will use to adjust for measurement error. Appendix Table A2 correlates teacher content knowledge with teacher characteristics for public school teachers.<sup>20</sup> Teacher characteristics only explain 7% of the variation in content knowledge, and reassuringly, having a bachelor’s degree is robustly correlated with content knowledge. However, the correlation is relatively small (0.2 to 0.3sd), which could either reflect the quality of the degree or ‘learning on the job’ among those without a degree.<sup>21</sup> Teacher quality is identified following the TVA literature (Rockoff, 2004; Chetty et al., 2014a; and Kane and Staiger, 2008) by regressing student test scores on a function of their lagged test scores, round, grade, and teacher fixed effects. Teacher value-added is the estimated teacher fixed effect. The panel structure of the data, where both students and teachers are observed multiple times, is important for identification: to be included in the value-added calculations, students must be observed at least twice across consecutive years, since they require a lagged test score to control for selection. To separate correlation in student outcomes within years from TVA, at least some teachers must also be observed across years so that round fixed effects are identified. To estimate the variance of teacher value-added, we cannot simply take the variance of these value-added estimates, both because it would be biased by sampling bias and because we cannot separately identify classroom-level shocks from teacher value-added for teachers who are only observed once. While this does not bias our estimates when teacher value-added is an outcome variable, as long as classroom shocks are uncorrelated with the independent variables in the regression, it would bias estimates of the variance of teacher effects. Therefore, to estimate the variance of teacher value-added, we focus on the sample of teachers who are observed at least two times in the data.

We observe a total of 1,756 public school teachers in at least one round of the data linked to 22,857 unique public school students and 1,346 private school teachers linked to 9,741 unique students. Appendix Figures A5 and A6 document this variation in public and private schools. However, we are only able to estimate TVAs for 1,533 of the public teachers and 975 of the private teachers. This is primarily because the TVA estimation does not include students if they were not observed in the prior year. In particular, this means that we cannot calculate TVAs for teachers who were only observed in the first round of the data. Table 2 provides more information on the sources of variation for the TVA calculations in both public and private schools. When we correlate TVAs with teacher characteristics, our sample is further reduced to 1,383 public and 294 private teachers. This is because detailed data on when a teacher started teaching, which allows us to include our experience controls, was only collected in the fourth round of the LEAPs study. One limitation of these data is that we cannot identify the association between teacher gender and TVA

> 20The regressions for private school teachers yield qualitatively similar results.

> 21Interestingly, gender is another covariate that consistently predicts teacher knowledge. Being a female teacher is associated with 0.15sd lower mean content knowledge.

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