1. In a 1996 article, Mark Wilhelm examined whether parents bequests are altruistic. 1 According to the altruistic model of bequests, a parent with several children would leave larger bequests to children with lower lifetime earnings. Define B ij as the bequest that the parents in family i give to child j in family i. Explanatory variables in the model include the projected present value of life-time earnings of each child (Earnings ij ), a dummy indicating whether the child is female (Female ij ), and whether the child has given birth to a grandchild (Granchild ij ). The model is estimated including a fixed effect for each family (u i ). The estimated model was: B ij = -.151Earnings ij -.016Female ij -.023Grandchild ij + u i + e ij Only earnings was statistically significant at the.05 level. a. Describe how the above regression could be estimated as a fixed effects model without using dummy variables for the family fixed effects. Leave no ambiguity in the description of your regression (e.g. undefined variables). b. The author mentions that parental income is available in the data set (i.e. the income of those making the bequests) and yet the variable is not included in his estimate of the fixed effects model. Why should you have expected this? Be precise in explaining the nature of the problem that would emerge. c. Suppose John and Sally are from the same family and the parents provide a total bequest of $1 million that must be divided between John and Sally. John is male, has no children, and has income of $50,000. Sally is female, has two kids, and income of $100,000. What s the predicted bequest for i. John ii. Sally Provide a brief outline of your calculations. d. If parents with only one child were dropped from the sample, how would this likely affect the estimated coefficients in the model with family fixed effects? Explain. e. Suppose the model above was estimated assuming that the u i could be captured by a random effects model. What do you expect would happen to the estimated coefficient on children earnings? Explain by describing the statistical conditions in the data that would cause the estimate to change in the predicted direction and provide a rationale for your assumptions about those statistical relationships (e.g. I think the covariance between x and y is positive because...). f. Compared to the fixed effects model, what are the advantages and disadvantages of the random effects model? How can you construct a test statistic to determine which model is appropriate? 1 Mark Wilhelm, Bequest Behavior and the Effect of Heirs Earnings: Testing the Altruistic Model of Bequests, American Economic Review, September 1996.
2. Autor (2003)2 examines whether state law regarding worker dismissal affects the share of workers in temporary employment. Common law is that employment is at will and an employer may dismiss a worker without cause. Over the years, courts in many states have limited employers discretion to terminate workers without justification. Some have argued that these limitations on the employment at will doctrine have encouraged firms to replace some of their workers by going to temporary help services (THS) for their workers since the firm need not justify the termination of a contract with a THS firm. To examine the effect of common law exceptions (i.e. different legal restrictions on the employer s ability to dismiss workers), Autor estimates the following model a. What variation in the data is being used to estimate the effect of the common law exceptions? Be sure to characterize this in the context of this specific problem. b. Suppose that union contracts make it difficult for an employer to use THS and that unions also pressure legislators to pass common law exceptions that protect workers from dismissal. How will the lack of control for each state s unionism affect the estimate of how common law exceptions affect THS? Also, to what extent will the inclusion of state fixed effects fix this problem? Explain. c. Suppose that state legislators are more likely to pass common law exceptions if their state has experienced and is expected to continue to experience long term employment growth since they are less fearful of job loss that might result from passage of such legislation. If this is true, is the estimated effect of common law exceptions on THS likely to be over- or under-stated by the diffin-diff model? Explain. 2 David H. Autor. Outsourcing at Will: The Contribution of Unjust Dismissal Doctrine to the Growth of Employment Outsourcing Journal of Labor Economics, Vol. 21, No. 1 (Jan., 2003), pp. 1-42.
3. Federal law allows restaurants to claim a tip credit against the federal minimum wage. This means that although the federal minimum wage is $7.25 per hour, a restaurant can pay as little as $2.13 per hour if the employees claim sufficient tips to make up the difference (i.e. the restaurant can claim a tip credit of $5.12 per hour). Some states have passed laws prohibiting restaurants from claiming a tip credit and require that the employer pay the full minimum wage. It seems reasonable that that a ban on the tip credit (i.e. a requirement that restaurants pay the full minimum wage) would reduce the employment of waiters and waitresses (wait-staff). a. Describe a diff-in-diff procedure for estimating the effect of banning tip credits on the employment of wait-staff. Be specific in defining the dependent variable and the control variables for your equation, including any subscripts that you use. Be sure to describe any controls beyond that for a ban on the tip credit that you would want to include to improve the quality of your estimates. b. Explain the identifying assumption that is required to make it possible to estimate the effect of the ban on the tip credit. c. Is the standard OLS formula for estimation of standard errors appropriate? If not, what do you think would be a better method and why? d. If the identifying assumptions are not satisfied, would you over- or under-estimate the effect of the ban on wait-staff employment? Explain. e. Describe methods you could use to check the validity of the identifying assumptions.
4. 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. If the instruments are weak, what problem does it create for your IV estimation of the wage equation? g. If the over-identifying assumptions are rejected, what problem does it create for your IV estimation of the wage equation?
5 Von Wachter & Bender (2006) 3 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 y i = X i β + D i γ + ε i 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 γ 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 γ drops to -0.012 when IV is used. a. Given the change in the estimate of γ 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? 3 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.
5. Knapp and Seakes (1994) 4 examine the factors that influence whether a student defaults on a college student loan. Below is a summary of 4 probit models that they estimated where the dependent variable is one if the student defaults on the loan and is zero otherwise. Family status is a dummy that equals one if the student is from a twoparent family; graduation is a dummy that equals one if the person finished his/her degree. The other variable names are sufficiently descriptive for your task ere. a. Using specification (1), estimate the probability of default on a student loan for a person who has graduated (graduation=1), comes from a one-parent family (family status=0), whose race is not black, and whose parent's income is $50,000 (note: parent's income is measured in $1000s of dollars in this problem --- so $50,000 is measured as 50 in the regression). Show the basic steps involved in your calculation. b. Notice that at the bottom of the table there is a row indicating whether "school dummies" are included. This indicates whether the specification includes a set of dummies indicating which of 26 different schools the student attended. Also, K is the number of explanatory variables in the specification, and log(l) is the log of the likelihood function. Based on the information provided, construct a test-statistic of the null hypothesis that there is no difference in the probability of default across the 26 schools the students attended. Show how you construct your statistic, describe its distribution, and whether you would reject the null at the.05 level. c. Controlling for the characteristics in specification (2), how much more or less likely is a black student to default on a student loan than a white student? d. One might argue that the racial difference in default rates could be related to racial differences in earnings after leaving school. Suppose that you had data on earnings for each person and could re-estimate the models above. Explain how you could use that data to estimate how much of the racial difference in default rates is due to racial differences in earnings. Be as precise as possible in explaining how you would do the calculation. 4 Laura Greene Knapp and Terry G. Seaks. "An Analysis of the Probabilty of Default on Federally Guaranteed Student Loans." Review of Economics and Statistics, August 1992.
6. Using data from the March 2008 CPS, I estimated a probit model of whether a worker is a union member. I estimated a model for men and women combined, and then a separate model for each sex. All Men Women Age 0.01000 0.0107 0.0108 (9.61) (7.14) (7.27) Dummies for years of education 9-11 0.0484-0.146 0.186 (0.41) (-0.78) (1.21) 12 0.365 0.0825 0.570 (3.64) (0.53) (4.34) 13-15 0.433 0.144 0.682 (4.32) (0.92) (5.18) 16 0.395 0.297 0.450 (3.90) (1.89) (3.36) >16 0.698 0.798 0.505 (6.75) (5.01) (3.65) Married 0.105 0.0601 0.139 (3.77) (1.53) (3.45) Constant -1.973-1.854-2.110 (-18.4) (-11.1) (-14.9) Observations 14,468 7,256 7,212 Log-likelihood -5736.601-2693.3881-2977.6747 a. Using the specification for all workers, compute the probability that a 40 year old single person is a union member. Show how you derived your answer. b. For the person described in (a), compute the marginal effect of being married on the probability of union coverage. Provide a brief description of how you performed the calculation. c. Perform a test of the hypothesis that the probit coefficients are identical for men and women. Explain how you calculated the test, its distribution, and whether you reject the null at the.05 level.