What happens when employers screen their employees but only observe a subset of output? We specify a model with heterogeneous employees and show that their response to the screening affects output in both the probationary period and the post-probationary period. The post-probationary impact is due to their heterogeneous responses affecting which individuals are retained and hence the screening efficiency. We show that the impact of the endogenous response on both the unobserved outcome and screening efficiency depends on whether increased effort on one task increases or decreases the marginal cost of effort on the other task. If the response decreases unobserved output in the probationary period then it increases the screening efficiency, and vice versa. We then assess these predictions empirically by studying a change to teacher tenure policy in New York City, which increased the role that a single measure -- test score value-added -- played in tenure decisions. We show that in response to the policy teachers increased test score value-added and decreased output that did not enter the tenure decision. The increase in test score value-added was largest for the teachers with more ability to improve students' untargeted outcomes, increasing their likelihood of getting tenure. We estimate that the endogenous response to the policy announcement reduced the screening efficiency gap -- defined as the reduction of screening efficiency stemming from the partial observability of output -- by 28%, effectively shifting some of the cost of partial observability from the post-tenure period to the pre-tenure period.
We study whether reallocating existing teachers across schools within a district can increase student achievement, and what policies would help achieve these gains. Using a model of multi-dimensional value-added, we find meaningful achievement gains from reallocating teachers within a district. Using an estimated equilibrium model of the teacher labor market, we find that achieving most of these gains requires directly affecting teachers' preferences over schools. In contrast, directly affecting principals' selection of teachers can lower student achievement. Our analysis highlights the importance of equilibrium and second-best reasoning in analyzing teacher labor market policies.