You are interested in estimating the return to education Bi for the men born in the U.S. between 1944 and 1950 the birth cohorts from which the Vietnam War veterans were selected. Specifically, you wish to estimate the model: log(wage) = Bo + Bieduc + Bevet + ajregionl + azregion2 + azregion3 + u, where wage is the person's hourly wage in USD, educ is number of years of completed education, vet is a dummy variable equal to 1 if the person is a Vietnam War veteran, and zero otherwise, and regionl region 4 are four mutually exclusive dummy variables for different regions of residence in the U.S. (Regions 2 and 3 are considered poor regions.) 1a. How would you design a random experiment to measure the effect of years of schooling on earnings? Would you get an unbiased estimate of the slope on educ, from this experiment? Explain. (5 points) I 1b. In the absence of a random experiment, what is the most likely direction of the omitted variable bias of the OLS estimate of the return to education, if any? Why? Be very specific: provide a variable in this context and comment on the direction of the bias. (3 points) 1c. Suppose now that you have data on each of the Vietnam War draft-eligible men's Differential Abilities Scale (DAS) test score when they were 4 years old. (The DAS test focuses on cognitive ability and concepts, such as abstract reasoning and critical thinking, and can be used for individuals aged 2 to 17.) How would you use the DAS test score (variable DAS) to improve your estimate of the true causal effect of education on earnings? Make sure your answer addresses the following: What is the model you would estimate (you must write down this model)? What is the estimation method you would use? Why do you think this model is an improvement over the original model? (9 points) I 1d. Recall from class that some researchers have suggested using an instrumental variable (IV) approach to estimating B1. Specifically, they suggested using variable lottery the Vietnam War draft lottery number that determined the sequence in which the draft eligible-men were called for military service - as an instrumental variable for educ. (Recall that each of the 366 days of the year, including February 29, was randomly assigned a lottery number; all draft- eligible men with the same lottery number (and, therefore, the same birth date) would be called to serve at once.) . What condition(s) does variable lottery have to satisfy in order to be a valid instrumental variable for educ? Do you think these conditions hold? Explain. Which one of these condition(s) is testable, and how would you test it? Be very specific: write down the model you would estimate in order to test it, the estimation method you would use, the hypothesis you would test in terms of the parameters of this model, and the result you hope to obtain from this test. (10 points) le. Assume for the purposes of this question that lottery is a valid IV for education in the original model. Below is Stata output from IV estimation of this model. ivreg lwage (educ=lottery) vet regionl region2 region3 region4 Instrumental variables (2SLS) regression Source SS df MS Model Residual 1193.80181 20056.0917 2 26,860 596.900905 .746689938 Number of obs (2, 26860) Prob > F R-squared Adj R-squared Root MSE 26,863 24.54 0.0000 0.0562 0.0561 .86411 Total 1 21249.8935 26,862 791076374 lwage Coef. Std. Err. t p>It! [95% Conf. Interval) educ vet regioni region2 region3 region4 cons .0399335 - .0165251 244217 289017 028035 0 9.189991 . 1921492 1.0025792 10047 .01971 .01424 (omitted) 2.567908 0.21 -6.41 2.43 14.66 1.97 0.835 0.000 0.015 0.000 0.049 -.3366889 -.0215805 . 0471798 2503526 .0000958 . 416556 -.0114698 .4412545 . 3276827 .0559761 3.58 0.000 4.156756 14.22323 Instrumented: Instruments: educ vet lottery Answer the following questions: . How do you interpret the IV estimate of the slope on educ? (3 points) Based on the IV results, do you reject the null hypothesis that education is unrelated to wages, at the 5%significance level? Why or why not? Is the result what you would have expected? (3 points) Explain why region4 is omitted from the estimation. (3 points) 1f. Assume for the purposes of this question that lottery is a valid IV for education in the original model. Suppose that someone suggests that you do not need to control for variable vet, and that, instead of estimating the original model, you should use the IV estimation method to estimate the model: log(wage) = Bo + Bieduc + ajregionl + azregion2 + azregion3 + v. Do you think this is a good idea in terms of the validity of your IV lottery? Explain. (7 points) 1g. Suppose now that you find data on each person's age, so you decide to include this variable in the original model: log(wage) = Bo + Bieduc + Buvet + Bzage+ ajregionl + azregion2 + azregion3 + u. Answer the following questions: How would you rewrite this model if you knew that the effect of education depends on the person's age (age)? For instance, suppose that one more year of education is more valuable when you are 25 years old than when you are 55 years old. (4 points) . What is the partial effect of education in the model you wrote in terms of the parameters of that model? Show formally. (4 points) 1h. Lastly, consider again the original model: log(wage) = Bo + Bieduc + Bevet + ajregionl + azregion2 + azregion3 + u, Write down the following null hypotheses in terms of the parameters of the model above: Conditional on veteran status and educational attainment, men in region 1 have the same log-earnings, on average, as men in region 4. (4 points) On average, all men in poor regions earn the same log-wages, conditional on veteran status and educational attainment. (5 points) You are interested in estimating the return to education Bi for the men born in the U.S. between 1944 and 1950 the birth cohorts from which the Vietnam War veterans were selected. Specifically, you wish to estimate the model: log(wage) = Bo + Bieduc + Bevet + ajregionl + azregion2 + azregion3 + u, where wage is the person's hourly wage in USD, educ is number of years of completed education, vet is a dummy variable equal to 1 if the person is a Vietnam War veteran, and zero otherwise, and regionl region 4 are four mutually exclusive dummy variables for different regions of residence in the U.S. (Regions 2 and 3 are considered poor regions.) 1a. How would you design a random experiment to measure the effect of years of schooling on earnings? Would you get an unbiased estimate of the slope on educ, from this experiment? Explain. (5 points) I 1b. In the absence of a random experiment, what is the most likely direction of the omitted variable bias of the OLS estimate of the return to education, if any? Why? Be very specific: provide a variable in this context and comment on the direction of the bias. (3 points) 1c. Suppose now that you have data on each of the Vietnam War draft-eligible men's Differential Abilities Scale (DAS) test score when they were 4 years old. (The DAS test focuses on cognitive ability and concepts, such as abstract reasoning and critical thinking, and can be used for individuals aged 2 to 17.) How would you use the DAS test score (variable DAS) to improve your estimate of the true causal effect of education on earnings? Make sure your answer addresses the following: What is the model you would estimate (you must write down this model)? What is the estimation method you would use? Why do you think this model is an improvement over the original model? (9 points) I 1d. Recall from class that some researchers have suggested using an instrumental variable (IV) approach to estimating B1. Specifically, they suggested using variable lottery the Vietnam War draft lottery number that determined the sequence in which the draft eligible-men were called for military service - as an instrumental variable for educ. (Recall that each of the 366 days of the year, including February 29, was randomly assigned a lottery number; all draft- eligible men with the same lottery number (and, therefore, the same birth date) would be called to serve at once.) . What condition(s) does variable lottery have to satisfy in order to be a valid instrumental variable for educ? Do you think these conditions hold? Explain. Which one of these condition(s) is testable, and how would you test it? Be very specific: write down the model you would estimate in order to test it, the estimation method you would use, the hypothesis you would test in terms of the parameters of this model, and the result you hope to obtain from this test. (10 points) le. Assume for the purposes of this question that lottery is a valid IV for education in the original model. Below is Stata output from IV estimation of this model. ivreg lwage (educ=lottery) vet regionl region2 region3 region4 Instrumental variables (2SLS) regression Source SS df MS Model Residual 1193.80181 20056.0917 2 26,860 596.900905 .746689938 Number of obs (2, 26860) Prob > F R-squared Adj R-squared Root MSE 26,863 24.54 0.0000 0.0562 0.0561 .86411 Total 1 21249.8935 26,862 791076374 lwage Coef. Std. Err. t p>It! [95% Conf. Interval) educ vet regioni region2 region3 region4 cons .0399335 - .0165251 244217 289017 028035 0 9.189991 . 1921492 1.0025792 10047 .01971 .01424 (omitted) 2.567908 0.21 -6.41 2.43 14.66 1.97 0.835 0.000 0.015 0.000 0.049 -.3366889 -.0215805 . 0471798 2503526 .0000958 . 416556 -.0114698 .4412545 . 3276827 .0559761 3.58 0.000 4.156756 14.22323 Instrumented: Instruments: educ vet lottery Answer the following questions: . How do you interpret the IV estimate of the slope on educ? (3 points) Based on the IV results, do you reject the null hypothesis that education is unrelated to wages, at the 5%significance level? Why or why not? Is the result what you would have expected? (3 points) Explain why region4 is omitted from the estimation. (3 points) 1f. Assume for the purposes of this question that lottery is a valid IV for education in the original model. Suppose that someone suggests that you do not need to control for variable vet, and that, instead of estimating the original model, you should use the IV estimation method to estimate the model: log(wage) = Bo + Bieduc + ajregionl + azregion2 + azregion3 + v. Do you think this is a good idea in terms of the validity of your IV lottery? Explain. (7 points) 1g. Suppose now that you find data on each person's age, so you decide to include this variable in the original model: log(wage) = Bo + Bieduc + Buvet + Bzage+ ajregionl + azregion2 + azregion3 + u. Answer the following questions: How would you rewrite this model if you knew that the effect of education depends on the person's age (age)? For instance, suppose that one more year of education is more valuable when you are 25 years old than when you are 55 years old. (4 points) . What is the partial effect of education in the model you wrote in terms of the parameters of that model? Show formally. (4 points) 1h. Lastly, consider again the original model: log(wage) = Bo + Bieduc + Bevet + ajregionl + azregion2 + azregion3 + u, Write down the following null hypotheses in terms of the parameters of the model above: Conditional on veteran status and educational attainment, men in region 1 have the same log-earnings, on average, as men in region 4. (4 points) On average, all men in poor regions earn the same log-wages, conditional on veteran status and educational attainment. (5 points)