1. Using data from the Survey of Consumers Finances between 1983 and 2007 (the surveys are done every 3 years), I used OLS to examine the determinants of a household s credit card debt. Credit card debt was converted into 1983 dollars using the CPI. The results from two specifications are presented in the table below. Coefficients are from an OLS regression of a household s real credit card balance. Numbers in parentheses are t-statistics. Year dummies (1983 excluded) (1) (2) 1986 309.5 308.1 (3.666) (3.655) 1989 266.8 276.9 (4.404) (4.577) 1992 318.8 351.6 (5.367) (5.927) 1995 518.5 558.9 (8.800) (9.492) 1998 749.1 800.6 (12.71) (13.59) 2001 657.9 706.9 (11.19) (12.02) 2004 839.6 894.6 (14.28) (15.22) 2007 1290 1343 (21.88) (22.78) Real income -4.31e-05-4.58e-05 (-15.28) (-16.22) Age -11.79-14.01 (-21.48) (-22.32) Years of education 68.40 58.64 (22.46) (18.93) Black -211.0-92.56 (-6.940) (-3.000) Marital status dummies (never married omitted) Married -- 550.6 (20.69) Widowed -- 200.2 (4.569) Divorced -- 340.2 (10.28) Constant 40.34-168.6 (0.556) (-2.292) Observations 152055 152055 1
a. Based upon specification (1), if the omitted year dummy was switched from 1983 to 2007, what would be the new estimate of i. the coefficient on the constant term? ii. the coefficient on the 2004 dummy? b. When controls for marital status are added to the regression, the coefficient on the black dummy drops from -211 to -92.56. What does this tell you about the relationship between marital status and race? Justify your conclusion. c. Suppose you want to test the hypothesis that, controlling for observed characteristics, credit card balances are significantly different in the last three years of the sample (2001, 2004, and 2007). Explain how you could construct a Wald test of this hypothesis (not an F-test). In your answer, define the matrices that you use in describing the construction of the Wald statistic, the degrees of freedom for the test statistic, and the values of the test statistic that would lead you to reject the null hypothesis. In writing out your matrices, assume that the order in the coefficient vector (call it B) and the corresponding variance-covariance matrix (call it V) matches that given in the above table (i.e. coefficient on 1986 year dummy is first element of B, coefficient on intercept is last element in B, etc.) d. Suppose that many people in the SCF had their income imputed (i.e. estimated) and thus income is reported with som error, explain how this would bias i. the estimated effect of income on credit card debt. ii. the estimated effect of education on credit card debt. Provide the basis for your conclusions about the direction of the biases. e. After estimating the 2 nd specification, I generated predicted residuals, squared them, and then estimated a regression with the squared-residual as the dependent variable and the black dummy variable as a control. Results are in the table below with t-statistics in parentheses. black -6036706 (-4.071) Constant 1.25e+07 (26.92) Observations 152055 Based on the results of this supplementary regression, why should you be concerned about the validity of the OLS estimates provided in the first table? What can you do to address these concerns? 2
2. In a recent article, von Wachter & Bender (2006) 1 examine the effect of job displacement on subsequent job earnings. The sample includes a group of workers who all complete training at one of many possible firms. Upon completion of the training, the firm decides whether to retain the worker. The regression used to determine the impact of retention on worker earnings is where y i is the natural log of earnings 5 years after completing training, X i is a vector of controls for human capital characteristics (age, experience, education, etc.) and D i is a dummy variable that equals one if the worker is displaced (i.e. not retained by the training firm) after training is complete. In the simplest OLS model, the estimate of is -0.072 implying that, controlling for human capital characteristics, a job-displacement reduces earnings by 7.2 percent. The article points out that job displacement could be endogenous and pursues an instrumental variables estimation strategy. The estimate of drops to -0.012 when IV is used. a. Given the change in the estimate of when switching from OLS to IV, what does this say about the nature of the endogeneity? Give both a statistical and behavioral description of the problem. A statistical description will rely on a discussion of covariances between different elements in the model. A behavioral description relies upon a discussion of the process that generates job displacements. b. The instrumental variables used to estimate the IV model included employment growth associated in the industry that worker was employed when the training was completed. Call this variable G. Under what conditions is G an appropriate instrument? c. How would you determine whether G is a weak instrument? d. What problem is there with the IV estimates if it is a weak instrument? Explain. e. Would a Sargan test be useful in this application? If so, how? If not, why not? 1 Till von Wachter & Stefan Bender. In the right place at the wrong time: the role of firms and lcuk in young workers careers, American Economic Review, December 2006: 1679-1705. 3
3. Using data from the March 2009 Current Population Survey, I created a sample of full-time full-year workers who were employed in the private sector, or state or local government. 2 Federal and selfemployed workers were omitted from the sample. I was interested in the effect of public sector (state/local) employment on total compensation. Total compensation is measured as annual earnings including the value of employer provided health insurance and pension benefits. I present the results of a few different specifications in the table below. The variable definitions should be obvious from the table. The dependent variable is the natural log of total compensation. T-statistics are in parentheses. Specifications (1) and (2) are identical except that (2) includes hours worked last year. Specifications (2) and (3) are identical except that the t-statistics in (3) are based on standard errors for clustering by state. (1) (2) (3) Public sector employee -0.0359-0.0225-0.0225 (-5.12) (-3.24) (-1.46) Hours worked last year -- 0.000298 0.000298 (33.2) (21.4) Age 0.0604 0.0572 0.0572 (39.5) (37.8) (40.8) Age 2-0.000588-0.000554-0.000554 (-33.1) (-31.5) (-34.3) Black -0.163-0.150-0.150 (-19.5) (-18.3) (-12.9) Hispanic -0.127-0.113-0.113 (-15.9) (-14.4) (-5.80) Female -0.316-0.281-0.281 (-58.7) (-52.0) (-36.7) Education dummies High school degree 0.274 0.267 0.267 (22.4) (22.1) (18.6) Some college 0.409 0.395 0.395 (32.2) (31.4) (25.5) Associates degree 0.491 0.480 0.480 (35.6) (35.3) (31.2) Bachelors degree 0.756 0.726 0.726 (58.3) (56.6) (44.3) Masters degree 0.935 0.891 0.891 (62.8) (60.2) (42.8) Professional degree 1.314 1.223 1.223 (46.2) (44.1) (50.2) Doctorate 1.134 1.050 1.050 (38.9) (36.2) (38.1) Firm size dummies 100-499 0.157 0.152 0.152 (19.2) (19.0) (19.0) 500-999 0.202 0.198 0.198 (18.4) (18.4) (19.6) 1000+ 0.210 0.206 0.206 (32.1) (32.0) (25.9) Constant 9.021 8.430 8.430 (273) (231) (154) Observations 59,667 59,667 59,667 R-squared 0.35 0.38 0.38 2 Full-time is defined as usually working 35 hours or more per week, full-year is defined as working 50 or more weeks per year. 4
a) Using specification (1), provide an economic interpretation of the coefficient on the public sector dummy. That is, what does this coefficient tell you about the earnings of public versus private sector workers? (Do not interpret in terms of the effect on log-earnings). b) When hours worked last year is added in specification (2), the coefficient on public moves toward zero. What does this tell you about the work hours of private versus public sector workers? Explain the basis for your conclusion. c) Suppose that hours worked is measured with error in specification (2). How is this likely to bias the coefficient on the public sector dummy? Explain the basis for your conclusion. d) Based on specification (2), what is the predicted earnings difference between a person with a bachelors degree and a high school degree? Be sure to describe what units you are using to measure the earnings difference. e) If the regression specification (2) switched the omitted dummy for education from less than high school to bachelors degree and omitted the largest firm size category instead of the smallest, i. what would be the new coefficient on high school graduate? (Just provide a number no explanation required). ii. what would be the new coefficient on the constant term? f) Specification (2) does not correct for clustering of errors by state whereas specification (3) does. Notice that correcting for clustering causes the t-statistic for the Hispanic dummy to drop to less than one-half of its original value whereas the t-statistics on the firm size and education dummies change only slightly. Why would you expect this? Be sure to justify your explanation with a discussion of the factors that determine how correcting for clustering affects t-statistics. g) In the 1990s, evidence was presented that the public sector premium/penalty differs across workers. For example, one study showed that the public sector underpays those with higher levels of education and overpays those with a high school degree or less. 3 How could you modify the above regression to test whether the public premium/penalty differs across education groups in 2009? Be precise in the description of what regressions you would run, what test statistic you would generate, and the distribution (including degrees of freedom) for this test statistic. h) A comparison of simple sample means reveals that state and local workers earn nearly 7 percent more than private sector workers. Several studies point out that educational attainment differs between the private and public sector. This is confirmed by a comparison of public/private sample means for the education dummies below. (Remember that the dummy for less than high school is omitted.) public hsgrad somecoll assoc bachelor master profdeg doctorate 0.30.18.11.21.07.02.01 1.19.15.11.27.20.02.03 3 Katz, L. and Krueger, A. 1992. Changes in the structure of wages in the public and private sectors. NBER Working Paper 3667. 5
Use the above information to compute how much more or less public sector workers should earn on the basis of education differences. Show how you derived your answer and interpret your result. 4. Using data from 1999 Current Population Survey, I estimated regression models with the log(wage) [log=natural log] as the dependent variable. The explanatory variables in the regression include the person's age and dummy variables listed below that should be self-explanatory. The first specification of the model does not include education controls, but the second does. Dependent variable: log(wage) Specification (1) Coefficient t-statistic Speciation (2) Coefficient t-statistic Intercept 2.2 72.16 1.82 61.7 age 0.008 15.27 0.005 11.12 female -0.24-22.19-0.24-24.64 Smoking status (never smoked omitted) quit smoking currently smoke 0.03-0.08 1.85-6.26 0.05 0.02 3.97 1.56 Marital status (divorced/separated omitted) married 0.23 15.83 0.2 15.09 single 0.11 5.75 0.13 7.31 Race (other race omitted) white black -0.03-0.09-1.11-3.07 0.02 0.00 0.7-0.28 Education (less than high school omitted) High school graduate 0.29 18.02 Some college 0.42 25.13 Bachelor s degree 0.74 41.43 Master's degree 0.88 36.45 Professional degree 0.99 21.06 Doctorate 1.08 21.76 Sample size 10976 10976 Sum of squared errors 3525.95 2817.81 a. Using specification (1), what is the predicted wage for a 30 year old white divorced male who currently smokes and has a high school degree. Note that the regression uses log(wage) as the dependent variable. b. Based on specification (2), other things being the same, how do the wages of smokers and quitters compare i. in percentage terms ii. in $ terms c. If specification (1) was reestimated with a dummy for "never smoked" included and the dummy for "quit smoking" excluded, i.. what would be the value of the coefficient on the intercept? (no explanation required.) ii. what would be the value of the coefficient on the "never smoked" dummy? (no explanation required.) d. Comparing specification (1) and (2), it's clear that inclusion of the education controls has a substantial effect on the coefficient for the "currently smoke" dummy. What does this tell you about the relationship between education and smoking? Explain the basis for your answer. 6
e. If age was measured with error, how would this likely affect the estimated effect of smoking on earnings? Justify your answer and describe any assumptions that you must make to decide on the direction of the bias. f. Using the information provided in the table, provide a test statistic for the hypothesis that wages are identical across education g groups. Provide a numerical value for the relevant test statistic, describe its distribution, and indicate whether you reject the null at the.05 level of significance. g. Suppose that you want to test whether the effect of smoking on earnings differs across education groups. Explain how you test this hypothesis (regressions you would estimate, the restrictions you would test, the resulting test statistic and degrees of freedom). 5. There is an extensive literature examining the effect of substance use (alcohol, drugs, tobacco) on a variety of socio-economic variables. For example, several studies examine the effect of alcohol consumption on earnings. A simplistic approach estimates a standard log-wage equation controlling for human capital variables (age, education, etc.) and a variable measuring a person s alcohol consumption. Some people argue that this is a naïve approach because alcohol consumption is likely to be endogenous in a wage equation. a. Explain what it means for alcohol consumption to be endogenous in a wage equation from both a statistical and behavioral perspective. b. Do you think that the endogeneity of alcohol consumption would cause the OLS estimate of the impact of alcohol consumption on wages to be biased upward or downward? Justify your prediction using based upon a discussion of how human behavior might lead to the statistical properties that could cause the bias you describe. A wide variety of instruments have been used to resolve the issue of endogeneity. A sampling of such instruments includes a list of state-specific variables such as (1) state differences in the minimum legal drinking age (MLDA); (2) the blood alcohol concentration (BAC) threshold for driving under the influence of alcohol; or (3) state differences in the tax rate on alcohol (AT) c. Describe the necessary conditions for these variables to be valid instruments for alcohol consumption. d. Presumably, a state s laws are partly driven by voter preferences in the state. For example, one might think that if a large share of the population belongs to a religion that opposes drinking, it may vote for stricter rules on drinking. If this is true,does it imply that state laws are NOT a valid instrument for alcohol consumption of the people in the state in the context of this problem (i.e..where we are trying to estimate the true effect of alcohol consumption on wages). EXPLAIN. e. Suppose that you estimate a log-wage (w) equation controlling for a vector of human capital characteristics (X) and a continuous variable representing ounces of alcohol consumed per week (A). Your instruments are MLDA, BAC, and AT. Using this notation, describe how you would i. test whether the instruments are weak ii. test whether the over-identifying assumptions are violated. (Be sure to describe the regression you would estimate and be precise about how the test statistics are created, the distribution of the test statistics, and how you would use the statistic to determine whether to reject the relevant hypothesis.) f. In words, what does it mean if the instruments are weak and what problem does it create for your IV estimation of the wage equation? g. In words, what does it mean if the over-identifying assumptions are rejected and what problem does it create for your IV estimation of the wage equation? 7