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Question: suppose you are interested in estimating how health affects productivity...

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Suppose you are interested in estimating how health affects productivity. Specifically, you are interested in the following regression: where Y, is a variable representing the income earned by person i, and X, is a variable representing the health of person i. Income is a fairly straightforward concept to capture. Health, however, is a more complex concept that can be measured in many different ways. For example, body mass index (BMI weight in kilograms, divided by height in meters squared) is one indicator of health. People with very low BMI are considered underweight (which is typically not healthy) and people with very high BMI are considered overweight (which is typically not healthy) 1. List a few other numerical measures of health that one could use as X\ in this regression. 2. Suppose we use BMI as our measure of X. What is the interpretation of βο? What is the interpretation of β? (Remember, Y; represents the income of person i-lets assume this is measured in dollars 3. BMI is a continuous variable. Suppose we instead replace it with a dummy variable, as in the following regression: Here, lets define D, as a dummy variable that is equal to 1 if individual i is underweight and 0 if they are not underweight. Now, what is the interpretation of β0? What is the interpretation of β? 4. A researcher estimated regression 1 (not regression 2) using data from the United States and obtained a negative, significant B1. What does this imply about the relationship between BMI and income in the United States? Does this make sense to you? Why or why not? 5. A researcher estimated regression 1 (not regression 2) using data from India and obtained a positive B1, but it was not significantly different from zero. What does this imply about the relationship between BMI and income in India?6. A researcher estimated regression 2 (not regression 1) using data from India and obtained a negative, significant B1. Describe in words what this means. Bonus Question: Do these results contradict the results of the previous regression (equation 1 for India)? Why or why not? Hint: Think about what the regression coefficient β1 would be if the true relationship between BMI and income looked something like the above figure. 7. Why might the relationship between BMI and income be different in the United States and India? 8. As stated at the beginning of the question, you are interested in how health affects produc- tivity i.e., the causal effect of health on productivity. However, you are concerned that your regression might have an endogeneity problem. What does endogeneity mean in this context? 9. Name one omitted variable that could be creating an endogeneity problem in regression 1 for the United States. If this particular omitted variable were the only issue, how would this affect your estimate of Bi (in particular, how would it compare to the true ) 10. Describe one way in which reverse causality might be a problem in these regressions?

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