Economics 1123 Midterm Exam Question & Answers Graded A

Department of Economics Economics 1123 Harvard University Fall 2022 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 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. 4) Using regressions (5): a) (5 points). Let the male licensing premium be the wage difference between a male worker having a license and not having a license. Compute a 95% confidence interval for the male licensing premium. The male licensing premium is license , the coefficient on license. The 95% confidence interval is 0.107 ±1.96×0.012 = (0.083, 0.131). b) (5 points). The licensing premium gap is the difference between the male licensing premium and the female licensing premium. Test the hypothesis that the licensing premium gap is zero for service workers, against the hypothesis that it is nonzero. The licensing premium for women is license  license female . Thus the licensing premium gap is license female . The estimate of the licensing premium gap in regression (5) is 0.006, SE = 0.016. The t-statistic is 0.006/0.016 < 1 so the hypothesis that the licensing premium gap is zero is not rejected at the 10% significance level. 5) (7 points) Regressions (5) and (6) have the same specification; (5) is estimated using service workers, (6) is estimated using construction workers. Test the hypothesis that the licensing premium gap is the same for service workers as it is for construction workers. State any additional assumptions you are making to perform this test. The difference between the licensing premium gaps is ˆservices  ˆconstruction . To compute license female license female the t-statistic you need the standard error of ˆservices  ˆconstruction . The variance of ˆservices  ˆconstruction is, license female license female license female license female varˆservices  ˆconstruction   varˆservices  varˆconstruction  2covˆservices , ˆconstruction  Estimates of the first two of the variances are obtained from the relevant standard errors in regressions (5) and (6): var ˆservices   SE ˆservices 2  0.0162 varˆconstruction   SE ˆconstruction   0.0562 The covariance is not given in the table. However under the data are a random sample of the population so that means the draws are independent. Because regressions (5) and (6) were estimated for different people, that means the samples are independent and thus the sampling distributions of ˆservices ˆconstruction license female are independent; thus covˆservices , ˆconstruction  = 0. Putting this all together, the t-statistic is, license female license female t   7.71 which exceeds 1.96, so the hypothesis is rejected at the 5% significance level. Part 2 – USE BLUE BOOK #2 (40 points) 6) (5 points) The only difference between regressions (5) and (7) is how the standard errors are computed. In regression (7), they are clustered at the state level (that is, workers in the same state are included in the cluster). Either provide a reason that justifies clustering at the state level, or provide an argument why clustering at the state level is unnecessary.