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1. 1 North Carolina data coverage: grades 4-5 from 1996-97 through 2010-11 and grade 3 from 1996-97 through 2009-10. The difference in sample sizes comparing columns (1) and (2) arises because we drop 1.37 million student-year observations that cannot be matched to their classroom teacher (see Appendix C for more detail).
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2. 2 LAUSD data coverage: grades 4-5 from 2003-04 through 2012-13 and 2015-16 through 2016-17 school years and third grade from 2003-04 through 2012-13.
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11. C Construction of the Teacher Value-Added Sample This appendix describes the construction of the final sample of students and teachers used for teacher VA estimation in both of our administrative datasets. Sample selection follows prior work (for instance, Chetty et al. (2014a,b)), the main requirements for inclusion in the sample being that the student has a valid score in a given subject both in the current and prior period, and can be matched to a teacher in that subject.
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16. Chetty, Raj, John N. Friedman, and Jonah E. Rockoff (2017), “Measuring the impacts of teachers: Reply.” American Economic Review, 107, 1685–1717.
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18. D Linking Long-run Outcomes to Teacher VA In Section 7.2, we referenced a method to link long-run outcomes with teacher VA proposed by Chetty et al. (2014b). In this appendix, for completeness, we describe the steps involved.
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19. Data on dropouts and graduations are available for school years 2011-12 through 2016-17. These data indicate whether students in the twelfth grade cohort of that year graduated or dropped out and so we restrict our data to cohorts in twelfth grade during this time period.61 We have a dropout or graduation record for 129,456 students – fifty percent of students from eligible cohorts.
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39. i yijt. The estimator commonly used in the literature is a method-ofmoments estimator proposed by Kane and Staiger (2008) under the additional assumption that αj ∼ N(0, σ2 α). Specifically, they propose the following estimators for the variance parameters: σ̂2 α = c cov(yjt, yjt−1) σ̂2 = b V (yijt) − σ̂2 α.
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40. In terms of demographics, we have information about parental education (six education groups, 1996-97 through 2005-06 only), economically disadvantaged status (1998-99 through 2010-11 only), ethnicity (six ethnic groups), gender, limited English status, disability status, academically gifted status and grade repetition. Besides missing data in some years for parental education and economically disadvantaged status, our demographic data cover over 99 percent of all student-year observations. When demographic information is missing, we create a missing indicator for that variable.
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41. Jackson, C. Kirabo (2018), “What do test scores miss? The importance of teacher effects on non-test score outcomes.” Journal of Political Economy, 126, 2072–2107.
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49. Koedel, Cory, Kata Mihaly, and Jonah E. Rockoff (2015), “Value-added modeling: A review.” Economics of Education Review, 47, 180–195.

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51. Koenker, Roger and Jiaying Gu (2017), “Rebayes: An R package for empirical Bayes mixture methods.” Journal of Statistical Software, 82, 1–26.
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55. LAUSD: California High School Exit Examination (CAHSEE) data cover 2003-04 through 201415. As the CAHSEE is normally first taken in tenth grade, we keep cohorts who were in tenth grade during this time period.60 If a student took the CAHSEE multiple times, we take the score from the student’s first sitting of the CAHSEE. We report the sum of CAHSEE scores from the mathematics and English sections and so CAHSEE scores range from 550 to 900. We have CAHSEE records for 184,128 students, covering sixty-seven percent of students from eligible cohorts.
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56. Li, Tong and Quang Vuong (1998), “Nonparametric estimation of the measurement error model using multiple indicators.” Journal of Multivariate Analysis, 65, 139–165.
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57. Lindsay, Bruce G. (1995), Mixture Models: Theory, Geometry, and Applications. Conference Board of the Mathematical Sciences: NSF-CBMS regional conference series in probability and statistics, Institute of Mathematical Statistics.
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58. McLachlan, G.J. (1987), “On bootstrapping likelihood ratio test statistics for the number of components in a normal mixture.” Journal of the Royal Statistical Society, Series C, 36, 318–324.
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59. North Carolina: Dropout data are available for school years 2003-04 through 2016-17. Given the majority of students drop out in tenth through twelfth grade, we ensure that our dropout data coverage starts in at least tenth grade for a given cohort65 and covers up to twelfth grade.66 Any student that has either not dropped out or moved out-of-state is coded as not being a high school dropout. We have dropout outcomes for 1,097,381 students.
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60. Other tests for normality are also possible. For instance, if α indeed follows a normal distribution N(0, σ2 α), then the logarithm of its characteristic function takes the form log φα(t) = −t2 /σ2 α , which implies that the first-order derivative with respect to t is of the form −t/σ2 α, which is a linear function of t. Since the distribution of α is identified (as established in Theorem 1), we can construct a consistent estimator for φα(t) and inspect linearity of the derivative of its logarithm transformation. Another specification test has been proposed in Bonhomme and Weidner (2019). We leave to future research a power comparison involving these and other specification tests. H Computation Appendix In this brief appendix, we discuss the estimation of F using NPMLE.
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61. Petek, Nathan and Nolan Pope (2018), “The multidimensional impact of teachers on students.” URL http://www.econweb.umd.edu/~pope/Nolan_Pope_JMP.pdf. Unpublished.
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62. Polyanskiy, Yury and Yihong Wu (2020), “Self-regularizing property of nonparametric maximum likelihood estimator in mixture models.” arXiv preprint arXiv:2008.08244.
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63. Proof. Using equation (2.64) in Rao (1992), we have log φα(t) = iE[α]t + Z t 0 ∂ ∂u log ψ(u, v) ψ(u, 0)ψ(0, v) u=0 dv. where i is the imaginary root. Using the fact that ∂ ∂u log ψ(u, v) ψ(u, 0)ψ(0, v) u=0 = ∂ψ(0, v)/∂u ψ(0, v) − ∂ψ(0, 0)/∂u ψ(0, 0) and that ∂ψ(0,0)/∂u ψ(0,0) = iE(Y1), we have log φα(t) = iE[α]t + Z t 0 ∂ψ(0, v)/∂u ψ(0, v) dv − iE(Y1)t = Z t 0 ∂ψ(0, v)/∂u ψ(0, v) dv, where the second equality holds because 1 has mean zero under Assumption 1. Additionally, under Assumptions 1- 3, we have ψ(u, v) = φα(u + v)φ(u)φ(v). Let u = 0, then φ(v) = ψ(0, v)/φα(v); and letting v = 0, then φ(v) = ψ(u, 0)/φα(u).
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64. PSAT data cover school years 2008-09 through 2016-17. As the PSAT is normally taken in tenth grade, we keep cohorts who were in tenth grade during this time period.62 We convert PSAT scores from the 2015-16 and 2016-17 administrations using the concordance tables provided by the College Board so that all PSAT scores are reported on a 600-2400 scale.63 If a student is recorded as receiving multiple administrations of the PSAT, we take the score from the first PSAT test taken by the student. We have PSAT records for 209,675 students, covering fifty-one percent of students from eligible cohorts.

65. PSAT data cover school years 2012-13 through 2016-17. As the PSAT is normally taken in tenth grade, we keep cohorts in tenth grade from 2012-13 through 2016-17.67 We convert PSAT scores from the 2015-16 and 2016-17 administrations using the concordance tables provided by the College Board, so that all PSAT scores are reported on a 600-2400 scale. If a student is recorded as receiving multiple administrations of the PSAT, we take the score from the first time the student took the PSAT test. We have PSAT records for 159,028 students, covering forty-four percent of students from eligible cohorts.
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66. Rao, B.L.S.P. (1992), Identifiability in Stochastic Models: Charaterization of Probability Distributions. Academic Press, United Kindom.
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67. Rivkin, Steven G., Eric A. Hanushek, and John F. Kain (2005), “Teachers, schools, and academic achievement.” Econometrica, 73, 417–458.
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68. Robbins, Herbert (1950), “A generalization of the method of maximum likelihood: Estimating a mixing distribution.” Annals of Mathematical Statistics, 21, 314–315.
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69. Robbins, Herbert (1956), “An empirical Bayes approach to statistics.” In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, volume I, University of California Press, Berkeley.
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70. Rockoff, Jonah E. (2004), “The impact of individual teachers on student achievement: Evidence from panel data.” American Economic Review, 94, 247–252.

71. Rothstein, Jesse (2017), “Measuring the impacts of teachers: Comment.” American Economic Review, 107, 1656–84.
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72. Saha, Sujayam and Adityanand Guntuboyina (2020), “On the nonparametric maximum likelihood estimator for gaussian location mixture densities with application to gaussian denoising.” Annals of Statistics, 48, 738–762.
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73. SAT data cover school years 2006-07 through 2016-17. As the SAT is usually taken in the eleventh or twelfth grade, we keep cohorts who attended both grades during the time period covered 60 We therefore drop fifth grade after 2009-10, fourth grade after 2008-09, and third grade after 2007-08. 61 We therefore drop fifth grade after 2008-09, fourth grade after 2007-08, and third grade after 2006-07. 62 We therefore drop fifth grade after 2011-12, fourth grade after 2010-11, and third grade after 2009-10. 63 Available at https://collegereadiness.collegeboard.org/pdf/2015-psat-nmsqt-concordance-tables.pdf.
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74. Suspension data are available from 2000-01 through 2016-17, although data from 2004-05 are missing. Here, we keep all data, since the suspension data cover a majority of high school years for all cohorts. We find the total number of days of out-of-school suspensions for each student occurring in middle or high school (grades 6-12). We top-code the number of days suspended in a school year at ten days. If we do not find the student in the suspension files, we assume that the student has never been suspended. Thus we have suspension outcomes for our full value-added sample of 1,386,555 students.
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75. Suspension data are available from 2003-04 through 2016-17. We restrict data to 2003-04 through 2009-10 to allow for sufficient time to elapse for students to receive suspensions after a student is taught by a given teacher. We calculate the total number of days of out-of-school suspensions for each student occurring in middle or high school (grades 6-12). If we do not find the student in the suspension files, we assume the student has never been suspended. Of note, the LAUSD embarked on an ambitious policy to eliminate “wilful defiance” suspensions, causing a large drop (almost seventy-five percent) in suspension rates in the 2010s, creating a low rate of suspension in the LAUSD data. We have suspension outcomes for our full value-added sample of 426,074 students from 2003-04 through 2009-10.
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76. van de Geer, Sara (1993), “Hellinger-consistency of certain nonparametric maximum likelihood estimators.” Annals of Statistics, 14–44.
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77. via the NPMLE of F performs similarly to the infeasible NPB estimator δNPB j defined in (2.6). It is surprising that the proposed estimator achieves this close approximation to the infeasible estimator, given the well-known fact that the NPMLE of F has a slow convergence rate (Fan, 1991). The key reason is that the nonparametric Bayes rule is a smooth functional of F, which can be estimated at a much better rate than the distribution F itself.
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78. We follow Chetty et al. (2014a,b) and parametrize the control function for lagged test scores f1g(yi,t−1) with a cubic polynomial in prior-year scores in mathematics and English and interact these cubics with the student’s grade level. When prior test scores in the other subject are missing, we set the other subject prior score to zero and include an indicator for missing data in the other subject interacted with the controls for prior own-subject test scores. We parametrize the control function for teacher experience f2(ej(i,g,t)) using dummies for years of experience from 0 to 5, the omitted group being teachers with 6 or more years of experience. The student-level control vector Xigt consists of the respective demographic variables in each dataset.
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79. We report the size and power performance of the proposed parametric bootstrap test in Table E.1 below, based on the following data generating process: Fix the sample size at n = 1000, and for a grid values of h ∈ {0, 0.4, 0.6, 0.8, 1}, sample individual αj’s from the following three-component normal distribution: 0.025N(−h, θh) + 0.95N(0, θh) + 0.025N(h, θh) with θh = 0.1−0.05h2. The design of θh is such that the variance of α is always 0.1; this is roughly the variance of the teacher effects in the LAUSD data. When h = 0, the latent effect αj follows a normal distribution, and the bootstrap test should reject with probability equal to nominal size.
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80. We then make several sample restrictions. First, we drop 100,000 student-year observations that cannot be matched to a teacher. Second, we drop 180,000 observations where we lack data on teacher experience; the data we drop here are over-represented in early years since we only have teacher experience data from 2007-08 onwards.58 Third, we only include classes with more than seven but fewer than forty students with valid current and lagged test scores in that subject, losing 11,000 observations. Fourth, we exclude 70,000 observations that lack a valid current or lagged test score in that subject.59 Our final sample is roughly 1.3 million student-year observations, covering roughly 660,000 million students and 11,000 teachers. Constructing Value-Added: With both samples in hand, we construct VA estimates for each teacher by running the following regression: yigt = f1g(yi,t−1) + f2(ej(i,g,t)) + φ1Xigt + φ2X̄c(i,g,t) + vj + igt .
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15. JAQ of All Trades: Job Mismatch, Firm Productivity and Managerial Quality. (2022). Tåg, Joacim ; Scognamiglio, Annalisa ; Pagano, Marco ; Coraggio, Luca ; Tg, Joacim.
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16. JAQ of All Trades: Job Mismatch, Firm Productivity and Managerial Quality. (2022). Tåg, Joacim ; Scognamiglio, Annalisa ; Pagano, Marco ; Coraggio, Luca ; Tg, Joacim.
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17. Asymptotic properties of the weighted-average least squares (WALS) estimator. (2022). Peracchi, Franco ; De Luca, Giuseppe ; Magnus, Jan R.
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18. Documento de Trabalho 03/2022 - Aprendizado de máquina e antitruste. (2022). de Macedo, Tatiana.
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19. Economic Predictions With Big Data: The Illusion of Sparsity. (2021). Primiceri, Giorgio ; Lenza, Michele ; Giannone, Domenico.
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20. A Model of Scientific Communication. (2021). Shapiro, Jesse ; Andrews, Isaiah.
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21. A Nonparametric Method for Estimating Teacher Value-Added. (2021). Gilraine, Michael ; McMillan, Robert ; Gu, Jiaying.
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22. Robust Empirical Bayes Confidence Intervals. (2021). Armstrong, Timothy ; Plagborg-Mller, Mikkel ; Kolesr, Michal.
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23. Is It Possible to Forecast the Price of Bitcoin?. (2021). Goutte, Stéphane ; Chevallier, Julien ; Guegan, Dominique.
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24. Is It Possible to Forecast the Price of Bitcoin?. (2021). Goutte, Stéphane ; Guegan, Dominique ; Chevallier, Julien.
    In: Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers).
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25. Is It Possible to Forecast the Price of Bitcoin?. (2021). Goutte, Stéphane ; Chevallier, Julien ; Guegan, Dominique.
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26. Market stability with machine learning agents. (2021). Pereira, Javier ; Georges, Christophre.
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27. Targeting humanitarian aid using administrative data: Model design and validation. (2021). O'Connell, Stephen ; Altindag, Onur ; Cadoni, Paola ; Jerneck, Matilda ; Amaz, Aytu ; Foong, Aimee Kunze ; Balciolu, Zeynep.
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28. Economic predictions with big data: the illusion of sparsity. (2021). Primiceri, Giorgio ; Lenza, Michele ; Giannone, Domenico.
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29. Social capital determinants and labor market networks. (2021). Neumark, David ; Kutzbach, Mark ; Asquith, Brian ; Hellerstein, Judith K.
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30. A New Method for Estimating Teacher Value-Added. (2020). Gilraine, Michael ; McMillan, Robert ; Gu, Jiaying.
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31. A Model of Scientific Communication. (2020). Shapiro, Jesse ; Andrews, Isaiah.
    In: NBER Working Papers.
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32. Targeting humanitarian aid using administrative data: model design and validation. (2020). O'Connell, Stephen ; Altindag, Onur ; Balciolu, Zeynep ; Cadoni, Paola ; Jerneck, Matilda ; Amaz, Aytu ; Foong, Aimee Kunze.
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33. Asymptotic analysis of statistical decision rules in econometrics. (2020). Porter, Jack R ; Hirano, Keisuke.
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34. How is Machine Learning Useful for Macroeconomic Forecasting?. (2020). Stevanovic, Dalibor ; Goulet Coulombe, Philippe ; Surprenant, Stephane ; Leroux, Maxime.
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35. How is Machine Learning Useful for Macroeconomic Forecasting?. (2020). Stevanovic, Dalibor ; Goulet Coulombe, Philippe ; Surprenant, St'Ephane ; Leroux, Maxime.
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