Suppose that you wanted to study the effect of prison sentence length on recidivism rates (e.g., whether a former inmate is arrested again within years of being released). Suppose that you wanted to implement a regression discontinuity design. Assume that the data exist to do this.
a. Explain in general terms how you would perform this study—i.e., identify the treatment group and the control group, etc. Be as specific as you can, assuming that the crucial source of variation that you need does exist but describing it in as much detail as possible. In your answer, describe the ideal data set that you would use to perform this study.
b. What is the identifying assumption of the RD design in this context? What type of behavior would violate those assumptions in the study you’re suggesting in part a? More importantly, how could you test to see if the identifying assumptions are supported? Be as specific and comprehensive as possible.
c. Suppose that you want to allow for a quadratic function of the running variable, and want to allow for a different quadratic functional form on either side of the cutoff. What is the equation that you want to estimate? What is the variable of interest?
d. Suppose that this is a fuzzy RD design. Evaluate whether it would be a good or bad solution to drop all observed noncompliers from the data set.

Suppose now you want to study the same question, except you wish to use an instrumental variable strategy. You propose using assigned judge at trial as an instrument. You learn that initial assignment of the entire year’s cases to Judge A or Judge B is random, and Judge B assigns harsher penalties, on average. However, after the initial assignment they realized two judges were not enough, and then Judge C came along and non-randomly took on cases assigned to the other two judges.
a. What instrument would be valid to use this in this case? Explain.
b. What empirical tests could you do to show the validity of the instrument? Explain.
c. If you were to estimate the effect using reduced form IV, describe the sample you would use (e.g., only cases actually handled by Judges A and B, all cases assigned, only cases handled by Judge C, etc.) and the equation you would estimate and how you would interpret the results.
d. If you were to estimate this with 2SLS, what are the equations for the 1st and 2nd stages? Describe also the sample you would use.
e. How do you expect your 2SLS estimates to compare to estimates obtained from a naïve OLS regression given what you consider to be the major types of selection bias in this case? Explain, give examples, and be specific (e.g., OLS estimates are biased upward on number line, or are biased toward zero, or could be biased in either direction, etc.).