An additional year of education for a person with 16 years of education is expected to increase wage by $2.85. The linear model in (b) suggested that an additional year of education is expected to increase wage by $2.40 regardless of the number of years of education attained. That is, the rate of change was constant. The quadratic model suggests that the effect of an additional year of education on wage increases with the level of education already attained.
(f)
Figure xr2.28(f) Quadratic and linear equations for wage on education
The quadratic model appears to fit the data slightly better than the linear equation, especially at lower levels of education.
Exercise 2.29
(a)
variableNmeanmedianminmaxskewnesskurtosis
ln(WAGE)12002.99942.96011.37125.39860.23062.6846
Figure xr2.29(a) Histogram and statistics for ln(WAGE)
The histogram shows the distribution of ln(WAGE) to be almost symmetrical. Note that the mean and median are similar, which is not the case for skewed distributions. The skewness coefficient is not quite zero. Similarly, the kurtosis is not quite three, as it should be for a normal distribution.
(b) The OLS estimates are
We estimate that each additional year of education predicts a 9.87% higher wage, all else held constant.
(c) The antilogarithm is . For someone with 12 years of education the predicted value is and for someone with 16 years of education it is .
(d) The marginal effect in the log-linear model , ignoring the error term, is . For individuals with 12 and 16 years of education, respectively, these values are $1.5948 and $2.3673. These are the estimated marginal effects of education on expected wage in this log-linear model.
Exercise 2.29 (continued)
(e)
Figure xr2.29(e) Observations with linear and loglinear fitted lines
The log-linear model fits the data better at low levels of education.
(f) A more objective measure of fit is . For the log-linear model this value is 228,573.5 and for the linear model 220,062.3. Based on this measure the linear model fits the data better than the linear model.