Department of Economics Economics 1123
Harvard University Fall 2020
Midterm Exam
9:00 a.m., Wednesday October 24, 2018
PACKET 2
Solutions in Calibri 11
Part 1 – USE BLUE BOOK #1 (60 points)
1) Consider regressions (1) and (2).
a) (5 points) Interpret the coefficient on license in regression (1), in which the
dependent variable is wage.
The mean wage for workers with a license is $2.76 higher than for those without a license.
b) (5 points) Based on the evidence in the tables and/or figures, is the regression error
in regression (1) heteroskedastic or homoskedastic? Briefly explain.
Heteroskedasticity is when the variance of the regression error depends on X. In regression (1),
heteroskedasticity arises when the variance of wages differs for those with and without a
license. Figure 1 suggests that it does, in particular the spread of the distribution of wages for
those with a license exceeds that for those without a license.
c) (5 points) Interpret the coefficient on license in regression (2), in which the
dependent variable is log(wage). Is this coefficient large or small in a real-world
sense?
The mean wage for workers with a license is 18.9% higher than for those without a license. That
is a lot in a real world sense, a 20% “raise”.
2) Consider regression (8), in which the dependent variable is the binary variable license.
a) (5 points). There are only four education variables in this regression, but there are five
in Table 1: the regression omits highschool. What would happen if highschool were
included?
There would be perfect multicolinearity, because the sum of the education variables would
1
, equal 1, which is the constant regressor.
b) (4 points). Interpret the coefficient on hsdropout.
Holding constant sex, workers who did not finish high school have a 4.5 percentage points lower
probability of having a license than workers with (only) a high school diploma.
c) (4 points). Interpret the pattern of coefficients on the four education variables.
The probability of having a required occupational license increases with the amount of
education.
3) Consider regression (3).
a) (5 points). The coefficient on license changes substantially from regression (2) to (3).
Explain why the coefficient changed and provide an explanation of the sign (direction)
of the change.
Regression (2) evidently suffered from omitted variable bias. In particular, the amount of
education is positively correlated with license (regression (8)) and workers with higher education
earn more generally (that is the pattern of the coefficients on education in regression (3)). Thus,
when the amount of education is omitted in (2), license captures both the effect of a license and
the effect of having more education.
b) (5 points). Test the hypothesis that all four of the education variables have
coefficients that equal zero, against the alternative that at least one is nonzero.
This is done using the F-test on the four education coefficients. The F-statsitic is 1234 which has
a p-value <0.000 so the hypothesis is rejected at the 1% significance level.
c) (5 points). Sketch the shape of the estimated relation between log(wage) and age. Briefly
interpret this shape in words. NOTE: You need not evaluate the regression function with
a calculator or have numerical values on the axis – just sketch the shape.
2
Harvard University Fall 2020
Midterm Exam
9:00 a.m., Wednesday October 24, 2018
PACKET 2
Solutions in Calibri 11
Part 1 – USE BLUE BOOK #1 (60 points)
1) Consider regressions (1) and (2).
a) (5 points) Interpret the coefficient on license in regression (1), in which the
dependent variable is wage.
The mean wage for workers with a license is $2.76 higher than for those without a license.
b) (5 points) Based on the evidence in the tables and/or figures, is the regression error
in regression (1) heteroskedastic or homoskedastic? Briefly explain.
Heteroskedasticity is when the variance of the regression error depends on X. In regression (1),
heteroskedasticity arises when the variance of wages differs for those with and without a
license. Figure 1 suggests that it does, in particular the spread of the distribution of wages for
those with a license exceeds that for those without a license.
c) (5 points) Interpret the coefficient on license in regression (2), in which the
dependent variable is log(wage). Is this coefficient large or small in a real-world
sense?
The mean wage for workers with a license is 18.9% higher than for those without a license. That
is a lot in a real world sense, a 20% “raise”.
2) Consider regression (8), in which the dependent variable is the binary variable license.
a) (5 points). There are only four education variables in this regression, but there are five
in Table 1: the regression omits highschool. What would happen if highschool were
included?
There would be perfect multicolinearity, because the sum of the education variables would
1
, equal 1, which is the constant regressor.
b) (4 points). Interpret the coefficient on hsdropout.
Holding constant sex, workers who did not finish high school have a 4.5 percentage points lower
probability of having a license than workers with (only) a high school diploma.
c) (4 points). Interpret the pattern of coefficients on the four education variables.
The probability of having a required occupational license increases with the amount of
education.
3) Consider regression (3).
a) (5 points). The coefficient on license changes substantially from regression (2) to (3).
Explain why the coefficient changed and provide an explanation of the sign (direction)
of the change.
Regression (2) evidently suffered from omitted variable bias. In particular, the amount of
education is positively correlated with license (regression (8)) and workers with higher education
earn more generally (that is the pattern of the coefficients on education in regression (3)). Thus,
when the amount of education is omitted in (2), license captures both the effect of a license and
the effect of having more education.
b) (5 points). Test the hypothesis that all four of the education variables have
coefficients that equal zero, against the alternative that at least one is nonzero.
This is done using the F-test on the four education coefficients. The F-statsitic is 1234 which has
a p-value <0.000 so the hypothesis is rejected at the 1% significance level.
c) (5 points). Sketch the shape of the estimated relation between log(wage) and age. Briefly
interpret this shape in words. NOTE: You need not evaluate the regression function with
a calculator or have numerical values on the axis – just sketch the shape.
2
