A. Suppose that you have a sample of some individuals who follow the "Paleo Diet" and some who do not. You are interested in the health effects of the Paleo Diet. Denote the health status of individual i by Yi.Xi is an indicator ("dummy") variable which equals 1 if individual i follows the Paleo Diet, and zero, if not. You are considering estimating the following model: Yi=+Xi+ei where i=1,2,,n. Assume that the errors are i.i.d. a. What needs to be true about the data for the OLS estimator of to be unbiased? b. Suppose that you are able to assign people randomly to the Paleo Diet or to a control group, which would allow them just to eat as they normally would. After the experiment, you calculate the average health status of the treatment group and that of the control group. Under what conditions would the difference between the average health of the treatment group and the average health of the control group give you the causal effect of the Paleo Diet on health? c. In the above question, instead of being randomly assigned, suppose that individuals who follow the Paleo Diet voluntarily chose to do so. Does this affect whether or not you should use OLS to estimate ? Is there a better approach? Explain. d. Suppose that you now get data on age for all individuals in your le, and denote the age of individual i by Ai. You hypothesize that health sample, and denote the status, Yi, is given by: Yi=+Xi+Ai+ei Your main interest is still about the health effects, if any, of the Paleo Diet, so you want to get an unbiased estimate of . If you only estimate Equation (2) using OLS, how would you calculate possible omitted variable bias? If you were interested in the sources of such bias, how would you isolate the individual effects of each source? B. It has been argued that Catholic parochial schools are more effective than nearby public schools at producing good educational outcomes that lead to high future incomes. Proponents of this viewpoint point to the fact that graduates of such schools have higher post-high-school incomes than do graduates of nearby public schools. To see this more precisely, consider Equation (2) above, and let Yi be the log of future income for individual i, and let Xi be a dummy variable that equals 1 if individual i went to a Catholic school, and is zero otherwise. Attendance at a Catholic school is, of course, voluntary. After estimating a regression similar to Equation (2), one researcher got an estimate of 0.12 for , meaning that attendance at a Catholic school is associated with a 12% higher income post-high-school. a. Is 12% an unbiased estimate of the Catholic school effect? Why, or why not? b. Suppose that families-Catholic or not- choose which neighborhoods to live in based on proximity to jobs, and that schools have nothing to do with location decisions. Could you then use proximity to Catholic schools as instruments for Catholic school attrendance? c. Now assume that Catholic families tend to live in neighborhoods that are near Catholic schools. Would you use proximity to Catholic schools as an instrument for Xi in this case? d. In Table 2.1 in AP (page 53), suppose that student 1 has 1996 earnings of $100,000 (instead of $110,000 ) and that student 5 has 1996 earnings of $50,000 (instead of $30,000). Calculate the average incomes of private college graduates and public college graduates. e. Using the logic of Table 2.1, how would you estimate the causal effect on 1996 earnings of going to a private college, as opposed to going to a public college? Note: Enrollment decisions are highlighted in gray. A. Suppose that you have a sample of some individuals who follow the "Paleo Diet" and some who do not. You are interested in the health effects of the Paleo Diet. Denote the health status of individual i by Yi.Xi is an indicator ("dummy") variable which equals 1 if individual i follows the Paleo Diet, and zero, if not. You are considering estimating the following model: Yi=+Xi+ei where i=1,2,,n. Assume that the errors are i.i.d. a. What needs to be true about the data for the OLS estimator of to be unbiased? b. Suppose that you are able to assign people randomly to the Paleo Diet or to a control group, which would allow them just to eat as they normally would. After the experiment, you calculate the average health status of the treatment group and that of the control group. Under what conditions would the difference between the average health of the treatment group and the average health of the control group give you the causal effect of the Paleo Diet on health? c. In the above question, instead of being randomly assigned, suppose that individuals who follow the Paleo Diet voluntarily chose to do so. Does this affect whether or not you should use OLS to estimate ? Is there a better approach? Explain. d. Suppose that you now get data on age for all individuals in your le, and denote the age of individual i by Ai. You hypothesize that health sample, and denote the status, Yi, is given by: Yi=+Xi+Ai+ei Your main interest is still about the health effects, if any, of the Paleo Diet, so you want to get an unbiased estimate of . If you only estimate Equation (2) using OLS, how would you calculate possible omitted variable bias? If you were interested in the sources of such bias, how would you isolate the individual effects of each source? B. It has been argued that Catholic parochial schools are more effective than nearby public schools at producing good educational outcomes that lead to high future incomes. Proponents of this viewpoint point to the fact that graduates of such schools have higher post-high-school incomes than do graduates of nearby public schools. To see this more precisely, consider Equation (2) above, and let Yi be the log of future income for individual i, and let Xi be a dummy variable that equals 1 if individual i went to a Catholic school, and is zero otherwise. Attendance at a Catholic school is, of course, voluntary. After estimating a regression similar to Equation (2), one researcher got an estimate of 0.12 for , meaning that attendance at a Catholic school is associated with a 12% higher income post-high-school. a. Is 12% an unbiased estimate of the Catholic school effect? Why, or why not? b. Suppose that families-Catholic or not- choose which neighborhoods to live in based on proximity to jobs, and that schools have nothing to do with location decisions. Could you then use proximity to Catholic schools as instruments for Catholic school attrendance? c. Now assume that Catholic families tend to live in neighborhoods that are near Catholic schools. Would you use proximity to Catholic schools as an instrument for Xi in this case? d. In Table 2.1 in AP (page 53), suppose that student 1 has 1996 earnings of $100,000 (instead of $110,000 ) and that student 5 has 1996 earnings of $50,000 (instead of $30,000). Calculate the average incomes of private college graduates and public college graduates. e. Using the logic of Table 2.1, how would you estimate the causal effect on 1996 earnings of going to a private college, as opposed to going to a public college? Note: Enrollment decisions are highlighted in gray