Question

Two-stage Least Squares (2SLS). You have cross-sectional data on individuals’ log earnings Yi, years of schooling Si, and an instrument Zi equal to the (demeaned) distance to the nearest college. Assume Yi = α + βSi + ui, E[ui | Zi] = 0, and that Zi affects Yi only through its impact on Si (think about the usual regression here, what assumptions should you make regarding the error of this second regression?) (a) Write down the first-stage and reduced-form regressions. Briefly explain what each coef- ficient measures. (b) Show how to compute the 2SLS estimate of β as a function of the first-stage and reduced- form coefficients. (c) Describe how you would test the instrument for relevance in the first stage, and state a rule-of-thumb threshold for the F -statistic. (d) Suppose the first-stage F -statistic is 5. What concerns does this raise for your 2SLS estimates? (e) Interpret the parameter identified by 2SLS in this context. Which subpopulation does it refer to?

          Two-stage Least Squares (2SLS). You have cross-sectional data on individuals’ log earnings Yi, years of schooling Si, and an instrument Zi equal to the (demeaned) distance to the nearest college. Assume Yi = α + βSi + ui, E[ui | Zi] = 0, and that Zi affects Yi only through its impact on Si (think about the usual regression here, what assumptions should you make regarding the error of this second regression?) (a) Write down the first-stage and reduced-form regressions. Briefly explain what each coef- ficient measures. (b) Show how to compute the 2SLS estimate of β as a function of the first-stage and reduced- form coefficients. (c) Describe how you would test the instrument for relevance in the first stage, and state a rule-of-thumb threshold for the F -statistic. (d) Suppose the first-stage F -statistic is 5. What concerns does this raise for your 2SLS estimates? (e) Interpret the parameter identified by 2SLS in this context. Which subpopulation does it refer to?

Added by Taylor F.

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