Labor Field Exam Aug 19

Labor Field Exam August 2019 There are two parts to this exam. Each should take about one hour. Calculators are not necessary. Please explain your notation in order to maximize chances of partial credit. Write your answer to each part in a separate book. Part I (244) The work of Chetty, Friedman, and Rockoff (2014) has generated substantial interest in identifying and firing bad teachers. Suppose we have a dataset on teacher “mistakes” which might correspond to instances where a teacher fails to meet a performance goal. We suppose the number of mistakes Y j ≥ 0 that teacher j makes in a year can be modeled as the outcome of a series of binomial trials. That is, that: Y j iid ∼ Bin ( n, p j ) for j = 1 , ..., J, where p j is teacher j ’s mistake probability and n is the number of performance goals teachers face in a year, which we assume is constant across teachers. Assume further that there are only two types of teachers: “good” teachers with mistake probability p g and “bad” teachers with mistake probability p b > p g . We formalize this with the following mixture representation: p j iid ∼ πBin ( n, p b ) + (1 - π ) Bin ( n, p g ) , where π gives the share of teachers that are bad. 1. Derive an expression for the likelihood of observing the event Y j = y . 2. Let B j be an indicator for being a bad teacher. Derive an expression for the probability Pr ( B j = 1 | Y j = y ) that teacher j is bad given that she made y mistakes. Your expression should involve only the parameters ( n, p b , p w , π ). 3. Suppose teacher j makes y mistakes while teacher k makes z < y mistakes. Given this evidence, what is the probability that teacher k has a lower probability of making mistakes than teacher j (i.e., that p k < p j )? 4. What is the probability that teacher k has a higher probability of making mistakes than teacher j (i.e., that p k > p j )? Prove that this quantity is smaller than your answer to question #3. 5. How would you estimate the parameters ( p b , p g , π ) from a dataset { Y j } J j =1 of teacher mistakes? What conditions would you need to hold for this estimator to be consistent? 6. What concerns might you have with linking your estimates of this model to personnel decisions involving teachers? 1