Essays in the Economics of Education UMI Number: 3183857 by Jesse Morris Rothstein A.B (Harvard University) 1995 A dissertation submitted in partial satisfaction of the requirements for the degree of Copyright 2003 by Rothstein, Jesse Morris All rights reserved Doctor of Philosophy in Economics in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY UMI Microform 3183857 Committee in charge: Professor David Card, Chair Professor John M Quigley Professor Steven Raphael Spring 2003 Copyright 2005 by ProQuest Information and Learning Company All rights reserved This microform edition is protected against unauthorized copying under Title 17, United States Code ProQuest Information and Learning Company 300 North Zeeb Road P.O Box 1346 Ann Arbor, MI 48106-1346 Abstract Essays in the Economics of Education by Jesse Morris Rothstein Doctor of Philosophy in Economics Essays in the Economics of Education University of California, Berkeley Professor David Card, Chair Copyright 2003 by Three essays consider implications of the strong association between student background characteristics and academic performance Jesse Morris Rothstein Chapter One considers the incentives that school choice policies might create for the efficient management of schools These incentives would be diluted if parents prefer schools with desirable peer groups to those with inferior peers but better policies and instruction I model a “Tiebout choice” housing market in which schools differ in both peer group and effectiveness If parental preferences depend primarily on school effectiveness, we should expect both that wealthy parents purchase houses near effective schools and that decentralization of educational governance facilitates this residential sorting On the other hand, if the peer group dominates effectiveness in parental preferences, wealthy families will still cluster together in equilibrium but not necessarily at effective schools I use a large sample of SAT-takers to examine the distribution of student outcomes across schools within metropolitan areas that differ in the structure of educational governance, and find little evidence that parents choose schools for characteristics other than peer groups This result suggests that competition may not induce improvements in educational productivity, and indeed I not obtain Hoxby’s (2000a) claimed relationship between school decentralization and student performance I address this discrepancy in Chapter Two Using Hoxby’s own data and specification, as described in her published paper, I am unable to replicate her positive estimate, and I find several reasons for concern about the validity of her conclusions Chapter Three considers the role of admissions tests in predictions of student collegiate performance Traditional predictive validity studies suffer from two important shortcomings First, they not adequately account for issues of sample selection Second, they ignore a wide class of student background variables that covary with both test scores To Joanie, for everything and collegiate success I propose an omitted variables estimator that is consistent under restrictive but sometimes plausible sample selection assumptions Using this estimator and data from the University of California, I find that school-level demographic characteristics account for a large portion of the SAT’s apparent predictive power This result casts doubt on the meritocratic foundations of exam-based admissions rules i College Performance Predictions and the SAT Contents List of Figures iv List of Tables v Preface vi Acknowledgements x Good Principals or Good Peers? Parental Valuation of School Characteristics, Tiebout Equilibrium, and the Incentive Effects of Competition among Jurisdictions 1.1 Introduction 1.2 Tiebout Sorting and the Role of Peer Groups: Intuition 10 1.3 A Model of Tiebout Sorting on Exogenous Community Attributes 15 1.3.1 Graphical illustration of market equilibrium 21 1.3.2 Simulation of expanding choice 24 1.3.3 Allocative implications and endogenous school effectiveness 27 1.4 Data .28 1.4.1 Measuring market concentration 28 1.4.2 Does district structure matter to school-level choice? 30 1.4.3 SAT data 34 1.5 Empirical Results: Choice and Effectiveness Sorting .37 1.5.1 Nonparametric estimates 38 1.5.2 Regression estimates of linear models 39 1.6 Empirical Results: Choice and Average SAT Scores 49 1.7 Conclusion 51 Tables and Figures for Chapter 55 Does Competition Among Public Schools Really Benefit Students? A Reappraisal of Hoxby (2000) References 128 Appendices 135 Appendix A Choice and School-Level Stratification .135 Appendix B Potential Endogeneity of Market Structure 137 Appendix C Selection into SAT-Taking 141 Appendix D Proofs of Results in Chapter 1, Section 144 Tables and Figures for Appendices 153 69 2.1 Introduction .69 2.2 Data and Methods 72 2.2.1 Econometric framework 76 2.3 Replication 78 2.4 Sensitivity to Geographic Match 80 2.5 Are Estimates From the Public Sector Biased? 82 2.6 Improved Estimation of Appropriate Standard Errors .85 2.7 Conclusion 88 Tables and Figures for Chapter 90 ii 97 3.1 Introduction .97 3.2 The Validity Model 100 3.2.1 Restriction of range corrections 101 3.2.2 The logical inconsistency of range corrections 102 3.3 Data 104 3.3.1 UC admissions processes and eligible subsample construction 106 3.4 Validity Estimates: Sparse Model .107 3.5 Possible Endogeneity of Matriculation, Campus, and Major 110 3.6 Decomposing the SAT’s Predictive Power 114 3.7 Discussion 119 Tables and Figures for Chapter 122 iii List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 List of Tables Schematic: Illustrative allocations of effective schools in Tiebout equilibrium, by size of peer effect and number of districts 62 Simulations: Average effectiveness of equilibrium schools in 3and 10-district markets, by income and importance of peer group 63 Simulations: Slope of effectiveness with respect to average income in Tiebout equilibrium, by market structure and importance of peer group 64 Distribution of district-level choice indices across 318 U.S metropolitan areas 65 Student characteristics and average SAT scores, school level 66 Nonparametric estimates of the school-level SAT score-peer group relationship, by choice quartile 67 “Upper limit” effect of fully decentralizing Miami’s school governance on the across-school distribution of SAT scores 68 3.1 Conditional expectation of SAT given HSGPA, three samples .127 B1 C1 D1 Number of school districts over time 160 SAT-taking rates and average SAT scores across MSAs 161 Illustration of single-crossing: Indifference curves in q-h space 161 1.1 1.2 1.3 1.4 1.5 Summary statistics for U.S MSAs 55 Effect of district-level choice index on income and racial stratification 56 Summary statistics for SAT sample 57 Effect of Tiebout choice on the school-level SAT score-peer group gradient 58 Effect of Tiebout choice on the school-level SAT score-peer group gradient: Alternative specifications 59 Effect of Tiebout choice on the school-level SAT score-peer group gradient: Evidence from the NELS and the CCD 60 Effect of Tiebout choice on average SAT scores across MSAs .61 1.6 1.7 2.1 2.2 First-stage models for the district-level choice index 90 Basic models for NELS 8th grade reading score, Hoxby (2000b) and replication 91 Effect of varying the sample definition on the estimated choice effect 92 Models that control for the MSA private enrollment share 93 Estimated choice effect when sample includes private schools 94 Alternative estimators of the choice effect sampling error, base replication sample .95 Estimates of Hoxby’s specification on SAT data 96 2.3 2.4 2.5 2.6 2.7 3.1 3.2 3.3 3.4 Summary statistics for UC matriculant and SAT-taker samples 122 Basic validity models, traditional and proposed models 123 Specification checks 124 Individual and school characteristics as determinants of SAT scores and GPAs 125 Accounting for individual and school characteristics in FGPA prediction 126 3.5 A1 A2 Evidence on choice-stratification relationship: Additional measures 153 Alternative measures of Tiebout choice: Effects on segregation and stratification .154 Effect of district-level choice on tract-level income and racial stratification 155 First-stage models for MSA choice index 156 2SLS estimates of effect of Tiebout choice .157 Sensitivity of individual and school average SAT variation to assumed selection parameter 158 Stability of school mean SAT score and peer group background characteristics over time .158 Effect of Tiebout choice on the school-level SAT score-peer group gradient: Estimates from class rank-reweighted sample 159 A3 B1 B2 C1 C2 C3 iv v insights into the underlying processes and new ways of thinking about the available policy Preface options The first two chapters consider parents’ choice of schools for their children The It is a well-established fact that students’ socioeconomic background has substantial predictive power for their educational outcomes Children whose parents are highly claim that parental choice can create incentives for schools to become more productive is a educated, whose households are stable, and whose families have high incomes substantially tenet of the neoclassical analysis of education It relies crucially on the assumption that outperform their less advantaged peers on every measure of educational output parents will choose effective, productive schools This is far from obvious—if peer effects With nearly as long a pedigree is the idea that these family background effects may are important, parents may be perfectly rational in preferring wealthy, ineffective schools to operate above the individual level The school-level association between average student competitors that are less advantaged but more effective, and even if there are no peer effects, background and average performance is typically much stronger than is the same association the strong association between school average test scores and student composition may at the individual level The interpretation of school-level correlations is nevertheless make it difficult for parents to assess a school’s effectiveness But if parents, in practice controversial: They may arise because academic outcome measures are noisy, implying that even if not by intent, choose schools primarily on the basis of their student composition group means are more reliable than are individual scores; because students with rather than for their effectiveness, the incentives created for school administrators will be unobservably attentive parents disproportionately attend schools that enroll observably diluted advantaged students; because the system of education funding assigns greater resources to Chapter One develops this idea and implements tests of the hypothesis that school schools in wealthy neighborhoods; or because there really are peer effects in educational effectiveness is an important determinant of residential choices among local-monopoly production school districts I model a “Tiebout”-style housing market in which house prices ration For many purposes, however, one need not know why it is that schools with access to desirable schools, which may be desirable either because they are particularly advantaged students outscore those with disadvantaged students; the fact that they is effective or because they enroll a desirable set of students I develop observable implications itself of substantial importance This dissertation focuses on two such topics: The of these two hypotheses for the degree of stratification of student test scores across schools, competitive impacts of school choice programs, and the design of college admissions rules and I look for evidence of these implications in data on the joint distribution of student In each case, when I incorporate into the standard analysis the key fact that student characteristics and SAT scores I find strong evidence that schools are an important composition may function as a signal of student performance (and vice versa), I obtain new component of the residential choice and that housing markets create sorting by family income across schools Tests of the hypothesis that this sorting is driven by parental pursuit vi vii of effective schools, however, come up empty This suggests that residential choice implement an omitted variables estimator that is unbiased under restrictive, but sometimes processes–and possibly, although the analogy is not particularly strong, non-residential plausible, assumptions about the selection process choice programs like vouchers—are unlikely to create incentives for schools to become more effective A second shortcoming of the validity literature is more fundamental In a world in which student background characteristics are known to be correlated with academic success This result conflicts with a well-known recent result from Hoxby (2000a), who (i.e with both SAT scores and collegiate grades), it is quite difficult to interpret validity argues that metropolitan areas with less centralized educational governance, and therefore estimates that fail to take account of these background characteristics A study can identify a more competition among local school districts, produce better student outcomes at lower test as predictively valid without being informative about whether the test provides an cost In Chapter Two, I attempt to get to the bottom of the discrepancy I reanalyze a independent measure of academic preparedness or simply proxies for the excluded portion of Hoxby’s data, and find reason to suspect the validity of her conclusions I am background characteristics unable to reproduce her results, which appear to be quite sensitive to the exact sample and In University of California data, I find evidence that observable background specification used I find suggestive evidence, however, that her estimates, from a sample of characteristics—particularly those describing the composition of the school, rather than the public school students, are upward biased by selection into private schools Moreover, an individual’s own background—are strong predictors of both SAT scores and collegiate investigation of the sampling variability of Hoxby’s estimates leads to the conclusion that her performance, and that much of the SAT’s apparent predictive power derives from its standard errors are understated, and that even her own point estimates of the competitive association with these background characteristics This suggests that the SAT may not be a effect are not significantly different from zero crucial part of the performance-maximizing admissions rule, as the background variables Chapter Three turns to a wholly different, but not unrelated, topic, the role of themselves provide nearly all the information contained in SAT scores It also suggests that admissions exam scores in the identification of well-prepared students in the college existing predictive validity evidence does not establish the frequent claim that the SAT is a admissions process The case for using such exams is often made with “validity” studies, meritocratic admissions tool, unless demographic characteristics are seen as measures of which estimate the correlation between test scores and eventual collegiate grades, both with student merit and without controls for high school grade point average I argue that there are two fundamental problems with these studies as they are often carried out First, they not adequately account for the biases created by estimation from a selected sample of students whose collegiate grades are observable because they were granted admission I propose and viii ix that in Chapter by the Center for Studies in Higher Education David Card and Alan Acknowledgements Krueger provided the SAT data used throughout Cecilia Rouse provided the hard-to-obtain I am very much indebted to David Card, for limitless advice and support throughout School District Data Book used in Chapters and Saul Geiser and Roger Studley of the my graduate school career The research here has benefited in innumerable ways from his University of California Office of the President provided the student records that permitted many suggestions, as have I It is hard to imagine a better advisor the research in Chapter The usual disclaimer applies: Any opinions, findings, I am grateful to the members of my various committees—Alan Auerbach, John conclusions or recommendations expressed are my own and not necessarily reflect the Quigley, Steve Raphael, Emmanuel Saez, and Eugene Smolensky—for reading drafts that views of the National Science Foundation, the Fisher Center, the Center for Studies in were far too long and too unpolished, and for nevertheless finding many errors and Higher Education, the College Board, the UC Office of the President, or any of my omissions advisors I have benefited from discussions with David Autor, Jared Bernstein, Ken Chay, Last, but not least, there is a sense in which Larry Mishel deserves substantial credit Tom Davidoff, John DiNardo, Nada Eissa, Jonah Gelbach, Alan Krueger, David Lee, for my Ph.D., as without his determined efforts at persuasion, I would never have pursued it Darren Lubotsky, Rob McMillan, Jack Porter, and Diane Whitmore, and from participants at in the first place several seminars where I have presented versions of the work contained here I also thank my various officemates over the last five years, particularly Liz Cascio, Justin McCrary, Till von Wachter, and Eric Verhoogen, for many helpful conversations All of the research contained here has been much improved by my interactions with those mentioned above, and with others who I have surely neglected here One must live while conducting research I thank my family and friends for putting up with me these last five years and for helping me to stay sane throughout I hope that I have not been too unbearable Much of my graduate career was supported under a National Science Foundation Graduate Research Fellowship In addition, the research in Chapters and was partially supported by the Fisher Center for Real Estate and Urban Economics at U.C Berkeley and x xi The potential effects of school choice programs depend critically on what Chapter characteristics parents value in schools Hanushek, for example, notes that parents might not choose effective schools over others that are less effective but offer “pleasant Good Principals or Good Peers? Parental Valuation of School Characteristics, Tiebout Equilibrium, and the Incentive Effects of Competition among Jurisdictions surroundings, athletic facilities, [and] cultural advantages,” (1981, p 34) To the extent that parents choose productive schools, market discipline can induce greater productivity from school administrators and teachers If parents primarily value other features, however, market discipline may be less successful Hanushek cautions: “If the efficiency of our school systems is due to poor incentives for teachers and administrators coupled with poor decisionmaking by consumers, it would be unwise to expect much from programs that seek to 1.1 Introduction strengthen ‘market forces’ in the selection of schools,” (1981, p 34-35; emphasis added) Many analysts have identified principal-agent problems as a major source of underperformance in public education Public school administrators need not compete for customers and are therefore free of the market discipline that aligns producer incentives with consumer demand in private markets Chubb and Moe, for example, argue that the interests of parents and students “tend to be far outweighed by teachers’ unions, professional organizations, and other entrenched interests that, in practice, have traditionally dominated Moreover, if students’ outcomes depend importantly on the characteristics of their classmates (i.e if so-called “peer effects” are important components of educational production), even rational, fully informed, test-score-maximizing parents may prefer schools with poor management but desirable peer groups to better managed competitors that enroll less desirable students, and administrators may be more reliably rewarded for enrolling the right peer group than for offering effective instruction the politics of education,” (1990, p 31).1 One proposed solution—advocated by Friedman (1962) and others—is to allow dissatisfied parents to choose another school, and to link school administrators’ compensation to parents’ revealed demand This would strengthen parents relative to other actors, and might “encourage competition among schools, forcing them into higher productivity,” (Hoxby, 1994, p 1) Chubb and Moe also identify the school characteristics that parents would presumably choose, given more influence: “strong leadership, clear and ambitious goals, strong academic programs, teacher professionalism, shared influence, and staff harmony,” (p 187) See also Hanushek (1986) and Hanushek and Raymond (2001) The mechanisms typically proposed to increase parental choice—vouchers, charter schools, etc.—are not at present sufficiently widespread to permit decisive empirical tests either of parental revealed preferences or of their ultimate effects on school productivity.2 Economists have long argued, however, that housing markets represent a long established, potentially informative form of school choice (Tiebout, 1956; Brennan and Buchanan, 1980; Hsieh and Urquiola (2002) study a large-scale voucher program in Chile, but argue that effects on school productivity cannot be distinguished from the allocative efficiency effects of student stratification Oates, 1985; Hoxby, 2000a) Parents exert some control over their children’s school A second issue is that there is little or no threat of market entry when competition is assignment via their residential location decisions, and can exit undesirable schools by among geographically-based school districts In the absence of entry, administrators of moving to a neighborhood served by a different school district As U.S metropolitan areas undesirable districts are not likely to face substantial declines in enrollment Indeed, a vary dramatically in the amount of control over children’s school assignment that the reasonable first approximation is that total (public) school and district enrollments are residential decision affords to parents, one can hope to infer the effect of so-called Tiebout invariant to schools’ relative desirability.5 Instead, Tiebout choice works by rewarding the choice by comparing student outcomes across metropolitan housing markets (Borland and administrator of a preferred school with a better student body and with wealthier and more Howsen, 1992; Hoxby, 2000a).3 motivated parents There are obvious benefits for educational personnel in attracting an In this chapter, I use data on school assignments and outcomes of students across schools within different metropolitan housing markets to assess parents’ revealed advantaged population, and I assume throughout this chapter that the promise of such rewards can create meaningful incentives for school administrators preferences To preview the results, I find little evidence that parents use Tiebout choice to My analysis of parental choices focuses on the possibility that parents may choose select effective schools over those with desirable peers, or that schools are on average more schools partly on the basis of the peer group offered Although existing research does not effective in markets that offer more choice conclusively establish the causal contribution of peer group characteristics to student In modeling the effects of parental preferences on equilibrium outcomes under outcomes (see, e.g., Coleman et al., 1966; Hanushek, Kain, and Rivkin, 2001; Katz, Kling, Tiebout choice, it is important to account for two key issues that not arise under choice and Liebman, 2001), anecdotal evidence suggests that parents may place substantial weight programs like vouchers The first is that residential choice rations access to highly- on the peer group in their assessments of schools and neighborhoods Realtor.com, a web demanded schools by willingness-to-pay for local housing.4 As a result, both schools and site for house hunters, offers reports on several neighborhood characteristics that parents districts in high-choice markets (those with many competing school districts) are more apparently value These include a few variables that may be interpreted as measures of stratified than in low-choice markets Increased stratification can have allocative efficiency school resources or effectiveness (e.g class size and the number of computers); detailed consequences that confound estimates of the effect of choice on productive efficiency socioeconomic data (e.g educational attainment and income); and the average SAT score at the local high school Given similar average scores, test-score maximizers should prefer Hoxby argues that this sort of analysis can “demonstrate general properties of school choice that are helpful for thinking about reforms,” (2000a, p 1209) Belfield and Levin (2001) review other, similar studies Small-scale voucher programs may not have to ration desired schools, or may be able to use lotteries for this purpose One imagines that broader programs will use some form of price system, perhaps by allowing parents to “top up” their vouchers (Epple and Romano, 1998) Poor school management can, of course, lead parents to choose private schools, lowering public enrollment Similarly, areas with bad schools may disproportionately attract childless families These are likely secondorder effects The private option, in any case, is not the mechanism by which residential choice works but an alternative to it: Inter-jurisdictional competition has been found to lower private enrollment rates (Urquiola, 1999; Hoxby, 2000a) demographically unfavorable schools, as these must add more value to attain the same identical peer groups I allow a continuous distribution of student characteristics, which outcomes as their competitors with more advantaged students.6 While it is possible that forces parents to trade off peer group against effectiveness in their school choices This parents use the demographic data in this way, it seems more likely that home buyers prefer seems a more accurate characterization of Tiebout markets, as the median U.S metropolitan wealthier neighborhoods, even conditional on average student performance (Downes and area has fewer than a dozen school districts from which to choose It leads to a substantially Zabel, 1997).7 different understanding of the market dynamics, as Hoxy’s assumption of competing schools With several school characteristics over which parents may choose, understanding which schools are chosen and which administrators are rewarded requires a model of with identical peer groups eliminates the “stickiness” that concern for peer group can create and that is the primary focus here residential choice I build on the framework of so-called multicommunity models in the As in other multicommunity models, equilibrium in my model exhibits complete local public finance literature (Ross and Yinger, 1999), but I introduce a component of stratification: High-income families live in districts that are preferred to (and have higher school desirability that is exogenous to parental decisions, “effectiveness,” which is thought housing prices than) those where low-income families live That this must hold regardless of of as the portion of schools’ effects on student performance that does not depend on the what parents value points to a fundamental identification problem in housing price-based characteristics of enrolled students Parental preferences among districts depend on both estimates of parental valuations: Peer group and, by extension, average student peer group and effectiveness, and I consider the implications of varying the relative weights performance are endogenous to unobserved determinants of housing prices One of these characteristics for the rewards that accrue in equilibrium to administrators of estimation strategy that accommodates this endogeneity is that taken by Bayer, McMillan, effective schools and Reuben (2002), who estimate a structural model for housing prices and community Hoxby (1999b) also models Tiebout choice of schools, but she assumes a discrete composition in San Francisco distribution of student types and allows parents to choose only among schools offering I adopt a different strategy: I compare housing markets that differ in the strength of the residential location-school assignment link, and I develop simple reduced-form This does not rely on assumptions about the peer effect: The effect of individual characteristics on own test scores, distinct from any spillover effects, is not attributable to the school, and test-score-maximizing parents should penalize the average test scores of schools with advantaged students to remove this effect (Kain, Staiger, and Samms, 2002) Postsecondary education offers additional evidence of strong preferences over the peer group: Colleges frequently trumpet the SAT scores of their incoming students—the peer group—while data on graduates’ achievements relative to others with similar initial qualifications, which would arguably be more informative about the college’s contribution, are essentially non-existent Along these lines, Tracy and Waldfogel (1997) find that popular press rankings of business schools reflect the quality of incoming students more than the schools’ contributions to students’ eventual salaries (but see also Dale and Krueger, 1999, who obtain somewhat conflicting results at the undergraduate level) implications of parental valuations for the across-school distribution of student characteristics and educational outcomes as a function of the strength of this link This across-market approach has the advantage that it does not rely on strong exclusion restrictions or distributional assumptions My primary assumptions are that the causal effect Shepard (1999) reviews hedonic studies of housing markets of individual and peer characteristics on student outcomes does not vary systematically with moving to the next lower peer group district and thus reduces the probability that wealthy the structure of educational governance; that the peer effect can be summarized with a small families will be trapped in districts with ineffective schools number of moments of the within-school distribution of student characteristics; and that Effectiveness sorting should be observable as a magnification of the causal peer school effectiveness acts to shift the average student outcome independent of the set of effect, as it creates a positive correlation between the peer group and an omitted variable— students enrolled school effectiveness—in regression models for student outcomes.9 This provides my Like Baker, McMillan, and Reuben (2002), I identify parental valuations by the identification: I look for evidence that the apparent peer effect, the reduced-form gradient location of clusters of high income families: If parental preferences over communities depend of school average test scores with respect to student characteristics, is larger in high-choice exclusively on the effectiveness of the local schools, the most desirable—and therefore than in low-choice markets If parents select schools for effectiveness, wealthy parents wealthiest—communities are necessarily those with the most effective schools If peer should be better able to obtain effective schools in markets where decentralized governance group matters at all to parents, however, there can be “unsorted” equilibria in which facilitates the choice of schools through residential location, and student performance should communities with ineffective schools have the wealthiest residents and are the most be more tightly associated with peer characteristics in these markets If parents instead select preferred These equilibria result from coordination failures: The wealthy families in schools primarily for the peer group, there is no expectation that wealthy students will attend ineffective districts would collectively have the highest bids for houses assigned to more effective schools in equilibrium, regardless of market structure, and the peer group-student effective schools, but no individual family is willing to move alone to a district with performance relationship should not vary systematically with Tiebout choice undesirable peers I use a unique data set consisting of observations on more than 300,000 The more importance that parents attach to school effectiveness, the more likely we metropolitan SAT takers from the 1994 cohort, matched to the high schools that students are to observe equilibria in which wealthy students attend more effective schools than attended The size of this sample permits accurate estimation of both peer quality and lower-income students Moreover, if parental concern for peer group is not too large, the average performance for the great majority of high schools in each of 177 metropolitan model predicts that this equilibrium effectiveness sorting will tend to be more complete in housing markets I find no evidence that the association between peer group and student high-choice markets, those with many small school districts, than in markets with more performance is stronger in high-choice than in low-choice markets This result is robust to centralized governance This is because higher choice markets divide the income distribution into smaller bins, which reduces the cost (in peer quality) that families pay for Willms and Echols (1992, 1993) are the first authors of whom I am aware to note the importance of the distinction between preferences for peer group and for effective schools They use hierarchical linear modeling techniques (Raudenbush and Willms, 1995; Raudenbush and Bryk, 2002), and estimate school effectiveness as the residual from a regression of total school effects on peer group This is appropriate if there is no effectiveness sorting; otherwise, it may understate the importance of effectiveness in output and in parental choices nonlinearity in the causal effects of the peer group as well as to several specifications of the does matter for student performance, but that it does not matter greatly to parental educational production function Moreover, although there is no other suitable data set with residential choices.11 This could be because effectiveness is swamped by the peer group in nearly the coverage of the SAT sample, the basic conclusions are supported by models parental preferences or because it is difficult to observe directly In either case, estimated both on administrative data measuring high school completion rates and on the administrators who pursue unproductive policies are unlikely to be disciplined by parental National Education Longitudinal Study (NELS) sample exit and Tiebout choice can create only weak incentives for productive school management This result calls the incentive effects of Tiebout choice into question, as it indicates that administrators of effective schools are no more likely to be rewarded with high demand 1.2 for local housing in high-choice than in low-choice markets To explore this further, I estimate models for the effect of Tiebout choice on mean scores across metropolitan areas Tiebout Sorting and the Role of Peer Groups: Intuition In this section I describe the Tiebout choice process and its observable implications in the context of a very simple educational technology with peer effects Let t ij = x ij β + x j γ + µ j + ε ij Consistent with the earlier results, I find no evidence that high-choice markets produce higher average SAT scores Together with the within-market estimates, this calls into question Hoxby’s (1999a, 2000a) conclusion that Tiebout choice induces higher productivity from school administrators.10 There are three plausible explanations for the pattern of findings presented here (1) be a reduced-form representation of the production function, where t ij is the test score (or other outcome measure) of student i when he or she attends school j ; x ij is an index of the student’s background characteristics; x j is the average background index among students at First, it may be that school and district policies are not responsible for a large share of the school j ; and µ j —which need not be orthogonal to x j —measures the “effectiveness” of extant across-school variation in student performance We would not then expect to school j, its policies and practices that contribute to student performance.12 observe effectiveness sorting, regardless of its extent, in the distribution of student SAT scores Second, the number of school districts may not capture variation in parents’ ability In fact, the main empirical approach cannot well distinguish between the case where parents value effectiveness to the exclusion of all else and that where they ignore effectiveness entirely, as in either case effectiveness sorting may not depend on the market structure The former hypothesis seems implausible on prior grounds, however 12 In the empirical application in Section 1.5, I allow for more general technologies in which the effects of individual or peer characteristics are arbitrarily nonlinear or higher moments of the peer group distribution enter the production function The key assumption is that all families agree on the relative importance of peer group and school effectiveness This rules out some forms of interactions between x ij and ( x j , µ j ) 11 to exercise Tiebout choice Results presented in Section 1.4.2 offer suggestive evidence against this interpretation, but not rule it out A final explanation is that effectiveness 10 Hoxby (2000a) argues that market structure is endogenous to school quality Instrumenting for it and using relatively sparse data from the NELS and the National Longitudinal Survey of Youth, she finds a positive effect of choice on mean scores across markets I discuss the endogeneity issue in Appendix B, and consider several instrumentation strategies As none indicate substantial bias in OLS results, the main discussion here treats market structure as exogenous Chapter investigates Hoxby’s results in greater detail in (1) The assumption of similar preference structures is common in studies of consumer demand, and in particular underlies both the multicommunity and hedonic literatures If it is violated, of course, the motivating question of whether parents prefer good principals or good peers is not well posed In view of the vast literature documenting the important role of family background 10 unions.14 It is worth noting that the relative magnitude of µ j may be quite modest Family characteristics—e.g ethnicity, parental income and education—in student achievement background variables typically explain the vast majority of the differences in average student (Coleman et al., 1966; Phillips et al., 1998; Bowen and Bok, 1998), I assume that x ij is test scores across schools, potentially leaving relatively little room for efficiency (or school positively correlated with willingness-to-pay for educational quality In the empirical analysis “value added”) effects.15 Nevertheless, most observers believe that public school efficiency below, I also estimate specifications that allow willingness-to-pay to depend on family is important, that it exerts a non-trivial role on the educational outcomes of students, and income while other characteristics have direct effects on student achievement that it varies substantially across schools Since model (1) excludes school resources, the term x j γ potentially captures both The potential efficiency-enhancing effects of increased Tiebout choice operate conventional peer group effects and other indirect effects associated with the family through the assumption that parents prefer schools with µ j -promoting policies To the background characteristics of students at school j For example, wealthy parents may be extent that this is true, Tiebout choice induces a positive correlation between µ j and x j , more likely to volunteer in their children’s schools, or to vote for increased tax rates to since high- x i families will outbid lower- x i families for homes near the most preferred support education They may also be more effective at exerting “voice” to manage agent behavior, even without the exit option that school choice policies provide (Hirschman, schools Thus, active Tiebout choice can magnify the apparent impact of peer groups on student outcomes in analyses that neglect administrative quality Formally, 1970) Finally, student composition may operate as an employment amenity for teachers and [ teachers that can be hired for any fixed salary (Antos and Rosen, 1975).13 ] [ ] E t j |x j = x j (β + γ ) + E µ j |x j , administrators, reducing the salaries that the school must pay and increasing the quality of (2) or, simplifying to a linear projection, [ ] E * t j |x j = x j (β + γ + θ * ), The effectiveness parameter in (1), µ j , encompasses the effects of any differences (3) across schools that not depend on the characteristics of students that they enroll It may include, for example, the ability and effort levels of local administrators, their choice of curricula, or their effectiveness in resisting the demands of bureaucrats and teacher’s 13 The distinction between direct and indirect effects of school composition is not always clear in discussions of peer effects Studies that use transitory within-school variation in the composition of the peer group (Hoxby, 2000b; Angrist and Lang, 2002; Hanushek, Kain, and Rivkin, 2001) likely estimate only the direct peer effect, while those that use the assignment of students to schools (Evans, Oates, and Schwab, 1992; Katz, Kling, and Liebman, 2001) likely estimate something closer to the full reduced-form effect of school composition 11 More precisely, ability and effort of school personnel is included in µ only to the extent that a good peer group does not enable a school to bid the best employees away from low- x schools A wealthy, involved population may not ensure high-quality, high-effort staff if agency problems produce district hiring policies that not reflect parents’ preferences (Chubb and Moe, 1990), or if it is difficult to enforce contracts over unobservable components of administrator actions (Hoxby, 1999b) 15 In the SAT data used here, a regression of school mean scores on average student characteristics has an R2 of 0.74 The correlation is substantially stronger in California’s school accountability data (Technical Design Group, 2000) Of course, these raw correlations may overstate the causal importance of peer group if there is effectiveness sorting 14 12 ( ) ( ) where θ * ≡ cov x j , µ j var x j represents the degree of effectiveness sorting in the local concerned only with school effectiveness, high- µ schools attract high- x families regardless market (For notational simplicity, I neglect the intercept in both test scores and school of the market structure, and θ * need not vary with local competition Similarly, when effectiveness.) The stronger are parental preferences for effective schools (relative to parental concern for peer group is large enough, even in highly competitive markets high- x schools with other desired attributes), the more actively will high- x i families seek out families are not drawn to high- µ schools, and again θ * is largely independent of market neighborhoods in effective districts, and the larger will θ * tend to be in Tiebout equilibrium The weaker are parental preferences for µ j relative to other factors, the smaller will θ structure This idea forms the basis of my empirical strategy In essence, I compare the sorting * parameter θ * in equation (3) across metropolitan housing markets with greater and lesser tend to be Importantly, one would expect the degree of local competition in public schooling [ degrees of residential school choice Let θ = θ (c , δ ) = E θ * |c , δ ] be the average (i.e the number of school districts in the local area among which parents can choose) to effectiveness sorting of markets characterized by the parameters c and δ , where c is the affect the magnitude of θ * whenever parents care both about peer groups and school degree of jurisdictional competition (i.e the number of competing districts from which effectiveness The reasoning is simple: If there are only a small number of local districts and parents can choose, adjusted for their relative sizes) and δ is the importance that parents parents value the peer group, they may be “stuck” with a high- x /low- µ school, even in place on peer group relative to effectiveness.17 The argument above, supported by the housing market equilibrium, by their unwillingness to sacrifice peer group in a move to a theoretical model developed in the next section, predicts that ∂θ ∂c > for moderate values more effective school district These coordination failures are less likely in markets with of δ but that ∂θ ∂c = when δ is zero or large (i.e when parents care only about more interjurisdictional competition, as in these markets there are always alternative districts that are relatively similar in the peer group offered, and parents are able to select effective schools without paying a steep price in reduced peer quality.16 When parental concern for peer group is moderate, then, a high degree of public school choice is needed to ensure that high- µ schools attract high- x families, and θ * tends to be larger in high-choice than in low-choice markets On the other hand, when parents are 16 effectiveness or only about peer group) To the extent that θ tends to increase with choice, then, we can infer that parents’ peer group preferences are small enough to prevent a breakdown in high-choice markets of the sorting mechanism that rewards high- µ administrators with high- x students On the other hand, if θ is no larger in high-choice 17 θ ( c , δ ) is treated as a random variable, as there can be multiple equilibria in these markets My empirical strategy assumes that δ is constant across markets, and that a sample of markets with the same c parameter * will trace out the distribution of θ An equilibrium selection model in which families could somehow coordinate on the most efficient equilibrium would violate this assumption * In the high choice limit, this is analogous to Hoxby’s (1999b) model of choice among schools with identical peers 13 14 than in low-choice cities it is more difficult to draw inferences about parental valuations, My model is a much simplified version of so-called “multicommunity” models I which may be characterized either by very small or very large δ In either case, however, we maintain the usual assumptions that the number of communities is fixed and finite, and that can expect little effect of expansions of Tiebout choice on school efficiency, as in the former access to desirable communities is rationed through the real estate market.19 There is no even markets with only a few districts can provide market discipline and in the latter no private sector that would de-link school quality from residential location Although some plausible amount of governmental fragmentation will create efficiency-enhancing incentives authors (i.e Epple and Zelenitz, 1981) include a supply side of the housing market, I assume for school administrators that communities are endowed with perfectly inelastic stocks of identical houses 20 Communities differ in three dimensions: The average income of their residents and the 1.3 A Model of Tiebout Sorting on Exogenous Community Attributes rental price of housing, both endogenous, and the effectiveness of the local schools.21 An important omission is of all non-school exogenous amenities like beaches, parks, In this section, I build a formal model of the Tiebout sorting process described above As my interest is in the demand side of the market under full information, I treat the views, and air quality I develop here a “best case” for Tiebout choice, where schools are the distribution of school effectiveness as exogenous and known to all market participants.18 I only factors in neighborhood desirability Amenities could either increase or reduce the demonstrate that Tiebout equilibrium must be stratified as much as the market structure extent of effectiveness sorting relative to this pure case, though the latter seems more likely.22 allows: Wealthy families always attend schools that are preferred to those attended by low- If, as the hedonics literature implies, schools are one of the more important determinants of income families There can be multiple equilibria, however, and the allocation of effective neighborhood desirability (see, e.g., Reback, 2001; Bogart and Cromwell, 2000; Figlio and schools is not uniquely determined by the model’s parameters Conventional comparative 19 This does not rule out administrative responses to the incentives created by parental choices, as these are a higher order phenomenon, deriving from competition among schools to attract students rather than from reactions of school administrators to the realized desirability of their schools My discussion presumes, however, that competition does not serve to reduce variation in school effectiveness Where most models incorporate within-community voting processes for public good provision (Fernandez and Rogerson, 1996; Epple and Romano 1996; Epple, Filimon and Romer, 1993), income redistribution (Epple and Romer, 1991; Epple and Platt, 1998), or zoning rules (Fernandez and Rogerson, 1997; Hamilton, 1975), I simply allow for preferences over the mean income of one’s neighbors These preferences might derive either from the effects of community composition on voting outcomes or from reduced-form peer effects in education 20 Tiebout equilibria must evolve quickly to provide discipline to school administrators, whose careers are much shorter than the lifespan of houses Inelastic supply is probably realistic in the short term, except possibly at the urban fringe Nechyba (1997) points out that it is much easier to establish existence of equilibrium with fixed supply 21 The inclusion of any exogenous component of community desirability is not standard in multicommunity models, which, beginning with Tiebout’s (1956) seminal paper, have typically treated communities as ex ante interchangeable This leaves no room for managerial effort or quality except as a deterministic function of community composition, so is inappropriate for analyses of the incentives that the threat of mobility creates for public-sector administrators 22 Amenities might draw wealthy families to low-peer-group districts, improving those districts’ peer groups and reducing the costs borne by other families living there This could increase effectiveness sorting, although the effect would be weakened if there were a private school sector Offsetting this, amenities might also prevent families from exiting localities with ineffective schools, reducing effectiveness sorting just as does concern for peer group 15 16 statics analysis is not meaningful when equilibrium is non-unique, as the parental valuation parameter affects the set of possible equilibria rather than altering a particular equilibrium To better understand the relationships between parental valuations, market concentration, and the equilibrium allocation, the formal exposition of the model is followed by simulations of markets under illustrative parameter values 18 Lucas, 2000; Black, 1999), the existence of relatively unimportant amenities should not much jurisdiction j is U ij = U ( x i − h j , x j δ + µ j ) , where U is twice differentiable everywhere with alter the trends identified here U and U both positive.25 I make the usual assumption about the utility function: Turning to the formal exposition, assume that a local housing market—a Single Crossing Property: U 12U1 − U 11U > everywhere metropolitan area—contains a finite number of jurisdictions, J, and a population of N Single crossing ensures that if any family prefers one school quality-price families, N >> J Each jurisdiction, indexed by j, contains n identical houses and is endowed with an exogenous effectiveness parameter, µ j No two jurisdictions have combination to another with lower quality—where quality is q j ≡ x j δ + µ j —all higherincome families as well; if any family prefers a district to another offering higher quality identical effectiveness Each family must rent a house There are enough houses to go around but not so many that there can be empty communities: n( J − 1) < N < nJ 23 All homes are owned by absentee landlords, perhaps a previous generation of parents, who have no current use for education, all lower-income families also (This is proved in Appendix D.) As in other multicommunity models, the single crossing assumption drives the stratification results outlined below Market equilibrium is defined as a set of housing prices and a rule assigning families them These owners will rent for any nonnegative price, although they will charge positive prices if the market will support them There is no possibility for collusion among landlords Housing supply in each community is thus perfectly inelastic: In quantity-price space, it is a to districts on the basis of their income that is consistent with individual family preferences, taking all other families’ decisions as fixed: vertical line extending upward from (n , ) { } Definition: An equilibrium for a market defined by δ ; J ; µ1 , K , µ J ; and F { consists of a set of nonnegative housing prices h1 , K , h J Family i ’s exogenous income is x i > ; the income distribution is bounded and has } and an allocation rule G : R + a Z J that satisfy the following conditions (where distribution function F, with F ' ( x ) > whenever < F ( x ) < 24 Families derive utility x j ≡ ∫ (G( x ) = j )x dF( x ) from school quality and from numeraire consumption, and take community composition EQ1 and housing prices as given Let x j denote the mean income of families in community j, EQ2 ∫ 1(G( x ) = j )dF( x ) ): No district is over-full For each j, ∫ (G( x ) = j )dF ( x ) ≤ n N Nash equilibrium At the specified prices and with the current distribution of peer groups, no family would prefer a district other than the one to which it and let h j be the rental price of local housing The utility that family i would obtain in The model is a “musical chairs” game, and the upper constraint serves to tie prices down, while the lower constraint avoids the need to define the peer group offered by a community with no residents 24 Of course, the income distribution cannot be continuous for finite N Relaxing the treatment to allow a discrete distribution would add notational complexity and introduce some indeterminacy in equilibrium housing prices, but would not change the basic sorting results 23 25 I might allow U ij = U ( x i − h j , Q ( x j , µ j )) , with Q1 ≥ and Q > , without changing the basic results; δ then corresponds to Q1 Q The key assumption is that all families share the same U and Q functions, with all differences in their behavior resulting from differences in their budget constraints (i.e from x i ) 17 18 ( ) is assigned: U x i − hG( x i ) , x G( x i )δ + µG( x i ) ≥ U (x i − hk , x kδ + µk ) for all i Note that Theorem does not rule out equilibria in which some families live in EQ3 and all k Normalization of housing prices h j = whenever lower- µ than some higher-income families I refer to these as unsorted (or imperfectly ∫ 1(G( x ) = j )dF ( x ) < n N sorted) equilibria They arise when the peer group advantage of high-income communities EQ4 No ties in realized quality For any j, k, x j δ + µ j ≠ x kδ + µk 26 over low-income communities is large enough to overcome deficits in school effectiveness.28 The following results are proved in Appendix D: For fixed income and effectiveness distributions, unsorted equilibria become harder to Theorem Equilibrium exists maintain as the weight that families place on peer group relative to school quality falls: Theorem Any equilibrium is perfectly stratified, in the sense that no family lives Corollary 2.3 Let G be an assignment rule satisfying Corollary 2.1 under which in a higher-quality, higher-price, or higher-peer-group district than does any higher there exist communities j and k satisfying µ j < µk but x j > x k Then for income family C ≡ max x k 0, Corollary 2.1 In any equilibrium, the n families with incomes greater than i F −1 (1 − n N ) live in the same community, which has higher quality ( xδ + µ ) than ( housing prices with which G is an equilibrium) ) any other The next n families, with incomes in F −1 (1 − 2n N ), F −1 (1 − n N ) , live in the community ranked second in quality This continues down the distribution: For ( ( { each j ≤ J , the families with incomes in F −1 max − jn Whenever δ > C , G is an equilibrium allocation (i.e there exist }), F −1 (1 − ( j −1)n N )) N ,0 ii Whenever δ < C , G is not an equilibrium allocation iii If δ = C , G can satisfy requirements EQ1-EQ3 for equilibrium, but violates EQ4 live in the community with the j th ranked schools.27 I not present formal results on the implications of increases in J for effectiveness Corollary 2.2 If δ = , equilibrium is unique Condition EQ4 corresponds to the “stability” notion of Fernandez and Rogerson (1996; 1997) Arrangements that satisfy EQ1 through EQ3 but not EQ4 are unstable, and perturbations in one of the tied communities’ effectiveness or peer group would lead to non-negligible differences between the communities as families adjust With EQ4, equilibria are locally stable 27 I neglect families precisely at the boundary between income bins (i.e those with incomes satisfying 26 sorting, as much depends on the µ j ’s assigned to the new districts Informally, however, Corollary 2.3 suggests that for a stable µ distribution, increasing the number of districts F (x ) = − N for some j) I demonstrate in the Appendix that families at boundary points are indifferent between the two communities in equilibrium As the income distribution approaches continuity, the potential importance of boundary families declines to zero It need not be true that unsorted equilibria are less efficient than the perfectly sorted equilibrium: If the marginal utility of school quality declines quickly enough, it can be more efficient to assign effective schools to low-income bins than to the wealthiest students In any case, concern for peer group amounts to an externality, and there is no assurance that the efficient assignment of families to districts is an equilibrium at all It may be efficient to have heterogeneous income distributions at each school, for example, but this is never a decentralized equilibrium 19 20 jn 28 constrains the possibility of unsorted equilibria: With more districts, the distance between another house-district take their “peer group” with them Regardless of parental valuations, the average incomes of districts that are adjacent in the quality distribution is smaller As C then, families always prefer a high- µ house to one with lower µ Because willingness-to- depends on this distance, a higher J reduces the amount by which a low-income district’s pay for a preferred school is increasing in x, equilibrium is unique, with the ranking of effectiveness parameter can exceed that of the next-wealthier district before the wealthier districts by effectiveness is identical to that by the income of the resident family Panels A families will bid away houses in the more effective district and B of Figure 1.1 graph the equilibrium allocations of effectiveness ( µ j ) and district This tendency is at the core of my empirical strategy To clarify it, I present next to a simulation exercise that demonstrates the impact of market structure (J) on effectiveness sorting under different assumptions about the importance of peer group to parental desirability ( x j δ + µ j ) as functions of family income when parents have no concern for peer group ( δ = , Panel A) and when concern for peer group is moderate ( δ = 1.5 , Panel B) preferences (δ), and thus about the “stickiness” of residential assignments I begin by describing the allocation of effectiveness in illustrative equilibria, then describe the simulation and its results Finally, at the end of this section I return to the basic model to discuss its allocative implications and the likely effects of endogenizing school effectiveness 1.3.1 Graphical illustration of market equilibrium The competitive case serves as a baseline, but it is not a realistic description of choice in the presence of peer group externalities I next consider a market with ten equally-sized districts, a degree of Tiebout choice that, as is discussed below in Section 1.4, corresponds roughly to the 80th percentile U.S metropolitan area Assume that J = 10 , n = N 10 , and µj = From Theorem and its corollaries, the income distribution in any equilibrium is divided into J quantiles, with wealthier quantiles living in more preferred—higher x j δ + µ j —districts In Appendix D, I show that this necessary condition is also sufficient j 10 , j = 1, K , 10 Panel C of Figure 1.1 displays the unique, perfectly sorted equilibrium when δ = Families in the j th decile of the income distribution live in the district with the j th most effective schools When parental concern for peer group is introduced, the perfectly sorted equilibrium for an assignment rule to be an equilibrium allocation Here, I use these results to construct possible equilibria under different (δ , J ) combinations is no longer unique It is now possible for ineffective districts to retain wealthy peer groups in equilibrium, as long as they are not so ineffective that families would prefer a lower- x , It is helpful to begin by considering a Tiebout market that approximates perfect competition Assume that there are as many districts as there are families, with only a single house in each district, and suppose that both family income and school effectiveness are uniformly distributed on [0, 1] There is no peer group externality, as families that move to higher- µ district One imperfectly sorted equilibrium is displayed in Panel D Note that district desirability is monotonically increasing in district average income, as Theorem requires that the desirability and income rankings be identical in equilibrium Effectiveness is not monotonic in family income, however: Some families live in districts that are less 21 22 effective than those where some poorer families live Effectiveness sorting nevertheless peer characteristics to student performance by one In the imperfectly sorted markets remains substantial, and effectiveness is highly correlated with peer group average income displayed in Panels D and F, however, the magnification effect is smaller: θ * = 0.9 in D and Finally, we consider the case where the housing market gives parents few options, with only three equally-sized districts ( J = , n = N ) This corresponds roughly to the 40th percentile of the U.S distribution Suppose here that µ j = j , j = 1, 2, When there are no peer effects (Panel E), equilibrium is again unique and is perfectly sorted on 0.5 in F The simulations below suggest that this tendency for effectiveness sorting and magnification to depend on the number of districts when parents care about both peer group and effectiveness holds generally, as long as concern for peer group ( δ ) is moderate When δ is large, however, even markets with many districts can have unsorted equilibria, [ and there is no tendency for E θ * |δ , J effectiveness When we add concern for peer group to the three-district market, there is ] to increase with J, at least in the ranges considered here.29 substantially more potential for mis-sortings than even in the ten-district case The gap in peer quality between adjacent districts has grown substantially, and families therefore require a much larger µ return to justify a move from one district to another whose current residents are lower in the x distribution Indeed, with the parameter values used here, there is no allocation of x terciles to districts in which any family would willingly move to a lower- x district; all six of the possible permutations are equilibria Panel F illustrates one possibility Here, the most effective district is rewarded with the wealthiest students, but the 1.3.2 Simulation of expanding choice In this subsection, I describe simulations of a hypothetical regional economy under several combinations of (δ , J ) As δ grows, the relative importance of school effectiveness diminishes and the likelihood of unsorted equilibria expands By the logic above, for any fixed δ we might expect unsorted equilibria to be less prominent with many districts than with few Where Figure 1.1 used uniform, nonstochastic distributions for both income and two remaining districts are mis-sorted Recall equation (3), which suggested that a naïve estimate of the peer effect is magnified by effectiveness sorting, with the degree of magnification being ( ) ( ) θ * ≡ cov x j , µ j var x j , the coefficient from a regression of µ j on x j across all * districts in the market θ = in the perfectly sorted markets displayed in Panels A, B, C, and E of Figure 1.1, indicating that the slope of school-level average test scores with respect to student characteristics in these markets will overstate the contribution of individual and 23 effectiveness, here I adopt the slightly more realistic assumption that income has a normal distribution and I draw random effectiveness parameters from the same distribution.30 For For any δ , there is some J for which effectiveness sorting will increase: The perfectly competitive case in Panels A and B would be perfectly sorted for any δ I simulate only markets with J ≤ 10 —the computational burden increases with the factorial of J—though this is easily enough to reveal the general trend 30 Analysis of varying δ subsumes the variance of the µ ’s: Increased variation in school effectiveness is j 29 equivalent, for the purpose of the sorting process, to increased parental valuation of a district with high effectiveness relative to one with a desirable peer group (i.e to a reduction in δ ) A normal (rather than log- 24 freshman major fixed effects This is in the spirit of a matching procedure used by Goldman 3.3.1 UC Admissions Processes and Eligible Subsample Construction and Widawski (1976) to correct the FGPA for departmental differences in grading standards, The UC system’s mandate is to admit the top 12.5 percent of California high school though the lack of course-level enrollment information prevents implementation of their full graduates each year Admissions decisions are made in several stages First, a central office procedure Campus and major effects are interpreted as criterion adjustments, not as determines whether each applicant is eligible to the UC (that is, whether she is in the top 12.5 explained variance, so are excluded in calculations of goodness-of-fit statistics In keeping percent) All eligible students are guaranteed admission to at least one campus Second, with this interpretation, they are constrained to remain unchanged in estimates of the each of the campuses to which the student applied decides whether to admit her.9 Campus restricted models (4B) and (4C) Of course, both campus and major are potentially admissions give preferences to eligible students, and no campus may make more than six endogenous Several specification checks presented in Section 3.5 suggest that endogeneity percent of its admissions offers to ineligible applicants Because campus admissions are not of campus and major does not seriously bias prediction coefficients from the pooled sample centrally coordinated, some eligible applicants are admitted to several campuses while others Two auxiliary data sources are used in concert with the UC database First, data from the College Board, with observations on all California SAT-takers from the 1994-1998 are rejected from all the campuses to which they have applied In a third stage, the latter students are offered admission at one of the less selective campuses, frequently Riverside high school cohorts, are used to estimate Σ , the population variance-covariance matrix of The central eligibility determination, unlike campus admissions decisions, is based on SAT scores and high school GPAs Second, school-level demographic characteristics are published rules that in 1993 consisted primarily of a deterministic function of the HSGPA drawn from the California Department of Education’s Academic Performance Index (API) and composite SAT score (UC Office of the President, 1993) As a result, eligibility can be database.8 The API data cover only public schools, and analyses in Section 3.6 are therefore simulated for each student in the UC database A subsample is constructed consisting of the restricted to graduates from 671 public high schools (81 percent of the students in the UC 17,346 students (14,102 from public schools with API data) who are judged to have been database) Range corrections in that section extend results only to the population of public UC eligible 10 For students in the eligible subsample, admission was guaranteed, and sample school SAT-takers Table 3.1 reports summary statistics for the population of California SAT-takers, for selection came only from decisions to apply and to accept an admissions offer that may not the UC sample, and for a subsample consisting of “UC eligible students.” UC eligibility policies—the Four Percent Plan, for example, but not affirmative action, which was only ever practiced at the individual campus level—are thus about how to define the “top 12.5%.” These are 93.3% of the UC sample; slightly more than 6% of the students in the sample are ineligible This overrepresentation relative to the above-cited 6% limit likely reflects different matriculation rates—yields— among eligible and ineligible admitted students 10 The API data were first collected in 1999, and may measure 1993 school characteristics with error 105 106 have been at the preferred campus.11 If these decisions may be assumed uninformative The regression coefficients in Column A are potentially biased by endogenous about unobserved ability— ε i in (2)—the subsample is selected on observables and permits sample selection Column B repeats the models using only the UC-eligible subsample The consistent OLS estimation of (4A) Moreover, even if student enrollment decisions are eligibility determination considers only observables, so coefficients from the unrestricted informative about ability, the subsample is arguably representative (again, conditional on X model in Panel A should not be biased by eligibility-induced sample selection.12 The sample and S) of the population of interest to admissions offices, who presumably care only to of eligible students produces a 13% increase in the HSGPA coefficient, with a negligible accurately predict performance for students who might choose to enroll if admitted In the effect on the SAT coefficient (The differential effect probably reflects the greater next section, I present validity estimates, first using conventional, inconsistent methods and importance of HSGPA in the 1993 eligibility determination: Table 3.1 reveals that ineligible then taking advantage of the UC eligibility rule to obtain estimates that are consistent as long students have low HSGPAs but SAT scores comparable to those of eligible students.) Panels B and C of Column B present OLS estimates of the restricted models from as matriculation, campus, and major choices are ignorable Section 3.5 examines this assumption, presenting evidence that student self-selection into the UC and into particular the eligible subsample These estimates are inconsistent, as observed independent variables campuses does not substantially bias pooled-model coefficients in (4A) act as unobservables in (4B) and (4C) Recall the omitted variables formulation of (4B): 3.4 Validity Estimates: Sparse Model Table 3.2 presents several estimates of the basic validity model described in Section 3.2, using HSGPA as the only X variable The first column presents OLS estimates from the full UC sample Panels A through C report coefficients and range-corrected fit statistics from models (4A) through (4C), respectively The final rows report the SAT increment to goodness-of-fit, the difference between fit statistics in Panels A and B The usual validity methods would thus report the SAT’s raw validity as 0.490 and its incremental validity as 0.055 E[ y X ] = E[E[ y X , S ] X ] = α + Xβ + E[S X ]γ range), E[S X ] is likely to be substantially different in the sample than in the population Figure 3.1 displays kernel estimates of this conditional expectation in the SAT-taker population, in the full UC sample, and in the eligible subsample The uptick at the leftmost extreme of the subsample graph reflects the SAT-HSGPA tradeoff inherent in the eligibility rule The figure legend reports linear regressions of S on X; these differ across samples not only in their vertical positions—which would be absorbed by the intercept in (5)—but also in their slopes 12 11 (5) Because eligibility rules allow high SAT scores to compensate for low HSGPAs (in a narrow Leonard and Jiang (1999) note the utility of the same institutional feature for validity studies 107 In the presence of other forms of sample selection, these coefficients may yet be biased In Section 3.5, I offer evidence that endogenous matriculation and campus selection not introduce substantial bias 108 The linear models reported in Figure 3.1 are sufficient statistics for calculation of the restricted model (4B) The coefficient of a regression of S on X is ρ SX var (S ) var (X ) , estimated as well.) The same approach estimates γ and var ( y|S ) , though here the regression in the population data is of X on S Column C of Table 3.2 presents estimates of the restricted models using the omitted where ρSX is the correlation between S and X and all three statistics are calculated from the variables approach Coefficients of each are substantially higher than the OLS estimates in regression sample Using the linear projection E[S X ] ≈ E[S ] + Column B.13 This increases each variable’s raw validity and thereby reduces SAT’s + (X − E[X ])ρ SX var (S ) var (X ) , we obtain from (5) the omitted variables formulae: ( incremental validity The usual methods in the full UC sample understate SAT’s raw validity ) α = α + E[S ] − E[X ]ρ SX var( S ) var( X ) γ , β = β + ρ SX var( S ) var( X )γ , and )var(S ) var (y X ) = σ + γ 12 (1 − ρ SX by eight percent and overstate its incremental validity by 25 percent, relative to the same (6) As Figure 3.1 indicates, the UC subsample inconsistently estimates ρSX Eligibility rules that permit a high SAT score to compensate for a low HSGPA lead to a within-sample statistics estimated by the more defensible methods in Column C 3.5 Possible Endogeneity of Matriculation, Campus, and Major The estimates in the rightmost columns of Table 3.2 correct for eligibility-induced ρSX that is lower than its population value This correlation is 0.52 in the population of selection-on-unobservables and for inconsistencies in the usual treatment of selection-onSAT-takers, but only 0.43 in the UC sample and 0.38 in the eligible subsample As a result, the OLS estimates of restricted models in Panels B and C of Table 3.2 are inconsistent, even when (as in Column B) the sample construction permits consistent estimation of the unrestricted model Equation (6) suggests an estimator for α , β , and var ( y|X ) A consistent observables, but they not solve all selection problems Prediction coefficients still may be biased by selection coming from sources other than eligibility decisions, either from individual campus admissions decisions that select on unobservables or from student matriculation choices between the UC and private alternatives, among UC campuses, and among available majors.14 In this section, I present alternative specifications meant to assess estimate of ρ SX var (S ) var (X ) can be obtained by regressing S on X in the unselected the bias introduced by these forms of selection Recall that all eligible students’ choice sets SAT-taker data When this is inserted into (6) along with consistent estimates of β1 and γ include at least one UC campus I thus treat the student’s decision as occurring in two from the eligible subsample of the UC data, the resulting estimate of β is consistent Var ( y|X ) can be estimated similarly, using the population data for ρSX and var (S ) (Note that α is a nuisance parameter for goodness-of-fit statistics, though it could easily be 13 14 The effect is larger for the SAT coefficient in Panel C, again reflecting the high relative weight placed on HSGPAs in eligibility determination Decisions about where to submit applications present yet another selection margin As applications have relatively low cost, however, it seems reasonable to assume that students apply to all colleges at which they have a reasonable probability of admission and at which they would consider matriculating 109 110 distinct stages, first a choice of UC versus non-UC colleges, then of a campus within the UC observations near the middle, where the probability of selection into the sample plausibly system.15 approaches one To evaluate whether endogenous matriculation biases the estimates Consider first the admitted student’s decision about whether to attend the UC presented in Table 3.2, models were re-estimated on “trimmed” samples that delete the top Conditional on HSGPA and SAT, students with high unobserved ability might face better and bottom deciles of the eligible subsample.16 Columns B through D of Table 3.3 report non-UC alternatives than their peers with similar scores (Of course, they may also have estimates from samples trimmed along three dimensions: An average of SAT and HSGPA better within-UC choice sets, as they are likely admitted to more desirable campuses.) One corresponding to the UC eligibility rules, which weight HSGPA heavily; fitted values from might also imagine that very low ability students have increased costs or reduced benefits of the unrestricted model in Table 3.2, Column B; and the SAT score alone None of the attending UC campuses Either could induce endogenous sample selection that would bias trimmed samples produces substantially different prediction or validity estimates than does prediction coefficients toward zero These stories of endogenous matriculation are most the full sample, suggesting that student matriculation decisions not much bias validity compelling at the extremes of the UC applicant pool, where the UC competes with Stanford estimates for eligible students and other elite private colleges on the one hand, and with the California State University on Having decided to attend the UC, students choose a campus and then a major the other Students in the middle are likely to be admitted to a UC campus at approximately Campus assignment is a function both of admissions decisions and of students’ own their desired selectivity level and to face few comparable alternatives, an effect compounded preferences, as the latter must decide to which campuses they will apply and, if accepted to by the relative rarity of middle-tier private colleges in the western United States several, at which they will enroll In the extreme case, the system would offer a continuum By this logic, the UC-sample distribution of unobserved characteristics, conditional of campuses and admissions rules would perfectly stratify students by preparedness; all on observed variables, may be quite truncated at the extremes of the observable distribution, predictive power from the SAT score would be incorporated in the campus assignment while near the middle of this distribution there is likely to be little selection-on- Within campuses, variation of SAT scores and HSGPAs would be perfectly offset by unobservables Selection bias should thus be less severe in a trimmed sample that discards unobservables and the fixed-effects SAT and HSGPA coefficients would be zero A similar observations at the extremes of the UC observable distribution, where the decision to attend problem could arise from the use of fixed effects for student major, if students sort into the UC may be quite informative about unobserved motivation and ability, in favor of majors based on their ability 16 15 I not consider separately the individual campus admissions decisions, instead allowing these decisions to influence students’ choices at each stage through their choice sets An Appendix (available from the author) develops a more complete model of the sample selection process and argues for the estimation strategy taken here 111 I am not aware of previous uses of this sort of specification check, although it has similarities to Altonji, Elder, and Taber’s (2000) approach and to Heckman’s (1990) “identification at infinity.” The essential insight is that the unobservable distribution is most severely truncated near the selection margin, and that this should be observable in the conditional FGPA distribution Alternative tests of the same phenomenon might look for changes in the residual variance or in quantile regression slopes near the selection margin 112 This suggests that both campus and major are potentially endogenously assigned within the UC-eligible subsample To evaluate this, I estimated the basic fixed effects model HSGPA Finally, Columns G and H combine the two strategies, reporting IV models on the trimmed sample These echo the full-sample IV results by instrumental variables It seems likely that students prefer to attend campuses near their Taking all the alternative specifications in Table 3.3 together, the evidence suggests homes but that unobserved ability does not vary with geography Thus, the probability of that biases in the basic model in Table 3.2 are small and, if anything, lead to overstatement of attending a particular campus might be expected to fall with the distance from that campus the SAT’s role in predictions The SAT’s incremental validity seems at least one fifth smaller and to rise with the distance from other campuses Fifteen geographic variables were used (from Table 3.2) than would be indicated by the usual methods as instruments for students’ campus assignments: eight indicators for residence in the same county as one of the UC campuses and seven measures of the distance between the student’s home county and the UC campuses 17 3.6 The full set of major dummies could not be used in Decomposing the SAT’s Predictive Power Thus far, only HSGPA and SAT have been considered as FGPA predictors This the IV model, as certain majors exist only at a single campus and therefore perfectly predict necessarily overstates the predictive accuracy that would be lost were SAT scores unavailable campus assignment.18 Column E of Table 3.3 reports IV estimates of a model that excludes in admissions The SAT’s absence would be partly compensated by re-weighting other major effects, while in Column F majors are collapsed into five broad categories and fixed predictors like application essays or teacher recommendations.19 SAT-based and non-SAT- effects are included for four of them The instruments are quite powerful predictors of based predictions would be more accurate, and more similar to each other, than is indicated campus assignment, and the first stage coefficients (not reported) generally have the by sparse models expected signs F statistics on the exclusion of the instruments in the first stage regressions In this section, I consider the implications of individual- and school-level range from 22 to 201 The IV estimates for the SAT and HSGPA coefficients are notably demographic variables for FGPA prediction, examining whether sparse models overstate the smaller than in the base model in Column A, with a slightly larger effect on the SAT than the SAT’s importance by allowing it to proxy for these demographic characteristics Specifically, I use the background variables to generate a predicted SAT score for each student in the UC database, and ask whether the predicted score can account for the relationship between SAT The distance to UC Irvine is excluded: Five of the eight campuses are in Southern California, and the eight distance variables are highly collinear Note that the instruments have no predicted relationship with admissions decisions, only with campus selection conditional on admission Under the assumption of constant treatment effects, an instrument for student preferences is sufficient Students not admitted to a particular campus are “never takers,” and their probability of attendance is unaffected by the instruments IV estimates are identified from compliers, students who are admitted to both nearby and faraway campuses and whose choice depends on location (Angrist, Imbens, and Rubin, 1996) 18 Plausible instruments for major are not apparent Encouragingly, however, the major effects estimated in Table 3.2 are similar to more plausibly causal matching results reported by Elliot and Strenta (1988) 17 scores and FGPAs If the SAT is a socioeconomically neutral measure of student preparedness, its predictable portion should be no more or less strongly related to FGPA 19 Willingham (1985) investigates the supplementary application variables that are used in admissions but are unavailable in the UC data He finds them to be significant predictors of college success when SAT is controlled, but does not report their effect on the SAT’s incremental contribution 113 114 than is the unpredictable portion On the other hand, to the extent that SAT scores are socioeconomic variables, HSGPAs can be appropriately discounted without access to SAT serving to “launder” the demographic characteristics but not well measure preparedness scores conditional on observable student background variables, the fitted SAT score will have a larger coefficient than does the residual Columns A through C of Table 3.4 present several OLS regressions with SAT scores as the dependent variable Individual and school background characteristics are strong I consider two categories of student background characteristics The first consists of predictors of SATs, and together account for over one fifth of their variance in the individual race (Black, Hispanic, and Asian) and gender indicator variables The second California public school SAT-taking population Column D presents an analogous model describes the demographic makeup of the student’s high school Five variables are used: for HSGPA on the same sample Although coefficients on the background variables are all The fraction of students who are Black, Hispanic, and Asian; the fraction of students statistically significant in this model, they account for a much smaller share of the HSGPA 20 receiving subsidized lunches; and the average education of students’ parents Because the variance than of SATs Finally, Columns E through G present models estimated on the latter data are available only for public schools, analyses in this section restrict attention to sample of eligible public-school UC students These again indicate that SAT scores are public school students highly correlated with student background, much more so than are either HSGPAs or The use of school-level predictors is a function of data availability, but also has a FGPAs Other notable differences between the models are that female students have higher substantive justification: The SAT has long been advocated as a necessary check on HSGPAs and FGPAs but lower SATs than males, and that Asians have higher HSGPAs but potentially heterogeneous high school grading policies (Caperton, 2001), and the College lower SATs and FGPAs than whites Board argues that admissions rules which consider only variables under the high school’s The decompositions in Table 3.4 suggest that the SAT’s role in prediction models control induce high school grade inflation To the extent that the SAT’s role is to limit such may be quite sensitive to the inclusion of background variables, particularly school a tendency, it might be expected to have more predictive power across high schools than characteristics, as predictors School and individual demographic characteristics explain fully within The implications of this, however, depend on whether observable characteristics of 23 percent of the SAT variance in the UC sample, but account for only five percent of high schools can serve the same role If “grade inflation” is highly correlated with variance of each of the grade-based variables Models which replace the school characteristics with high school fixed effects, not reported in Table 3.4, indicate that observable demographics account for a substantial share of the across-school variation in 20 The results are insensitive to the exclusion of average parental education, the least widely available background variable, and also to the inclusion of the school’s exam-based API score as a measure of school effectiveness It would be desirable to test the importance of individual parental education, but this is not observed in the UC data 115 SAT scores 116 Table 3.5 considers whether the SAT’s heavy loading onto student background Coefficients on the background variables are generally what one might expect, with racial characteristics accounts for its predictive power for FGPA Column A repeats the sparse minorities and students from schools with high concentrations of Blacks, Hispanics, or low- FGPA model from Table 3.2 using only eligible students from public schools Columns B education parents earning lower FGPAs than white, upper-SES students even when SAT through D include as an additional FGPA predictor the fitted SAT score from Columns A scores and HSGPAs are controlled As other authors have found, women earn higher through C, respectively, of Table 3.4 FGPAs than expected given their HSGPAs and SATs (see, e.g., Leonard and Jiang, 1999).21 These models indicate that characteristics of students’ schools, though not individual One coefficient is somewhat surprising: Asian students earn lower FGPAs than otherwise race and gender, account for a large share of the SAT’s predictive power Consider Column similar Whites, although students from schools with many Asian students quite well (see D, which effectively decomposes SAT scores into a portion that reflects student and school also Young, 2001) There is a sense in which the inclusion of variables not used for admissions in Table characteristics and an individual innovation Two students who differ in background characteristics producing a 100-point gap in fitted SAT scores (a student from an all-white 3.5 can lead to understatement of the SAT’s role The two SAT subscores have reliabilities school and an otherwise identical student from a school that is nearly all black, for example), of about 0.9 (College Board, 2001) Saturation of prediction models concentrates the earn FGPAs that differ, on average, by 0.13 If we instead compare two students with unreliability, attenuating the SAT coefficient and inflating coefficients on variables correlated identical observable characteristics but SAT residuals that differ by 100 points—i.e students with SAT scores even more than in sparse models One would not want to correct ordinary with the same predicted SATs but different actual SATs—the expected gap in FGPAs is validity estimates for this—admissions offices not have access to a perfectly reliable SAT only 0.07 points The latter, the SAT coefficient when its easily observed correlates are score, only to the noisy one—but the saturated models risk overstating the coefficient on the controlled, characterizes the independent information provided by SAT scores The sparse fitted SAT relative to that on the actual SAT Table 3.5 reports, in square brackets, selected models used in the literature conflate the two SAT portions and predict a 0.09 point gap coefficients derived from a multivariate errors-in-variables correction (Greene, 2000, p 378), regardless of the source of SAT differences assuming a SAT reliability ratio of 0.9.22 The adjusted coefficients indicate that the decline in Columns E through G carry out a similar exercise, this time allowing student background characteristics to predict FGPAs directly F tests reject the restrictions imposed the SAT coefficient when individual race and gender are controlled is primarily a statistical artifact, but not affect the interpretation of the results for school characteristics in the earlier models at any reasonable confidence level, but the new models not change the substantive interpretation: SAT scores are less informative, net of the information they 21 22 provide about student background, about FGPAs than is implied by sparse models 117 This may account for the relatively small coefficient on the fitted SAT in Column B: The racial component of the fitted SAT appears strongly related to FGPA while the gender component seems negatively correlated This is probably a lower bound for the SAT composite’s reliability, as true scores are likely to be more highly correlated across the two subtests than is noise 118 The final rows of Table 3.5 present range-corrected goodness-of-fit statistics, based The selection results are interesting, but not particularly informative about on uncorrected coefficients, for both the full models and omitted-variables specifications admissions policy After all, an incremental validity of 0.044 may well be enough to justify that exclude the actual SAT score but retain the predicted score or demographic variables use of the SAT The second portion of the analysis, focusing on the role of demographic The latter measures might be used in predictions even if SAT scores were not, and their variables in FGPA prediction, is of more direct substantive interest predictive power is thus not attributable to the SAT score The inclusion of school The results in Tables 3.4 and 3.5 suggest that in sparse models the SAT serves in part demographic controls—either separately or through the predicted SAT score—lowers the to proxy for student background characteristics These variables account for a substantial SAT’s estimated incremental validity by about 50% share of the variance in SAT scores They are also strong predictors of FGPA in their own right—together with HSGPA, school and individual demographic variables explain 45 3.7 Discussion percent of the variance in FGPAs, about as much as SAT and HSGPA together in This study has addressed two methodological concerns generally ignored in the SAT models excluding background variables Moreover, fitted SATs predicted from student and validity literature First, it has embedded the sample selection problem in an explicit model, school demographic variables are more strongly related to FGPAs than are actual SATs.23 Table 3.5 indicates that admissions offices could admit better-prepared entering proposing a new estimator that is consistent under certain assumptions These assumptions are not general—the current analysis benefits from the UC’s unusual reliance on easily classes by giving explicit admissions preferences to high-SES students and to students from observed characteristics for eligibility decisions—but are more reasonable than the internally high-SES high schools.24 SAT scores would receive some weight in “best predictor” inconsistent assumptions needed to support usual practice Researchers working with data admissions rules, but considerably less than is indicated by sparse models Few would advocate this sort of admissions rule, which might be called “affirmative from other colleges will typically not be able to rely on the selection-on-observables assumption that permits the current analysis, but might consider using the “trimming” action for high SES children,” and even fewer would consider it meritocratic A decision approach from Section 3.5 along with the omitted variables estimator to assess the not to consider student background characteristics explicitly in prediction models used for magnitude of selection biases I estimate that the usual methods overstate the SAT’s admissions, however, does not justify excluding them from SAT validation models If incremental validity for University of California FGPAs by about one quarter relative to the One interpretation is that the SES gradient in SAT scores is too low: An increase in this gradient could permit the predictable and unpredictable parts of the SAT to have the same coefficient in FGPA models The high explanatory power of models for SAT in Table 3.4, however, suggests an arguably more reasonable interpretation: the SAT score captures background characteristics more than it independently measures student preparedness 24 Note that affirmative action typically assigns preferences in the opposite direction from that indicated by Table 3.5 23 selection-adjusted estimate Additional specifications not indicate substantial downward bias in the latter from non-eligibility-based forms of sample selection 119 120 background characteristics are not accounted for, the researcher will assign predictive power Tables and Figures for Chapter to any variable that correlates with the excluded variables, whether or not it conveys independent information about preparedness The results here indicate that the exclusion of Table 3.1 Summary statistics for UC matriculant and SAT-taker samples student background characteristics from prediction models inflates the SAT’s apparent validity, as the SAT score appears to be a more effective measure of the demographic All characteristics that predict UC FGPAs than it is of variations in preparedness conditional on Mean (A) student background A policymaker who preferred not to use demographic variation to identify students likely to succeed might want to build an admissions rule around the SAT and HSGPA coefficients in Table 3.5, while ignoring the coefficients on demographic control variables.25 Regardless of one’s view of the appropriate role of student background characteristics in admissions, the results here suggest that the SAT should be assigned less importance than is implied by the sparse, selection-biased models in the validity literature If one wishes to exploit the predictive power of student background, the background variables themselves can provide much of the information contained in the SAT score; if one does not wish to use background in prediction, the SAT’s contribution is smaller than the validity UC Sample UC Eligible Only Mean S.D (C) (D) S.D (B) CA SAT-takers, 1994-1998 Mean S.D (E) (F) Number of observations FGPA SAT HSGPA Black Hispanic Asian Female 18,587 2.84 0.64 1,091 179 3.75 0.46 4% 14% 39% 53% 17,346 2.88 0.62 1,104 172 3.81 0.39 3% 12% 40% 53% 904 3.24 7% 19% 22% 55% % with school data Skl: Frac Black Skl: Frac Hispanic Skl: Frac Asian Skl: Frac lunch Skl: Avg parental ed 81% 7% 27% 21% 25% 14.4 81% 7% 27% 21% 24% 14.5 76% 8% 30% 17% 28% 14.2 10% 23% 18% 22% 1.3 620,013 10% 22% 18% 21% 1.3 226 0.63 11% 24% 17% 22% 1.3 literature would suggest Comparing incremental validities in Tables 3.2 (Column A) and 3.5 (Column D), a conservative estimate is that traditional methods and sparse models overstate Notes : HSGPAs are calculated using the UC's weighting rule, which assigns points to an A grade and awards an extra point to grades earned in honors courses School variables report means and standard deviations of student-level measures among the public school students for which the data are available the SAT’s importance to predictive accuracy by 150 percent 25 A more extreme reaction would be to admit using only the residuals from models like those in Table 3.4, rather than the entire HSGPA and SAT (Studley, 2001) Percent plans, which base admissions on withinschool rank in class, essentially this 121 122 Table 3.2 Basic validity models, traditional and proposed methods Full Sample n=18,587 (A) Panel A: Both Predictors HSGPA SAT / 1000 R2 R Panel B: HSGPA only HSGPA R R Panel C: SAT only SAT / 1000 Table 3.3 Specification checks Eligible Subsample n=17,346 (B) (C) 0.507 (0.010) 0.930 (0.027) 0.409 0.639 0.571 (0.012) 0.928 (0.028) 0.454 0.674 0.662 (0.011) 0.342 0.585 0.726 (0.013) 0.392 0.626 0.744 (0.013) 0.396 0.630 1.414 (0.029) 0.228 0.478 1.758 (0.032) 0.284 0.533 0.062 0.047 0.058 0.044 1.485 (0.028) R 0.240 R 0.490 SAT increment to goodness-of-fit (Panel A - Panel B) R 0.067 R 0.055 Notes : Each column reports three regressions and associated goodness-of-fit statistics Panel A includes fixed effects for campuses and 18 freshman majors; Panels B and C constrain these effects to be the same as in A Models in Panel A and all models in Column A are estimated by OLS Restricted models in Columns B and C are estimated from Column B, Panel A by proposed omitted variables correction All fit statistics are corrected for restriction of range, extending models to all 620,013 California SAT-takers Campus and major effects are excluded from fit calculations See text for details 123 Sample trimmed on: Basic model Elig Pred SAT index FGPA (A) (B) (C) Prediction coefficients (unrestricted model) HSGPA 0.57 0.61 0.59 (0.01) (0.02) (0.02) SAT / 1000 0.93 0.92 0.95 (0.03) (0.03) (0.04) Campus FEs y y y Major FEs y y y Goodness-of-fit statistics (corrected for restriction of range) R2 A: SAT and HSGPA 0.454 0.472 0.465 B: HSGPA only 0.396 0.417 0.407 C: SAT only 0.284 0.286 0.289 A-B: SAT increment 0.058 0.054 0.058 R A: SAT and HSGPA 0.674 0.687 0.682 B: HSGPA only 0.630 0.646 0.638 C: SAT only 0.533 0.535 0.538 A-B: SAT increment 0.044 0.041 0.044 (D) Instrumental variables Full sample Trimmed on pred FGPA (E) (G) (F) (H) 0.58 (0.01) 0.89 (0.04) y y 0.45 0.49 0.44 0.48 (0.03) (0.04) (0.05) (0.05) 0.65 0.71 0.53 0.58 (0.08) (0.09) (0.11) (0.12) endog endog endog endog n y (4) n y (4) 0.457 0.403 0.280 0.054 0.676 0.635 0.529 0.041 0.259 0.231 0.155 0.028 0.509 0.481 0.393 0.028 0.290 0.258 0.173 0.031 0.538 0.508 0.416 0.030 0.204 0.187 0.114 0.017 0.452 0.432 0.337 0.019 0.229 0.210 0.127 0.019 0.478 0.458 0.357 0.021 Notes : Columns A through D are estimated by OLS on the UC-eligible subsample; columns E through H by IV Number of observations = 17,346 in A, E, F Columns B through D, G, and H delete observations in top and bottom deciles along listed index, retaining 13,879 observations Instruments In Columns E-H are indicators for residence in the same county as each of the UC campuses and the continuous distance between the home county and each campus (excluding Irvine) Fit statistics in all columns extend models to 620,013 SAT-takers, using omitted variables estimator and constraining fixed effects in restricted models 124 Table 3.4 Individual and school characteristics as determinants of SAT scores and GPAs Table 3.5 Accounting for individual and school characteristics in FGPA prediction Sample Dependent Variable HSGPA SAT-takers SAT/1000 (A) (B) (C) Intercept 0.979 0.049 (0.001) (0.008) Black -0.220 -0.151 (0.001) (0.001) Hispanic -0.175 -0.104 (0.001) (0.001) Asian -0.026 -0.010 (0.001) (0.001) Female -0.044 -0.040 (0.001) (0.001) Skl: Frac Black -0.253 -0.119 (0.003) (0.003) Skl: Frac Hispanic -0.023 0.057 (0.003) (0.003) Skl: Frac Asian 0.041 0.037 (0.002) (0.002) Skl: Frac free lunch 0.001 0.006 (0.003) (0.003) Skl: Avg parental ed 0.062 0.063 (0.001) (0.001) R 0.134 0.177 0.225 UC sample (UC elig only) HSGPA SAT/1000 HSGPA FGPA (D) (E) (F) (G) 3.011 0.488 4.231 2.532 (0.026) (0.039) (0.095) (0.156) -0.403 -0.105 -0.244 -0.291 (0.004) (0.008) (0.020) (0.033) -0.186 -0.105 -0.152 -0.248 (0.003) (0.005) (0.011) (0.019) 0.134 -0.012 0.091 -0.067 (0.002) (0.003) (0.008) (0.013) 0.120 -0.068 0.036 0.061 (0.002) (0.003) (0.006) (0.010) -0.159 -0.076 -0.039 -0.389 (0.010) (0.015) (0.037) (0.061) -0.084 0.061 -0.021 -0.099 (0.008) (0.012) (0.031) (0.050) -0.272 0.061 -0.184 0.195 (0.006) (0.008) (0.021) (0.034) 0.037 -0.022 -0.048 -0.017 (0.008) (0.012) (0.030) (0.049) 0.020 0.045 -0.026 0.033 (0.002) (0.002) (0.006) (0.010) 0.069 0.230 0.050 0.053 Notes : Sample includes students from public high schools with non-missing data only Number of observations = 473,758 in columns A-D; 14,102 in E-G FGPA in Column G is adjusted to remove estimated campus and major effects from Table 3.1, Column D Reported R is the traditional measure, estimated within-sample and unadjusted for restriction of range SAT/1000 (A) 0.583 (0.014) [0.556] 0.906 (0.031) [1.050] Demographic measures Predicted SAT/1000 (B) 0.581 (0.014) [0.554] 0.875 (0.034) [1.032] (C) 0.619 (0.014) [0.595] 0.694 (0.034) [0.832] (D) 0.602 (0.014) [0.578] 0.705 (0.036) [0.852] 0.187 (0.078) [0.066] 0.818 (0.055) [0.698] 0.635 (0.056) [0.511] Black (E) 0.554 (0.014) [0.526] 0.886 (0.034) [1.047] Asian Female Skl: Frac Black Skl: Frac Hispanic Skl: Frac Asian Skl: Frac free lunch Skl: Avg Parental Ed R (range corrected) Without SAT SAT Increment R (range corrected) Without SAT SAT Increment (G) 0.604 (0.014) [0.578] 0.725 (0.036) [0.880] -0.326 (0.053) -0.195 (0.045) 0.145 (0.028) 0.032 (0.044) 0.021 (0.009) 0.475 0.449 0.026 0.689 0.670 0.019 -0.073 (0.030) -0.082 (0.017) -0.122 (0.011) 0.100 (0.010) -0.296 (0.054) -0.146 (0.045) 0.265 (0.030) 0.026 (0.043) 0.016 (0.009) 0.481 0.454 0.027 0.694 0.674 0.020 -0.136 (0.030) -0.125 (0.017) -0.094 (0.010) 0.110 (0.010) Hispanic (F) 0.621 (0.014) [0.597] 0.694 (0.034) [0.833] 0.459 0.404 0.056 0.678 0.635 0.042 0.457 0.410 0.047 0.676 0.641 0.035 0.471 0.445 0.027 0.687 0.667 0.020 0.462 0.435 0.027 0.680 0.660 0.020 0.461 0.414 0.048 0.679 0.643 0.036 Notes : Sample in all columns consists of UC-eligible, public school graduates with non-missing data Number of observations = 14,102 All models include fixed effects for campuses and 18 freshman majors Standard errors are in parenthesis; errors-in-variables corrected coefficients, assuming SAT reliability = 0.9, are in square brackets Corrected demographic variable coefficients are not shown Predicted SAT is fitted value from regressions reported in Table 3.4, columns A through C, respectively Goodness-of-fit statistics extend results to 473,758 public school SAT-takers "Without SAT" statistics are based on unreported restricted models that exclude the SAT score and are calculated by omitted variables estimator described in text 125 126 Figure 3.1 Conditional expectation of SAT given HSGPA, three samples References 1600 1400 E[SAT | HSGPA] Altonji, Joseph G., Todd E Elder, and Christopher R Taber (2000) “Selection on Observed and Unobserved Variables: Assessing the Effectiveness of Catholic Schools.” Working Paper 7831, National Bureau of Economic Research, August UC eligibility cutoff All SAT-takers: SAT = 299 + 187*HSGPA + u Full UC sample: SAT = 465 + 167*HSGPA + u UC-elig subsample: SAT = 464 + 168*HSGPA + u 1200 Angrist, Joshua D., Guido W Imbens, and Donald B Rubin (1996) “Identification of Causal Effects using Instrumental Variables.” Journal of the American Statistical Association 91, 444-455 _ and Kevin Lang (2002) “How Important Are Classroom Peer Effects? Evidence from Boston’s Metco Program.” Unpublished manuscript, October 1000 Antos, Joseph R., and Sherwin Rosen (1975) “Discrimination in the Market for Public School Teachers.” Journal of Econometrics (2), 123-150 800 600 Atkinson, Richard C (2001) “Standardized Tests and Access to American Universities.” Atwell Lecture at 83rd Annual Meeting of the American Council of Education, Washington, D.C., Februrary 400 Barro, Robert J (2001) “Economic Viewpoint: Why Colleges Shouldn’t Dump the SAT.” Business Week, April 9, 20 0.0 1.0 2.0 3.0 4.0 5.0 HSGPA Notes : Figure displays kernel means, using Epanechnikov kernel (bandwidth = 0.15) Number of observations = 620,013 SAT-takers, 18,587 in full UC sample, and 17,346 in eligible subsample HSGPAs are calculated using the UC's weighting rule, which assigns points to A grades but awards an extra point to grades earned in designated honors courses Legend indicates linear regression coefficients for each sample Barrow, Lisa (2002) “School Choice Through Relocation: Evidence from the Washington, D.C Area.” Journal of Public Economics 86, 155-189 Bayer, Patrick, Robert McMillan, and Kim Rueben (2002) “An Equilibrium Model of Sorting in an Urban Housing Market: A Study of the Causes and Consequences of Residential Segregation.” Unpublished manuscript, August Belfield, Clive R., and Henry M Levin (2001) “The Effects of Competition on Educational Outcomes: A Review of U.S Evidence.” Occasional Paper No 35, National Center for the Study of Privatization in Education, Teachers College, Columbia University September Black, Dan, Gary Gates, Seth Sanders, and Lowell Taylor (2002) “Why Do Gay Men Live in San Francisco?” Journal of Urban Economics 51, 54-76 Black, Sandra E (1999) “Do Better Schools Matter? Parental Valuation of Elementary Education.” Quarterly Journal of Economics 114 (2), May, pp 577-599 Bogart, William, and Brian Cromwell (2000) “How Much is a Neighborhood School Worth?” Journal of Urban Economics 47 (2), March, 280-305 Borland, Melvin V., and Roy M Howsen (1992) “Student Academic Achievement and the Degree of Market Concentration in Education.” Economics of Education Review 11 (1), 31-39 127 128 Bowen, William G., and Derek Bok (1998) The Shape of the River: Long-Term Consequences of Considering Race in College and University Admissions Princeton, NJ: Princeton University Press Elliot, Roger, and A Christopher Strenta (1988) “Effects of Improving the Reliability of the GPA on Prediction Generally and on Comparative Predictions for Gender and Race Particularly.” Journal of Educational Measurement 25, 333-347 Breland, Hunter M (1979) Population Validity and College Entrance Measures New York: College Entrance Examination Board Epple, Dennis N., Radu Filimon and Thomas Romer (1993) “Existence of Voting and Housing Equilibrium in a System of Communities with Property Taxes.” Regional Science and Urban Economics 23 (5), November, 585– 610 Brennan, Geoffrey, and James Buchanan (1980) The Power to Tax: Analytical Foundation of a Fiscal Constitution Cambridge: Cambridge University Press Bridgeman, Brent., Laura McCamley-Jenkins, and Nancy Ervin (2000) “Predictions of Freshman Grade-Point Average From the Revised and Recentered SAT I: Reasoning Test.” Research Report 2000-1 New York: College Entrance Examination Board Camara, Wayne J (2001) “There is no Mystery When it Comes to the SAT I.” College Board News _ and Gary Echternacht (2000) “The SAT I and High School Grades: Utility in Predicting Success in College.” Research Report RN-10 New York: College Entrance Examination Board Caperton, Gaston (2001) “Toward a Holistic Definition of Merit: A Response to the UC Proposal.” College Board News Card, David, and A Abigail Payne (2002) “School Finance Reform, the Distribution of School Spending, and the Distribution of SAT Scores.” Journal of Public Economics 83, 49-82 Chubb, John, and Terry M Moe (1990) Politics, Markets, and America's Schools Washington, D.C.: The Brookings Institution _ and Glenn J Platt (1998) “Equilibrium and Local Redistribution in an Urban Economy When Households Differ in Both Preferences and Incomes.” Journal of Urban Economics 43 (1), January, 23–51 _ and Richard E Romano (1996) “Ends Against the Middle: Determining Public Service Provision When There Are Private Alternatives.” Journal of Public Economics 62, 297-325 _ and Thomas Romer (1991) “Mobility and Redistribution.” Journal of Political Economy 99, August, 828-58 _ and Holger Sieg (1999) “Estimating Equilibrium Models of Local Jurisdictions.” Journal of Political Economy 107 (4), August, 645-681 _ and Allen Zelenitz (1981) “The Implications of Competition Among Jurisdictions: Does Tiebout Need Politics?” Journal of Political Economy 89 (6), 1197-1217 Evans, William N., Wallace E Oates, and Robert M Schwab (1992) “Measuring Peer Group Effects: A Study of Teenage Behavior.” Journal of Political Economy 100 (5), October, 966-991 Coleman, James, et al (1966) Equality of Educational Opportunity Washington, DC: United States Office of Education Fernandez, Raquel, and Richard Rogerson (1996) “Income Distribution, Communities, and the Quality of Public Education.” Quarterly Journal of Economics 111 (1), February, 135-164 College Board (2001) College Bound Seniors 2001 New York: College Entrance Examination Board _ and _ (1997) “Keeping People Out: Income Distribution, Zoning, and the Quality of Public Education.” International Economic Review 38 (1), February, 23-42 Crouse, James, and Dale Trusheim (1988) The Case Against the SAT Chicago: University of Chicago Press Figlio, David N., and Maurice E Lucas (2000) “What’s in a Grade? School Report Cards and House Prices.” Working Paper #8019, National Bureau of Economic Research Cutler, David M., Edward L Glaeser, and Jacob L Vigdor, (1999) “The Rise and Decline of the American Ghetto.” Journal of Political Economy 107(3), 455-506 Friedman, Milton (1962) Capitalism and Freedom Chicago: University of Chicago Press Dale, Stacey Berg, and Alan Krueger (1998) “Estimating the Payoff to Attending a More Selective College: An Application of Selection on Observables and Unobservables.” Princeton University Industrial Relations Section Working Paper #409, December Geiser, Saul, and Roger Studley (2001) “UC and the SAT: Predictive Validity and Differential Impact of the SAT I and SAT II at the University of California.” Manuscript, University of California Office of the President Dee, Thomas S (1998) “Competition and the Quality of Public Schools.” Economics of Education Review 17 (4), 419-427 Goldman, Roy D., and Mel H Widawski (1976) “A Within-Subjects Technique for Comparing College Grading Standards: Implications in the Validity of the Evaluation of College Achievement.” Educational and Psychological Measurement 36, 381-390 Dynarski, Mark (1987) “The Scholastic Aptitude Test: Participation and Performance.” Economics of Education Review 6, 263-73 Gray, E R (1944) Governmental Units in the United States 1942 Washington D.C.: Bureau of the Census Eberts, R W., and T J Gronberg (1981) “Jurisdictional Homogeneity and the Tiebout Hypothesis.” Journal of Urban Economics 10, 227-239 Greene, William H (2000) Econometric Analysis, 4th ed New Jersey: Prentice Hall 129 130 Hamilton, B.W (1975) “Zoning and Property Taxation in a System of Local Governments.” Urban Studies 12, 205-211 Leonard, David K., and Jiming Jiang (1999) “Gender Bias and the College Predictions of the SATs: A Cry of Despair.” Research in Higher Education 40, 375-407 Hanushek, Eric A (1981) “Throwing Money at Schools.” Journal of Public Policy Analysis and Management (1), 19-41 Luizer, James, and Robert Thornton (1986) “Concentration in the Labor Market for Public School Teachers.” Industrial and Labor Relations Review 39 (4), 573-584 _ (1986) “The Economics of Schooling: Production and Efficiency in Public Schools.” Journal of Economic Literature 24 (3), September, 282-294 McWhorter, John H (2001) “Eliminating the SAT Could Derail the Progress of Minority Students.” Chronicle of Higher Education, March 9, B11-B12 _, John F Kain, and Steven G Rivkin (2001) “New Evidence about Brown v Board of Education: The Complex Effects of School Racial Composition on Achievement.” Working Paper #8741, National Bureau of Economic Research, December National Center for Education Statistics (1999) Digest of Education Statistics 1998 Washington D.C.: U.S Department of Education _ and Margaret E Raymond (2001), “The Confusing World of Educational Accountability.” National Tax Journal 54 (2), 365-384 Heckman, James (1979) “Sample Selection Bias as a Specification Error.” Econometrica 47, 153-161 _ (1990) “Varieties of Selection Bias.” American Economic Review 80 (2), 313-318 Hernandez, Michelle A (1997) A is for Admission: The Insider’s Guide to Getting into the Ivy League and Other Top Colleges New York: Warner Books Hirschman, Albert O (1970) Exit, Voice, and Loyalty: Responses to Decline in Firms, Organizations, and States, Cambridge, Mass.: Harvard University Press Hoxby, Caroline M (1994) “Do Private Schools Provide Competition for Public Schools?” Working Paper #4978, National Bureau of Economic Research _ (1999a) “The Effects of School Choice on Curriculum and Atmosphere.” In Mayer, Susan E., and Peterson, Paul E., eds., Earning and Learning: How Schools Matter Washington, DC: Brookings Institution Press Chapter 11, 281-316 _ (1999b) “The Productivity of Schools and Other Local Public Goods Producers.” Journal of Public Economics 74 (1), October, 1–30 _ (2000a) “Does Competition Among Public Schools Benefit Students and Taxpayers?” American Economic Review 90 (5), December, 1209-38 _ (2000b) “Peer Effects in the Classroom: Learning From Gender and Race Variation.” Working Paper #7867, National Bureau of Economic Research Hsieh, Chang-Tai, and Miguel Urquiola (2002) “When Schools Compete, How Do They Compete? An Assessment of Chile’s Nationwide School Voucher Program.” Unpublished manuscript, January Kane, Thomas J., Douglas O Staiger, and Gavin Samms (2002) “School Accountability Ratings and Housing Values.” Unpublished manuscript, October 20 Katz, Lawrence F., Jeffrey R Kling, and Jeffrey B Liebman (2001) “Moving to Opportunity in Boston: Early Results of a Randomized Mobility Experiment.” Quarterly Journal of Economics 116 (2), May, 607-654 Kenny, Lawrence W., and Amy B Schmidt (1994) “The Decline in the Number of School Districts in the U.S.: 1950–1980.” Public Choice 79 (1-2), April, 1–18 131 _ (2000) Private School Online Directory, v 1.00e Washington D.C.: U.S Department of Education Nechyba, Thomas J (1997) “Existence of Equilibrium and Stratification in Local and Hierarchical Tiebout Economies with Property Taxes and Voting.” Economic Theory 10, 277-307 Oates, Wallace E (1985) “Searching for Leviathan: An Empirical Study.” American Economic Review 75 (4), 748-757 Orfield, Gary (1983) Public School Desegregation in the United States, 1968-1980 Washington: Joint Center for Political Studies Phillips, Meredith, Jeanne Brooks-Gunn, Greg J Duncan, et al (1998) “Family Background, Parenting Practices, and the Black-White Test Score Gap.” In Jencks, Christopher, and Meredith Phillips, The Black-White Test Score Gap, Washington, D.C.: Brookings Institution Press Raudenbush, Stephen W., and Anthony S Bryk (2002) Hierarchical Linear Models : Applications and Data Analysis Methods Thousand Oaks, California: Sage Publications _ and J Douglas Willms (1985) “The Estimation of School Effects.” Journal of Educational and Behavioral Statistics 20 (4), Winter, 307-335 Reback, Randall (2001) “Capitalization Under School Choice Programs: Are the Winners Really the Losers?” University of Michigan Office of Tax Policy Research Working Paper 2001-21, November Ross, Steven, and John Yinger (1999) “Sorting and Voting.” In Cheshire, Paul, and Edwin S Mills, eds, Handbook of Regional and Urban Economics vol New York: NorthHolland Rothstein, Jesse M (2002) “College Performance Predictions and the SAT.” Working Paper #45, UC Berkeley Center for Labor Economics, March Sheppard, Stephen (1999) “Hedonic Analysis of Housing Markets.” In Cheshire, Paul, and Edwin S Mills, eds, Handbook of Regional and Urban Economics vol New York: North-Holland Stricker, Lawrence J (1991) Current Validity of 1975 and 1985 SATs: Implications for Validity Trends Since the Mid-1970s.” Journal of Educational Measurement 28, 9398 132 Studley, Roger (2001) “Inequality and College Admissions.” Unpublished manuscript Technical Design Group (2000) “Construction of California’s 1999 School Characteristics Index and Similar Schools Ranks.” PSAA Technical Report 00-1 Sacramento: California Department of Education Theil, Henri (1972) Statistical Decomposition Analysis London: North-Holland Young, John W (1993) “Grade Adjustment Methods.” Review of Educational Research 63, 151-165 _ (2001) “Differential Validity, Differential Prediction, and College Admissions Testing.” Presented at “Rethinking the SAT in University Admissions” conference, University of California at Santa Barbara, November 16 Tiebout, Charles M (1956) “A Pure Theory of Local Public Expenditures.” Journal of Political Economy, 64 (5), October, 416–24 Tracy, Joseph, and Joel Waldfogel (1997) “The Best Business Schools: A Market-Based Approach.” The Journal of Business 70 (1), January, 1-31 Urquiola, Miguel (1999) “Demand Matters: School District Concentration, Composition, and Educational Expenditure.” Working Paper #14, University of California, Berkeley, Center for Labor Economics, April United States Commission on Civil Rights (2000) “Toward an Understanding of Percentage Plans in Higher Education: Are They Effective Substitutes for Affirmative Action?” Washington, D.C University of California Office of the President (1993) “Introducing the University of California: Information for Prospective Students.” U.S Geological Survey (2002) Geographic Names Information System, State Gazetteer electronic data Welch, Finis, and Audrey Light (1987) New Evidence on School Desegregation United States Commission on Civil Rights Clearinghouse Publication 92, June Wells, Amy Stuart (1993) “The Sociology of School Choice.” In Rasell, E., and R Rothstein, eds., School Choice: Examining the Evidence Washington, DC: Economic Policy Institute Willingham, Warren W (1985) Success in College: The Role of Personal Qualities and Academic Ability New York: College Entrance Examination Board _ and Hunter M Breland (1982) Personal Qualities and College Admissions New York: College Entrance Examination Board _, C Lewis, R Morgan, and L Ramist (1990) Predicting College Grades: An Analysis of Institutional Trends over Two Decades Princeton, NJ: Educational Testing Service Willms, J Douglas, and Frank H Echols (1992) “Alert and Inert Clients: The Scottish Experience of Parental Choice of Schools.” Economics of Education Review 11 (4), 339-350 _ and _ (1993) “The Scottish Experience of Parental School Choice.” In Rasell, E., and R Rothstein, eds., School Choice: Examining the Evidence Washington, DC: Economic Policy Institute Wilson, Kenneth M (1983) A Review of Research on the Prediction of Academic Performance After the Freshman Year New York: College Entrance Examination Board 133 134 strong effect on this variable, one that is robust to the inclusion of controls for the school- Appendices level choice index and census tract-level segregation measures in Columns B As noted in the main text, however, there may be a mechanical relationship between district size and across-district sorting Columns C and D present estimates for the school-level white/non- Appendix A: Choice and School-Level Stratification Table 1.2 presents evidence for an effect of Tiebout choice over school districts on white isolation index The isolation index has the interpretation of the fraction non-white at the average non-white student’s school, normalized by the overall fraction non-white in the the segregation of students across both districts and schools There are several reasons to be metropolitan area (Cutler, Glaeser, and Vigdor, 1999) Again, we see a strong effect of interested in this relationship First, an effect of interjurisdictional competition on residential district structure that is unchanged by the inclusion of additional controls The remaining columns of Table A1 present models for an alternative measure of stratification is a clear implication of Tiebout-style multicommunity models, and indeed has been proposed as a test of the Tiebout hypothesis (Eberts and Gronberg, 1981; Epple and racial segregation, the Theil segregation measure (Theil, 1972) This measure has two Sieg, 1990) Second, the determinants of residential sorting are interesting in their own right advantages: It can accommodate multiple racial groups, which the two-group dissimilarity (Cutler, Glaeser, and Vigdor, 1999) Finally, the allocation effects of school choice policies and isolation indices cannot, and the school-level measure has a natural decomposition into must be weighed in any evaluation of their social benefits; many authors have argued that across-district and across-school, within-district components Column E presents estimates choice’s first-order effect will be to increase the inequality of peer group allocations (Wells, of the effect of choice on the school-level Theil index; Column F the effect on the across- 1993) district component; and Column G the effect on the across-school, within-district In this Appendix I present additional evidence that district-level choice has a strong, component District structure has a weak, but significant, effect on the school-level robust effect on student stratification across schools and districts Interestingly, this result is measure and a much stronger effect on the across-district component It also has a large, not affected by the inclusion of controls for micro-neighborhood segregation, and indeed negative effect on the within-district measure, perhaps resulting from the mechanical there seems to be no relationship between district structure and stratification measured at relationship discussed earlier or from district-level desegregation policies levels finer than the school Table A1 presents models similar to those in Table 1.2, but for several alternative measures of student stratification In Columns A and B, the dependent variable is the across-district share of variance of adult education The district-level choice index has a 135 I use racial groups to calculate the Theil index: Asian, Black, Hispanic, Native American, and White The school-level Theil measure in Column E is defined over both public and private schools, so cannot be literally decomposed into across- and within-district components The decompositions in Columns F and G use an alternate measure defined over public schools only; this correlates 0.998 with the public-plus-private measure 136 Hoxby (2000a, p 1236) argues that even at the school level there is a mechanical markets with particularly badly-run central city schools, suburban districts may have resisted relationship between the district-level choice measure and her measure of student pressures to consolidate, producing a negative effect of past school quality on current stratification, as the two are constructed similarly and “the similarity of construction creates Tiebout choice If quality is sufficiently stable over time, current choice may be endogenous spurious correlation.” This should not be a problem for the results presented here, as I use to current school quality, biasing estimates of choice’s effect on school performance (Table measures of stratification that not share the choice index’s construction Nevertheless, 1.7) downward following Hoxby, I present in Table A2 estimates of the choice-stratification relationship Figure B1 graphs the evolution of the number of districts over time in the U.S as a that use two alternative measures of district-level choice, the number of districts and the whole, and in the counties comprising the 1990 MSAs These data are drawn from a variety number of districts per 17-year-old resident Each specification indicates a strong of sources, using varying definitions, but the overall trends are clear The number of relationship, though the alternative measures of district competition produce somewhat less districts fell dramatically in the decades before 1970, but has been essentially unchanged in precise estimates than does the choice index These results are somewhat at odds with the three decades since The recent stability provides a reason to be suspicious of the claim Hoxby’s, as she finds no relationship between number of districts and racial segregation that district structure is endogenous to current quality: Unless quality is extraordinarily Finally, Table A3 presents estimates that use measures of segregation at the census tract level as the dependent variable The choice index coefficient is uniformly small, and is stable, any quality considerations that influenced consolidation during the 1960s are unlikely to be reflected in 1994 test scores insignificant for three of the four measures The basic estimates are unchanged in models Nevertheless, I present here instrumental variables estimates of the effect of choice that include a school-level choice index (not shown) Family sorting across schools and on student stratification, effectiveness sorting, and average SAT scores across MSAs I use districts evidently does not carry over into substantial sorting at the micro-neighborhood several instruments for current market structure that are likely to predate possible sources of level endogeneity One candidate is the number of rivers and streams in the area, which Hoxby (2000a) argues predicts current district structure because initial district boundaries are likely Appendix B: Potential Endogeneity of Market Structure There has been a broad trend in the U.S in the post-war era toward consolidation of to have been drawn along natural barriers to travel and to have persisted in current institutions A second strategy takes advantage of the fact that, even where districts are school districts, and there were only 19% as many districts in the country in 1990 as there were in 1950 (Kenny and Schmidt, 1994) Hoxby (2000a) argues that one factor that may independent entities, school districts rarely cross county lines Some MSAs have larger have influenced the degree of consolidation is the quality of local school administration: In counties, and consequently looser upper bounds for district size, than others I calculate 137 138 for each market a choice index for the distribution of population across counties, analogous expected sign When all of the instruments are included together, in columns F and G, all to the choice index for students across school districts As county boundaries are effectively but the streams variable remain significant.4 unchangeable in nearly all states, differences across MSAs in county choice should predate any endogenous influences.2 Table B2 reports two-stage least squares estimates of several specifications from the main text, using the same combinations of instruments as in Table B1 Each column reports Yet another strategy is to derive instruments from institutional characteristics governing the size of districts that are stable over time One such characteristic is an OLS estimates of the choice coefficient in the first row and different 2SLS estimates of the same coefficient in each succeeding row.5 indicator for whether the market is in one of the six states where districts have always been Columns A and B present 2SLS estimates of the segregation models from Table 1.2 functions of county government: As counties are relatively large, MSAs in these states have Although several 2SLS specifications yield imprecise estimates, the point estimates are always had fewer districts than other MSAs Another instrument is derived from a Census uniformly of the same sign as the OLS coefficients and generally are similar in magnitude Bureau tabulation of independent school districts by county in 1942 (Gray, 1944).3 I Eight of the twelve estimates in these columns reject zero effect of choice, while Hausman calculate for each 1990 metropolitan area an estimate of the area’s choice index in 1942, tests fail to reject equality of any of the 2SLS choice coefficients with their OLS treating all districts as equally sized and ignoring the primary-secondary distinction counterparts Additional IV estimates of the models in Appendix A, not shown here, A final strategy takes more direct account of the differences in state-level institutions: I instrument for each MSAs choice index with the average choice index of other MSAs in similarly support the OLS results There would seem to be little evidence that the choicesegregation relationship is biased by endogeneity of the choice index Column C of Table B2 presents 2SLS estimates of the choice-peer group interaction the same state, with the idea that institutions determining market structure in Boston (choice index=0.98) and Miami (choice index=0) are likely to be similar to those operative in coefficient in a model for the school average SAT score Here, the instruments are the Worcester, MA (0.96) and Fort Lauderdale, FL (0), respectively interactions between the variables listed and the school-level peer group background index Table B1 displays first stages for several combinations of the candidate instruments Each of the proposed instruments is highly significant in the first stage, and each has the The estimates are quite noisy, and most are positive None reject zero, however, and none of the Hausman tests reject the OLS estimate The county choice index depends on the intra-MSA distribution of growth as well as on the original county boundaries If MSA settlement patterns are endogenous to average school quality, the county choice index is an invalid instrument As it happens, results are largely insensitive to the particular choice of instruments In several states where districts are traditionally dependent upon town or county governments, the 1942 data list zero districts I assign to counties in these areas either a single district (where schooling is traditionally a county function) or as many districts as there are townships These imputation rules are quite accurate for later years in which counts of dependent districts are available Hoxby actually uses both a “smaller streams” and a “larger streams” variable, but has declined to make them available for this analysis My streams measure is constructed from the Geographic Names Information System (U.S Geological Survey, 2002) database following Hoxby’s description of her “smaller streams” variable, which she reports as by far the more powerful of her two instruments The coefficient in column B, 0.32, is nearly identical to that which Hoxby reports for the “smaller streams” variable in her first stage Of course, if all the instruments are indeed exogenous, the estimates in the final row—which optimally weight the entire set of instruments—are the most efficient Other rows are presented to demonstrate that the basic similarity between OLS and IV specifications does not rely on any single exogeneity assumption 139 140 Finally, Column D reports 2SLS estimates of the effect of choice on MSA mean SAT the SAT data in Chapter use only observations from SAT-state MSAs, and moreover scores, as discussed in Section 1.6 Once again, the estimates are somewhat noisy, but there control for the MSA SAT-taking rate In this appendix, I describe several additional tests is again no indication that high-choice MSAs produce higher SAT scores than low-choice that have been performed to gauge the degree to which selection into SAT-taking, and markets, once student background is controlled One Hausman test—for the model using particularly within-MSA selection, may bias the results above streams as the sole instrument—rejects the equality of OLS and 2SLS estimates, suggesting perhaps a larger negative effect of choice on average scores than is indicated by OLS Taking the instrumental variables estimates as a whole, there appears to be no reason The first form of analysis involves explicit models for the selection process Ideally, one would use a variable that predicts a student’s probability of taking the SAT but does not predict the student’s score conditional on test-taking It is difficult to think of an instrument to suspect serious endogeneity of the 1990 district-level choice index to any of the for this selection margin, however Instead, I attempted to use the school SAT-taking rate as dependent variables considered here I read this pattern of results as justification for my a summary of the factors that might determine sample selection Specifically, I estimated focus in the main text on the somewhat more precise OLS results models of the form [ ] ( ) E t ij |X ij , Z j , π j , i takes the SAT = a + X ij b + Z j c + λ π j d , Appendix C: Selection into SAT-taking The great limitation of the SAT data used in this thesis is that students self-select into taking the SAT Because SAT-taking rates vary considerably across states, estimates (C1) where X ij is a vector of individual characteristics for student i at school j; Z j is a vector of school-level measures, and π j is the SAT-taking rate at school j λ (⋅) is a “control ( ) ( ) based only on SAT-takers’ performance may not accurately describe patterns of student function,” which was specified as the inverse-Mills ratio, λ π j ≡ ϕ Φ −1(π j ) π j This performance in the entire population of students Figure C1 displays the relationship specification is appropriate for a conventional Heckman-style model of sample selection in between SAT-taking rates and average SAT scores across MSAs There is a clear negative which the factors determining SAT-taking are constant for all students at school j and relationship, indicating that at this macro level there is probably positive selection into SAT- residuals in selection and SAT-score equations are jointly normal (Heckman, 1979; Card and taking (Dynarski, 1987, and Card and Payne, 2002, present similar graphs) Payne, 2002) The picture is very different, however, when one distinguishes between MSAs in If students are positively selected into SAT-taking, we expect d > , as increases in a “SAT states,” indicated by solid diamonds, and MSAs not in SAT states, indicated by pluses school’s SAT-taking rate should reduce average scores Using a variety of peer group Within the SAT state sample, the correlation disappears: Markets with high participation measures in Z, OLS estimates of d were all large and negative, most likely indicating that this rates have average scores no lower than those with relatively low rates All analyses of cross-school comparison does not adequately control for the determinants of SAT scores 141 142 In an effort to obtain a more reasonable selection model, I also estimated versions of (C1) extent that selection into SAT-taking biases the school-level averages that are the focus of with school fixed effects, using data from the 1994 through 1998 SAT-taking cohorts and the analysis here, there is apparently very little variation in this selection across years identifying the selection parameter from within-school, across-year variation in SAT-taking Moreover, the correlations decay quite slowly over time, indicating that schools not This produced an estimated d with the correct sign, although the implied correlation between change rapidly and that much of the across-year variation in school averages likely derives test-taking propensity and the latent test score was almost implausibly small: ρˆ = 0.02 from transitory sampling error It is difficult to have much faith in estimates of selection models like (C1) without an In a final attempt to test the robustness of the basic results to selection into SAT- adequate instrument for selection To further explore the potential impact of selection, taking, I made use of a variable in the SAT data describing students’ self-reported rank (by individual SAT scores were adjusted according to model (C1) under several assumed ρ (and grade point average) within their high school classes Response categories correspond to top therefore d) values Table C1 reports the correlation of individual and school mean SAT decile, second decile, and second through fifth quintile, although the bottom categories are scores and student background indices across different choices of ρ These correlations are very rarely reported I used the class rank variable to “re-weight” the SAT data so that one- all quite large, indicating that school-level selection adjustments (at least using models like (C1)) are unlikely to affect results greatly Based on these correlations, the basic analyses in the main text were conducted using unadjusted SAT scores for the sample of 177 high-SATparticipation MSAs Exploratory analyses with adjusted scores (for moderate assumed ρ ) produced substantially similar results to those obtained from raw scores Table C2 offers further suggestive evidence that selection bias is not a major problem for the school-level analyses conducted here It displays the correlation across years in school-level average SAT scores and peer group background indices.7 The smallest sixth of the weighted SAT observations at each school come from each of the top two deciles and one-third come from each of the second and third quintiles (observations from the bottom two quintiles are dropped) Under the assumption that sample selection is random within each school-decile cell, these weights produce consistent estimates of average SAT scores and student characteristics for the 60 percent highest-ranked students at each school, and in particular produce averages that are comparable across schools Table C3 presents estimates of the SAT score-peer group gradient model—equation (7)—from the reweighted data The estimated models are nearly identical to those in Table 1.4 correlation coefficient here is 0.899, indicating that both measures are quite reliable: To the Appendix D: Proofs of Results in Section 1.3 There is almost certainly measurement error in school enrollment, and therefore in the school-level SATtaking rate One explanation for the small selection coefficient is attenuation from unreliability of withinschool changes in SAT-taking rates, which may contain very little signal but a good deal of noise The background index was estimated separately for each year, with a new set of weights for individual characteristics derived from a year-specific regression of SAT scores on individual characteristics with high school fixed effects 143 It is useful to begin with a Lemma that follows directly from the single crossing property: 144 U (x − hk , qk ) U (x − hk , qk ) h j − hk > ≥ > U (x − hk , qk ) U (x − hk , qk ) q j − qk Lemma Suppose that x j δ + µ j > x kδ + µk and h j > hk and assume the singlecrossing property: i If a family with income x (weakly) prefers community j to community k, then all families with x > x strictly prefer district j to district k ii If a family with income x (weakly) prefers community k to community j, then all families with x < x strictly prefer district k to district j Proof of Lemma (D3) An expansion similar to (D1) for family x easily establishes that ( ) U x − h j , q j > U (x − hk , qk ) Now suppose that districts j and k are discretely different The single-crossing ( property holds everywhere Consider family x’s indifference curve ( x > x )through q j , h j in q-h space (Refer to Figure D1.) We have shown that this curve passes below (q j − ε , h j −ν ) for small ε and ν such that U (x − h j +ν , q j − ε ) = U (x − h j , q j ) Because I prove part i; the remainder follows directly by a similar argument Define q j ≡ x j δ + µ j Suppose first that the two districts’ quality and housing prices are “close” to each other, so that first-order Taylor expansions are accurate Consider an expansion of the utility function around the utility that family x obtains in district k, evaluated at ( ) it crosses family x ’s indifference curve at q j , h j , it cannot cross anywhere else, so in ( ) particular must remain strictly below family x ’s at all points to the left of q j , h j As (qk , hk ) is one such point by assumption, and as family (x − h j , q j ): x ’s curve passes no higher than (qk , hk ) , family x must prefer district j to k  U (x − h j , q j ) − U ( x0 − hk , q k ) ≈ (D1) − (h j − hk )U ( x0 − hk , q k ) + (q j − q k )U ( x0 − hk , q k ) Proof of Theorem We prove the Theorem by construction First, without loss of generality, let the µ j s By the assumption that family x weakly prefers district j, the left-hand side must be non- be sorted in descending order: µ j > µ j +1 for all j < J Define an allocation rule: negative Rearranging terms, this implies that U (x − hk , qk ) ≥ >0 U (x − hk , qk ) q j − qk h j − hk j ~ G( y ) =  1 (D2) F (x ) ∈ [1 − whenever jn F (x ) = when N , − ( j −1)n N ), j = 1, K , J ; (D4) Note that the derivative of U (x − hk , qk ) U (x − hk , qk ) with respect to x is This rule assigns the n highest-income families to district 1—the district with the highest (U 21U1 − U11U ) U1 µ —the next n families to district 2; and so on To construct housing prices that make this As the denominator is always positive, the single crossing property ~ allocation an equilibrium, let h J = For j < J , let says that U (x − hk , qk ) U (x − hk , qk ) is strictly increasing in x If x > x , then, 145 146 ) (( ) ) ( U x j − h j +1 , q j +1 ~ ~ , h j = h j +1 + q j − q j +1 ( U x j − h j +1 , q j +1 ( ( ( where x j ≡ F −1 − ) jn N (D5) ) (Note that x( j = inf {x|G~(x ) = j }= sup{x|G~(x ) = j + 1}, by the ( } ~ ~ ~ I demonstrate that G (⋅) and h1 , K , h J are an equilibrium To begin, note that −1 −1 ( j −1)n jn n N )) − F (F (1 − N )) = N ∫ 1(G( x ) = j ) f ( x )dx = F (F (1 − ∫ 1(G( x ) = J ) f ( x )dx < n N , the latter a direct result of − Jn N ) ( ( ) ( ( ~ construction of G ) { ( ) ( ) ( ) ) ( ~ ( ~ ( ~ ~ ~ U x j − h j , q j ≈ U x j − h j +1 , q j +1 − h j − h j +1 U x j − h j +1 , q j +1 ( ~ + q j − q j +1 U x j − h j +1 , q j +1 ( ~ U x j − h j +1 , q j +1 ( ~ ( ~ = U x j − h j +1 , q j +1 − q j − q j +1 U1 x j − h j +1 , q j +1 (D6) ( ~ U1 x j − h j +1 , q j +1 ( ~ + q j − q j +1 U x j − h j +1 , q j +1 ( ~ = U x j − h j +1 , q j +1 ) (( ) ) ( ( ) ( ) ) ) ( ) ~ All that remains is to demonstrate that EQ4 is satisfied By definition of G (⋅) , for each j < J and that < EQ1 and EQ3 are thus x j > x k whenever j < k , which also implies that µ j > µk For any δ ≥ , then, x j δ + µ j > x kδ + µk , so in particular x j δ + µ j ≠ x kδ + µk  clearly satisfied What about EQ2? It suffices to show that for each district j, the ( “boundary” family—the family with income x j —is indifferent between districts j and j+1 ~ If this is true, Lemma provides that all families in districts k > j —who under G (⋅) have Proof of Theorem Consider the following statements: ( ~ incomes x < x j —will strictly prefer district j + to j under h , while all families in districts k < j + —other than the boundary family—will strictly prefer district j to j + Since this will be true for all j, there cannot be any family who prefers another district to the one to i x j δ + µ j > x kδ + µ k ; ii h j > hk ; iii x j > xk ; iv inf {x|G (x ) = j } ≥ sup{x|G (x ) = k} Given EQ1-EQ4, I show that (i) holds if and only if (ii) does; that (i) and (ii) imply (iii) and ~ which it is assigned by G (⋅) To demonstrate boundary indifference, plug the housing price equation (D5) into the ( ) ( ) ~ ~ first-order Taylor expansion of the utility function around q j , h j , evaluated at q j +1 , h j +1 : (iv), and that either (iii) or (iv) implies (i) By assumption, all families prefer a high-quality community to a low-quality community if there is no extra cost associated with it, and a low-priced community to a highpriced community if there is no loss of quality Thus, (i) must imply (ii) and vice versa, as no one would live in a low-quality community if houses were no more expensive in a higherquality community 147 148 Lemma tells us that if any family prefers community j to k when (i) and (ii) hold, all Proof of Corollary 2.2 When δ = , q j ≡ x j δ + µ j ≡ µ j , so the only possible quality ranking is the ranking higher-income families must as well There cannot, therefore, be any residents of community k who have incomes higher than any residents of district j, establishing both (iv) by effectiveness (When δ > , a high-income population can allow an ineffective school to and, trivially, (iii) This argument can be reversed: Let x j be the income of some family in outrank an effective one.) Corollary 2.1 thus describes the only possible allocation function: district j and x k the income of some family in k, with x j > x k If either (iii) or (iv) holds, The highest-income families must live in the district with the highest µ ; the next highest in there must be such a pair Now suppose that q j < qk Then it must be that h j < hk , else the next-most effective district; and so on Moreover, in order to maintain this allocation as x j would strictly prefer district k By Lemma 1, however, x k would also prefer district j in an equilibrium, housing prices must keep boundary families indifferent The price vector described in the proof of Theorem accomplishes this; because U > , no other price this situation Thus, q j > qk ; equality is ruled out by EQ4  vector can so.8 As an equilibrium is completely described by the allocation rule and price vector, it must be unique  Proof of Corollary 2.1 Before proving Corollary 2.3, it is useful to introduce an important Lemma: For finite J, in any equilibrium there must be one community that has higher quality than any other Theorem provides that every resident of this community has higher Lemma Let G be an assignment rule satisfying Corollary 2.1, and suppose that G assigns x to a more preferred district than that where x is assigned whenever x > x and the two are in different (n N ) income bins.9 Then there exist housing prices with which G is an equilibrium income than any resident of any other community As Theorem also establishes that the high-quality community has higher housing prices than any other, and as this can only occur when all homes are occupied, the community must contain the n highest-income families By definition of F, these are precisely those families with incomes above F −1 (1 − n N ) (As in the main text, I neglect families precisely at the boundary point.) Now consider the second-ranked district by quality Again, it has positive prices and higher income families than any district save the highest-ranked district, so must have ( ) families with incomes in F −1 (1 − 2n N ), F −1 (1 − n N ) The argument proceeds identically for the next-ranked district, and so on to the one of lowest quality  This is where the assumption of extra houses comes in; without it, the lowest-quality district could have positive prices, with a corresponding (but not necessarily identical) shift in each higher-quality district’s prices Formally, these conditions are: G (x ) = G (x ) whenever int (1 − F ( x )) N n = int (1 − F ( x )) N n , and i { ii } x G (x )δ + µG (x ) > x G (x )δ + µG (x ) whenever { } int{(1 − F ( x )) N n } < int{(1 − F ( x )) N n } 149 150 communities; that is, if and only if x ( j )δ + µ( j ) > x ( k )δ + µ( k ) for all j and all k > j Note Proof of Lemma Define r ( j ) as the index number of the j-th ranked district, where ranking is by ( xδ + µ Also, let x k be the lower bound of the kth (n N ) income bin: ( x k ≡ F −1 (1 − kn N ) , k = 1, K , J − Let housing prices be as follows: hr ( j ) 0  U2 = h +  r ( j +1) U1  (x( j − hr ( j +1) , qr ( j +1) ) (x( j − hr ( j +1) , qr ( j +1) ) (qr ( j ) − qr ( j +1) ) that the latter is equivalent to δ > µ( k ) − µ( j ) x( j ) − x(k ) C ≡ max j ,k > j for j=J for j j Recall that µ( k ) − µ( j ) x ( j ) − x (k ) It is immediately clear that when δ > C , assumption (ii) of Theorem is satisfied, so G is an equilibrium Similarly, when δ < C , there exist some j and some k > j such that x ( j )δ + µ ( j ) < x ( k )δ + µ ( k ) , violating Theorem 2, so G cannot be an equilibrium When δ = C , there are at least two districts for which x ( j )δ + µ( j ) = x ( k )δ + µ( k ) , violating EQ4, EQ2 is satisfied by the stated housing prices, it suffices to show that the family with income ( x j is indifferent between district r ( j ) and r ( j + 1) given G’s allocation of peer group and but otherwise the argument for Lemma could proceed  hr ( j ) and hr ( j +1) This is the result shown in (D6), above; it follows from a direct Taylor ( expansion of the utility function around family x j ’s consumption and quality in district r ( j + 1) Lemma then guarantees that no family in the districts {r (1), r (2 ), K , r ( j )} prefers any of the districts {r ( j + 1), r ( j + ), K , r ( J )} and vice versa As this must hold for each j, EQ2 must be satisfied  Proof of Corollary 2.3 Let x ( j ) denote the mean income of the j th bin, and let µ( j ) be the effectiveness of the community to which G assigns that income bin By Theorem and Lemma 2, G is an equilibrium assignment if and only if it assigns higher-income bins to higher-quality 151 (D8) 152 Table A2 Alternate measures of Tiebout choice: Effects on segregation and stratification Tables and Figures for Appendices Table A1 Evidence on choice-stratification relationship: Additional measures School-Level Racial Segregation Dissim Theil Index Measure (B) (C) Theil Segregation Measure AcrossDistrict Share of Variance: Adult Educ School-Level White/NonWhite Isolation Index (A) (B) 0.08 0.10 (0.01) (0.01) (C) (D) 0.07 0.06 (0.03) (0.03) ln(Population) / 100 0.05 0.53 (0.25) (0.34) 4.27 3.56 (0.80) (1.10) 0.60 (0.76) -0.18 (0.74) 1.73 (0.50) Number of districts (00s) Pop: Frac Black 0.03 0.03 (0.03) (0.03) 0.81 0.80 (0.09) (0.09) 0.19 (0.07) -0.07 (0.07) 0.07 (0.05) Pop: Frac Hispanic 0.04 0.03 (0.02) (0.02) 0.07 0.08 (0.06) (0.06) 0.06 (0.04) 0.05 (0.04) -0.01 (0.03) Districts per 17-yr-old population (* 10) ln(mean HH income) 0.02 0.02 (0.02) (0.02) 0.29 0.29 (0.06) (0.06) -0.05 (0.04) -0.12 (0.04) -0.01 (0.03) Gini coeff., HH income 0.50 0.46 (0.13) (0.13) 1.74 1.79 (0.41) (0.41) 0.35 (0.28) -0.15 (0.28) 0.28 (0.19) Pop: Frac BA+ 0.22 0.21 (0.04) (0.04) -0.47 -0.44 (0.12) (0.12) 0.26 (0.10) 0.42 (0.10) -0.03 (0.07) Foundation plan state / 100 0.17 0.17 (0.47) (0.46) -3.27 -3.28 (1.53) (1.53) 0.46 (0.96) 1.10 (0.93) 0.40 (0.63) -0.07 (0.04) 0.17 (0.10) 0.24 (0.08) -0.15 (0.08) 0.24 (0.06) Census tract- level segregation measures: Isolation index (white/non0.06 white) (0.03) 0.50 (0.10) 0.13 (0.08) 0.45 (0.08) -0.07 (0.05) Dependent Variable: Choice School-level choice index Across Across Across Schools, Schools Districts Within Districts (E) (F) (G) 0.06 0.26 -0.14 (0.02) (0.03) (0.02) Dissimilarity index (white/non-white) -0.04 (0.04) 0.27 (0.11) 0.47 (0.08) 0.25 (0.08) 0.15 (0.06) Across share of variance, education 0.52 (0.05) -0.44 (0.16) -0.36 (0.13) -0.51 (0.12) 0.04 (0.08) Across share of variance, HH inc -0.16 (0.05) 0.08 (0.16) 0.07 (0.13) 0.05 (0.13) 0.06 (0.09) 289 0.79 289 0.78 264 0.81 264 0.62 N R2 293 0.48 293 0.62 289 0.65 Isol Index (A) Tiebout Choice Measure District-level choice index 0.10 (0.02) 0.15 (0.04) 0.59 (0.30) 0.16 (0.02) 0.15 (0.04) 0.91 (0.31) 0.11 (0.02) 0.16 (0.03) 0.77 (0.25) Across-District Share of Variance Income Education (D) (E) 0.08 (0.01) 0.09 (0.01) 0.25 (0.10) 0.08 (0.01) 0.06 (0.01) 0.34 (0.11) Notes: Each entry is the coefficient on a single choice measure from a distinct MSA-level regression, with control variables as in Table 2, column C (except that the school-level choice index is excluded and population is entered here in levels rather than in logs) Number of observations = 289 for racial segregation measures; 293 for across-district analyses of variance Notes: Observations are unweighted MSAs/PMSAs Columns C-G exclude MSAs missing racial composition data for more than 20% of public enrollment Columns A, B, F, and G exclude MSAs with only one district See Theil (1972) for description of the Theil segregation measure, which is calculated over all schools in column E and over public districts and schools in F and G All columns include fixed effects for census divisions 154 153 Table A3 Effect of district-level choice on tract-level income and racial stratification Dependent Variable: Tract-Level Racial Segregation Dissimilarity Isolation (A) (B) Choice 0.00 -0.03 (0.02) (0.02) ln(Population) / 100 3.51 4.39 (0.65) (0.70) Pop: Frac Black 0.32 0.75 (0.07) (0.07) Pop: Frac Hispanic -0.03 0.00 (0.04) (0.05) ln(mean HH income) 0.31 0.41 (0.05) (0.06) Gini coeff., HH income 2.25 2.36 (0.33) (0.36) Pop: Frac BA+ -0.75 -0.77 (0.10) (0.11) Foundation plan state / 100 -4.03 -3.15 (1.27) (1.36) N R2 318 0.66 318 0.71 Across-Tract Share of Variance Income (C) -0.02 (0.01) 2.51 (0.28) 0.27 (0.03) 0.05 (0.02) 0.08 (0.02) 0.66 (0.14) 0.15 (0.04) -0.50 (0.54) Education (D) 0.01 (0.01) 1.24 (0.26) 0.11 (0.03) 0.12 (0.02) 0.02 (0.02) 0.71 (0.13) 0.37 (0.04) 0.31 (0.50) 318 0.70 318 0.68 Notes: Observations are MSAs/PMSAs, unweighted Each model includes fixed effects for census divisions Table B1 First stage models for MSA choice index (A) Instruments # of streams/1000 (B) (C) (D) (E) (F) 0.32 (0.08) County choice index (G) 0.01 (0.06) 0.41 (0.05) 0.19 0.18 (0.04) (0.05) Est 1942 choice index 0.62 (0.05) 0.50 0.50 (0.05) (0.05) County-district state indic -0.08 (0.04) -0.05 -0.05 (0.04) (0.04) Avg choice index, rest of state Controls ln(Population) 0.49 0.17 0.17 (0.07) (0.06) (0.06) 0.13 (0.02) 0.09 0.05 0.09 0.13 0.06 0.06 (0.02) (0.02) (0.01) (0.01) (0.01) (0.01) Pop: Frac Black 0.07 (0.17) 0.23 0.10 -0.14 -0.12 -0.14 -0.14 (0.17) (0.16) (0.13) (0.16) (0.13) (0.13) Pop: Frac Hispanic -0.16 (0.11) 0.01 0.08 -0.19 -0.22 -0.10 -0.09 (0.12) (0.11) (0.09) (0.11) (0.09) (0.09) ln(mean HH inc.) -0.40 (0.13) -0.28 -0.25 -0.13 -0.30 -0.08 -0.08 (0.13) (0.12) (0.10) (0.12) (0.10) (0.10) Gini, HH inc -2.88 (0.84) -3.16 -2.80 -1.29 -2.36 -1.38 -1.38 (0.82) (0.76) (0.64) (0.79) (0.62) (0.63) Pop: Frac BA+ 0.28 (0.26) 0.22 0.27 -0.18 0.14 -0.15 -0.15 (0.25) (0.23) (0.19) (0.24) (0.19) (0.19) Foundation plan state 0.01 (0.03) 0.01 -0.01 0.00 0.02 -0.01 -0.01 (0.03) (0.03) (0.02) (0.03) (0.02) (0.02) N 318 0.51 R F statistic, exclusion of instruments 318 0.54 17.7 318 0.60 64.3 318 0.73 122.0 315 0.58 54.1 315 0.75 72.2 315 0.75 57.6 Sources : Electronic Geographic Names Information System (Streams); 1990 Census STF-3C (County choice); Gray, 1944 (1942 choice index); Kenny and Schmidt, 1994 (County Districts); author's calculations Notes : Dependent variable is the district-level choice index Observations are MSAs All columns include fixed effects for census divisions Columns E, F, and G exclude MSAs for which there are no other MSAs in the same state 155 156 Table B2 2SLS Estimates of Effect of Tiebout Choice Table C1 Sensitivity of individual and school average SAT variation to assumed selection parameter Across-District Share of Variance, HH Income Dissimilarity Index SAT ScorePeer Group Gradient Avg SAT Score Table , Col C (A) 0.10 (0.01) Table , Col F (B) 0.10 (0.02) Table 4, Col E (C) -0.09 (0.15) Table 7, Col G (D) -14.1 (5.1) 0.13 (0.10) 0.17 (0.14) -0.27 (0.36) -55.9 (21.3) County choice 0.08 (0.06) 0.02 (0.08) 0.14 (0.40) -18.7 (15.1) Historical (1942 choice + county districts) 0.06 (0.03) 0.08 (0.03) 0.17 (0.25) -6.1 (7.3) Rest of state 0.16 (0.06) 0.16 (0.08) 1.27 (1.30) -35.0 (36.7) All but streams 0.07 (0.02) 0.07 (0.03) 0.12 (0.25) -5.7 (7.2) All 0.07 (0.02) 0.07 (0.03) 0.02 (0.23) -9.9 (7.0) Model: Source Table, Specification OLS 2SLS Streams Correlation between actual and selection-adjusted value Individual School average SAT score SAT score (A) (B) Assumed selection parameter ρ = 0.05 1.000 0.999 ρ = 0.1 0.999 0.998 ρ = 0.25 0.996 0.987 ρ = 0.5 0.983 0.956 ρ = 0.75 0.956 0.910 ρ = 0.9 0.930 0.873 Notes : Entries in table represent cross-sectional correlation between observed score (or average score) and that obtained by adjusting scores using the school-average SAT-taking rate and within-school selectivity described by the listed parameter Obser Table C2 Stability of school mean SAT score and peer group background characteristics over time Notes: Each entry represents the coefficient on the district-level choice index (or, in Column C, on the interaction between that index and the peer group background index) from a separate regression Specifications are the same as the OLS specification listed at top, but are estimated by instrumental variables Bold coefficient indicates that a Hausman test rejects equality of the 2SLS and OLS choice coefficients at the 5% level 1994 1994 1995 1996 1997 1998 0.957 0.957 0.955 0.952 1995 0.906 1996 0.908 0.912 0.961 0.959 0.957 0.963 0.961 1997 0.902 0.908 0.918 1998 0.899 0.909 0.915 0.921 0.963 Notes : Entries above diagonal represent correlations across years in schools' average SAT scores Entries below diagonal are correlations of school peer group background index values 158 157 Table C3 Effect of Tiebout choice on the school-level SAT score-peer group gradient: Estimates from class rank-reweighted sample (C) 1.40 (0.16) (D) -0.14 (0.24) (E) -5.18 (2.51) (F) -2.66 (2.79) 0.11 (0.22) -0.40 (0.15) -0.32 (0.12) -0.09 (0.17) -0.07 (0.18) 2.19 (0.52) 2.03 (0.45) 1.18 (0.46) 1.25 (0.49) 0.10 (0.02) 0.05 (0.02) 0.05 (0.03) * Pop: Frac Black -0.45 (0.37) -2.37 (1.28) * Pop: Frac Hispanic 0.02 (0.20) -1.47 (0.94) * ln(mean HH inc.) 0.42 (0.23) 0.28 (0.23) * Gini, HH inc 3.20 (1.56) 2.88 (1.77) * Pop: Frac BA+ 0.77 (0.56) 1.12 (0.69) 0.02 (0.07) 0.01 (0.06) * MSA SAT-taking rate * ln(Population) * Foundation plan state * Pop: Frac White -1.17 (0.76) * ln(Density) 0.01 (0.03) * Pop: Frac LTHS 0.39 (0.88) * Census division FEs R R , within MSAs n n y y y y 0.78 0.75 0.78 0.75 0.79 0.76 0.79 0.76 0.80 0.76 0.80 0.76 150,000 30,000 Entire continental U.S (left axis) Counties in 1990 MSAs (right axis) 100,000 50,000 1930 10,000 1940 1950 1960 1970 1980 1990 Year Sources: Statistics of state school systems , 1966: 1932, 1944, 1952, 1954, 1956, 1958, 1962, 1964, 1966 Gray, E.R., 1944, Governmental Units in the United States 1942: 1942 Governments in the United States 1957: 1957 Elsegis electronic file, ICPSR #2238: 1969, 1970, 1971, 1972 Common Core of Data: 1981 forward Notes : Sample in each column is 5,671 schools in 177 MSAs Dependent variable is the weighted mean SAT score at the school, with weights adjusted using students' self-reported rank in class to balance the first and second deciles and second and third quintiles within the school; students not reporting a class rank or reporting a rank in the bottom 40% are dropped Within MSAs, schools are weighted by the number of twelfth grade students; these are adjusted at the MSA level to make total MSA weights proportional to the 17-yr-old population All models include 177 MSA fixed effects, and standard errors are clustered at the MSA level 159 20,000 160 2000 # of Districts in MSAs (B) 1.70 (0.19) Avg student background index # of Districts in U.S (A) 1.79 (0.04) Interaction of student background average with: * Choice index Figure B1 Number of School Districts Over Time Figure C1 SAT-taking rates and average SAT scores across MSAs 1300 Non-Sample MSAs Average SAT Score 1200 Sample MSAs 1100 1000 900 800 0% 10% 20% 30% 40% SAT-Taking Rate 50% 60% 70% Notes : Sample MSAs are those used in main analysis (i.e those in states with SAT-taking rates above one third) Honolulu and Anchorage MSAs are excluded Figure D1 Illustration of single-crossing: Indifference curves in q-h space h x>x x0 x