Does School Quality Matter? a Travel Cost Approach

DOES SCHOOL QUALITY MATTER? A TRAVEL COST APPROACH

Jonathan Eyer Abstract Sol Price School of Public Although the majority of school districts in the United States as- Policy sign students to schools on the basis of geographic location, there University of Southern is increasing interest from parents and policy makers in school California choice programs. These programs allow parents (and children) to Los Angeles, CA 90089 choose their school. In many cases, when students opt to attend a [email protected] nonlocal school, the parent is responsible for transportation. This creates a trade-off between school quality and travel expenditure. This paper estimates the value of school quality in a random utility model framework, using data on rank-ordered preferences submitted in a school choice program in Garland, Texas. I find that a standard deviation of high school quality is valued between $495 and $783, substantially lower than hedonic estimates that include noneducational benefits associated with good schools. In addition to estimating the average value of school quality, this approach can also be used to estimate the value of school quality for demographic groups or for individual students.

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1. INTRODUCTION In the Unites States, and most other developed countries, elementary and secondary schooling is provided as a public good. Because schools are a public good, there is no price signal through which households can directly signal their preferences for educational quality, making it difficult to justify costly educational improvement. This has led to a large literature in economics that seeks to value school quality. In the United States, most students are assigned to schools based on their home residence, and households obtain high-quality education by moving to areas near good schools. This structure is beginning to change, though, as an increasing number of school districts and states allow for households to express preferences in the school assignment process without changing homes. In some cases, school choice programs require households to provide transportation to schools, implicitly placing a price on school attendance. As outlined by Hotelling (1949), and later formalized by Trice and Wood (1958), the travel cost associated with reaching an amenity can be used to value that amenity, and Hanemann (1978) showed that amenity attributes can be valued when travel cost is included in a random utility model. When multiple travel decisions are made, each choice can be used to enhance the precision of the estimate. In this paper I use data on ranked preferences of high schools in a school choice program in Garland, Texas, to estimate the value of school quality based on the trade-off between school quality and travel cost. This imposes less stringent data requirements than wage studies and avoids the need to control for neighborhood attributes, because neighborhood characteristics do not change when an individual student changes schools. The resulting estimate of the value of school quality is robust to the number of schools that students are required to rank in their school choice application, allowing for this approach to be used to value school quality under a wide range of school choice programs. Moreover, the travel cost model asks a different question than the frequently used hedonic models. Whereas hedonic models estimate how much more homes are worth near good schools, the travel cost model estimates how much households are willing to spend in order to attend a good school. This is a policy-relevant question in a number of contexts, particularly in the consideration of student-specific interventions, and these are not questions that hedonic models are well-suited to address. Further, to the ex- tent that policy makers are concerned about improving school quality for economically disadvantaged or minority students, this allows for an estimate of the value of school quality that is tailored to a particular demographic. The rest of the paper proceeds as follows: Section 2 discusses existing methods to value school quality; section 3 discusses school choice programs; section 4 describes the study area; section 5 lays out the methodological approach; section 6 describes the data; section 7 discusses the results and section 8 relates them to hedonic estimates; section 9 concludes the paper.

2. PREVIOUS VALUATIONS OF SCHOOL QUALITY The common approach to measuring the value of school quality relies on the hedonic pricing model, which has been applied to the question of school quality since at least Oates (1969). The hedonic model relies on the fact that school quality is generally driven by home location, and homes that are in the attendance zone of good schools are more

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valuable to households that value school quality than homes that convey attendance to a poor school. The premium that the housing market places on homes near good schools can be interpreted as the value of school quality. Moreover, because hedonic models consider the premium on all homes—not just homes with students—they capture not only the direct benefits associated with attending a good school but also the ancillary benefits associated with living in a neighborhood with good schools. The inclusion of these ancillary benefits means that the corresponding estimates of the value of school quality can be used to value policy improvements that affect an entire school or district. Good schools tend to be in neighborhoods that have other, enjoyable attributes, and these unobserved neighborhood attributes may bias upward estimates of the value of school quality if values of correlated neighborhood attributes are falsely assigned to school quality. A number of methodological approaches have been developed to address this concern, generally following Black (1999) by exploiting geographic boundaries of school attendance zones as a regression discontinuity. (See Nguyen-Hoang and Yinger 2011 for a thorough overview of the hedonic literature on the value of school quality following Black 1999.) Focusing on only the area immediately surrounding an attendance zone limits the sample to only areas that have the same level of unobservable nonschool-related amenities, isolating the value of school quality from the potential confounding amenities. There is concern, however, that the boundary approach will not completely remove the bias associated with neighborhood characteristics that are correlated to school quality. If school boundaries are consistent over time, households that value school quality will sort toward the side of the boundary that conveys access to the good school, potentially resulting in substantial differences between the neighborhoods and endogenously determined amenities on either side of the boundary (see Dhar and Ross 2012). Bayer, Ferreira, and McMillan (2007) find that after controlling for neighborhood characteristics on either side of a school attendance boundary, the estimated value of school quality falls substantially relative to a model that does not control for neighborhood characteristics. Given that this paper estimates the value of school quality in a discrete choice context, it is also worth highlighting a literature that estimates the value of school quality in a discrete choice framework in which households select over a set of homes with an array of attributes, including school quality and price. These models specify an indirect utility function over household and home characteristics and estimate the parameters of the utility function to maximize the likelihood that households select their house rather than the other homes in the choice set (see Nechyba and Strauss 1998). Characteristics of the home are generally represented by a house-specific fixed effect that captures the utility derived from the school quality for which the home is assigned, as well as other observable and unobservable characteristics of the home. The house-specific fixed effect is then regressed in a second stage on school quality and other attributes to isolate the effect of school quality on utility. The discrete choice approach conveys an important benefit relative to the traditional hedonic approach because it allows for the value of school quality to vary across the observable characteristics of individuals. Barrow (2002), for example, finds that white households in Washing- ton, DC, respond to school quality in their housing selection decisions whereas black households do not. Again, the issue of correlation with unobserved neighborhood attributes arises because a home’s price is likely correlated with the unobserved house and

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neighborhood attributes. Housing prices are then isolated from house characteristics in an instrumental variables framework using observable attributes of distant houses (see Bayer, Ferreira, and McMillan 2007).

3. SCHOOL CHOICE Although students are generally assigned to a school based on the location of their home, there is growing interest in programs that allow households some degree of choice in the school assignment process. Proponents of these programs argue that, because these programs de-couple educational quality and home prices, they improve educational access for low-income households (Sugarman 2004). Indeed, several states target school choice programs specifically at low-income households or in areas with under-performing schools. Despite the interest in school choice programs, there is mixed evidence of the efficacy of these programs (see, e.g., Cullen, Jacob, and Levitt 2003 and Deming et al. 2014). At their most basic, school choice programs—depending on state and local laws— allow students to request a transfer to a nonlocal school, either elsewhere in the school district or in the state. In these programs, admission to the requested school is not guaranteed, and if a transfer is declined the student continues to attend her neighborhood school. In other programs, students do not receive a default, location-based school assignment, but rather all students in a school district are assigned based on student preferences. Such programs are generally enacted at the school district level and several major U.S. school districts (including Boston, MA; Berkeley, CA; and San Francisco, CA) currently assign all students based on preferences. Because demand for high-quality schools frequently exceeds the number of available seats, students may be asked to submit a ranking of several schools in order of preference, and students are assigned to a school based on an assignment mechanism. These mechanisms consider student rankings, school capacity constraints, and, if ap- plicable, student priorities, in order to assign students to schools. Student priorities provide a rule by which over-requested school seats are assigned. Priorities may be based, for example, on test scores, sibling attendance, racial diversity requirements, or residence in a walking zone. In the absence of school priorities, ties are generally broken by lottery. A wide range of studies have examined school choice programs in the context of these mechanisms, particularly with respect to designing incentive-compatible allocation mechanisms (see, e.g., Abdulkadirogluˇ et al. 2005; Chen and Sönmez 2006; Pathak 2011). School quality and proximity are generally found to be considered in student rankings of schools in choice programs, although other considerations also matter. Hast- ings, Kane, and Staiger (2009) use data on student rankings of schools in Charlotte, NC, in a multinomial logit framework to examine the components of schools that in- fluence household decisions. Importantly, they also find a large degree of heterogeneity across households, particularly that preferences over school quality and proximity are negatively correlated. This suggests that households placing a high value on school quality are willing to travel to obtain it. Similarly, Burgess et al. (2015) examine public school choice in England and find that changes in rankings are most responsive to changes in school quality, with distance having the second largest effect on rankings among the considered covariates. Racial composition of schools also enters the

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household decision-making process, with households preferring schools in which their race constitutes the majority (Hastings, Kane, and Staiger 2009). These results generally align with survey-based studies on the trade-off between public and private schools (see Altenhofen, Berends, and White 2016 for a review). Information about the quality of schools is important in the choice process as well, leading more students to opt out of their assigned school and into a school choice program that allows access to higher- quality schools, although the effect of information seems to be largest when the set of potential schools is very large (Hastings and Weinstein 2008). Also notable, Reback (2005) finds substantial increases in home values in neighborhoods with low-quality schools following the introduction of an interdistrict school choice program. This provides some evidence that households expect to experience higher school quality when school choice programs are in place.

4. GARLAND INDEPENDENT SCHOOL DISTRICT The data for this paper come from the Garland Independent School District (GISD), which serves the city of Garland, Texas (a large suburb of Dallas), and portions of two surrounding cities. Approximately 50 percent of students in the district are considered economically disadvantaged and 94 percent of the student body is nonwhite. The GISD comprises seven high schools, twelve middle schools, and forty-seven primary schools serving 56,000 students. Under the 1964 Civil Rights Act, the GISD must meet deseg- regation targets and the GISD is one of the few districts to use a school choice program to meet these requirements. In the GISD, students submit rank-ordered choices of three of the seven high schools and are assigned to a school on the basis of their submitted rankings and student priorities. Each student has a level of priority at each school, based on walking zones, busing zones, and sibling attendance. Students participate in the school choice program each time they transition between schools (elementary, middle, and high school), so students submit rankings at least three times throughout their academic career—by the time students apply for high schools, they are experienced with the choice process. In years that a student is not transitioning between school levels, the student may either keep her current assignment or reenter the school choice program and be reassigned. The GISD uses the Boston Mechanism. This mechanism first attempts to assign students to their first choice of school and, when the demand for a school exceeds the number of available seats, assigns the available seats to the students with the highest priority. The mechanism then attempts to assign the remaining students—those who were not assigned to their first choice—on the basis of their second choice, again assigning scarce seats to the student with the highest priority. Abdulkadirogluˇ and Sön- mez (2003) show that the Boston Mechanism is not incentive-compatible. Note that if a student has low priority in her first and second choices, she can falsely rank her second choice as her favorite school and compete only with those who also listed that school as their first choice. This behavior is unlikely to be a concern in this case, however, because over 97 percent of GISD students receive their first choice, and households should be aware they have a high probability of receiving admission to their first choice, thereby submitting their truly preferred school as their first choice.

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Figure 1. Garland Independent School District Busing Zones.

Although there are magnet programs offered at three of the seven high schools in the GISD providing a targeted education based on student preferences and capability, these programs have a separate application process from the school choice submis- sions, and entry criteria vary between magnet programs. Although there are seats at each school not designated for the magnet program, the GISD specifically advises parents to ignore the magnet programs in school choice rankings, and entry to the magnet programs is not related to a student’s school choice assignment. The submitted rankings, therefore, are in response to the quality of non-magnet program education. The GISD covers approximately 100 square miles spread across the cities of Gar- land, Sasche, and Rowlett. Bus transportation is generally not provided to students who live fewer than two miles from the closest school. In general, if students live more than two miles from the closest school, busing is provided to the closest available school, although the district occasionally deviates from the two-mile rule and the GISD makes available maps that delineate busing zones. If a student is eligible for busing to a particular school but requests and attends a different school, the student is responsible for his own transportation. A map of the GISD and corresponding bussing zones is provided in figure 1. The Texas Assessment of Knowledge and Skills (TAKS) exam is a standardized achievement test used to test Texas students’ proficiency in English, math, science, and social studies. Students must pass all four portions of the TAKS exam by eleventh grade in order to graduate from high school.1 The high schools in the GISD are generally of average quality compared with other schools in Texas, but there is a large amount of heterogeneity in quality between the schools. Scores range from a 50 percent pass rate among tenth graders at South Garland High School to a 93 percent pass rate among

1. This requirement is currently being phased out in favor of an alternative exam. At the time that rankings were submitted, however, TAKS was the pertinent achievement exam.

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Table 1. Garland Independent School District School TAKS Scores (%)

School Tenth Grade Eleventh Grade

South Garland 50 81 Naaman Forest 54 85 Garland 62 84 Lakeview Centennial 69 82 North Garland 72 91 Sasche 75 88 Rowlett 76 93 District average 65.4 86.3 District standard deviation 10.3 4.5 State average 65 85

eleventh graders at Rowlett High School. Tenth- and eleventh-grade TAKS scores for each of the seven high schools in the district are presented in table 1.

5. METHODOLOGY The rank-ordered logit model derived by Hausman and Ruud (1987) provides a framework in which ranked data can be incorporated in a multinomial logit context. The utility derived from each option is specified as a function of individual and alternative specific traits. Individuals are assumed to submit rankings that reflect their utility— the alternative that yields the highest utility is ranked first, the alternative that yields the second-highest utility is ranked second, and so forth. For each alternative that is ranked, the utility derived from that alternative must exceed the utility from all alternatives that receive a lower ranking. The parameters of the utility function are estimated to maximize the likelihood that the utility function would result in the observed rankings. In this case, households are assumed to have utility over school quality and expenditure. For each student i, the utility derived from attending school j is given by

Ui, j = αq j + βdi, j + i, j, (1)

where qj is the quality at school j, di,j is the distance that student i must travel to reach school j,andεi,j is an idiosyncratic error term. Under the assumption that εi,j is dis- tributed Extreme Value Type II, the associated likelihood function is given by

N U , eUi,i1 eUi,i2 e i i3 L = , (2) U , U , U , ∈ e i n ∈ \ e i n ∈ \ , e i n i=1 n N n N i1 n N i1 i2

where U(.) is the utility function described above, i1,i2, and i3 are individual i’s first, second, and third choices, and N is the set of all schools. School quality and travel distance need not be the only characteristics of a school that households value. Normally, an econometrician would include an alternative-specific constant in the utility function and regress the alternative-specific constant utility on school quality. This allows for school-specific preferences that are not related to school quality to be included in the utility function. Unfortunately, this approach requires a

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large set of alternatives in order to accurately estimate the eﬀect of school quality on utility, and because there are only seven schools (and seven alternative speciﬁc constants) this approach is not feasible in this context.2 One might be concerned that households prefer neighborhood schools for reasons other than distance. In this case, the relative importance of commuting would be over- stated because the preference to attend a neighborhood school would be falsely assigned to the proximity of that school. This would lead to an underestimate of the value of school quality because the value of school quality is derived from the relative importance of school quality to distance. Relatedly, because the assignment mechanism takes into account the walking zones around schools, a household might move to areas that are close to good schools in order to increase the probability of receiving admission to the high-quality school. This would mean that a household’s payment for school quality would be composed of both the travel expenditure as well as the traditional hedonic expenditure that increases the probability of acceptance into a good school, and the travel cost approach would underestimate the true willingness to pay for school quality.3 I address this concern by removing the local school from the choice set and estimating the value of school quality based on only the nonlocal schools. I remove the local school from the student’s choice set for each of her preferences (i.e., reducing the de- nominator of the likelihood function from the product of utility at seven schools to the product of utility at six schools), and in the event the local school is selected as one of the student’s choices, I remove that observation from the data. For example, if a student ranked her local school as her second choice, her contribution to the likelihood would be

U , eUi,i1 e i i3 . (3) eUi,n eUi,n n∈N\local n∈N\i1,local Because I do not observe the home address of students, I first define a student’s neighborhood school as a school that shares her home ZIP code. Some ZIP codes have no neighborhood schools and students in these ZIP codes have the full choice set of seven schools. Two ZIP codes have two schools and students in these ZIP codes have both schools removed from their choice set so they choose from five possible schools. Alternatively, I remove the geographically closest school from each student’s choice set, as defined by the population-weighted centroid of each ZIP code. The former approach assumes that neighborhood characteristics stop at the ZIP code boundary and that every school has a neighborhood, and the latter approach assumes that every person has a neighborhood that is centered around the closest school. There may, of course, be other school-specific student preferences that are unrelated to school quality or travel distance. For example, A student may prefer a school that her sibling attends or one that has a good sports team. Normally these sorts of preferences would be captured in an alternative-specific constant. Although I cannot separately identify the value of school quality from alternative-specific constants because of the size of the choice set, I partially address these concerns by reestimating

2. Although the standard errors are unreliable, I report the results of the alternative speciﬁc constant approach in Appendix B. The implied value of school quality is comparable to the main results. 3. According to Zillow (an online real estate site), home prices do tend to be higher in the 75048 ZIP code than in the others. This ZIP code is home to Sasche, which is among the best schools in the district.

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the model using only the second and third choices. This assumes that students can have idiosyncratic preferences that are unrelated to school quality or travel distance, but limits these idiosyncratic preferences to a single school. The student’s preferences over her second and third choices will then be based only on the trade-off between school quality and travel cost. Also note that this approach addresses some of the concerns surrounding sorting behavior. If households sort to neighborhoods to be close to a school or the amenities that are correlated with good schools, this school will likely be their first choice and removed from the choice set. On the other hand, the vast majority of students in the GISD are assigned to their first choice of school. One might be concerned, then, that households do not put in as much effort in ranking their second and third choices as they put in to their first choice. To account for this possibility, the model is reestimated using only the top-ranked school for each student. Finally, I estimate a mixed logit model on the full set of rankings. The logit model carries the well-known independence of irrelevant alternatives (IIA) assumption, which requires that the inclusion or exclusion of an alternative will not alter the relative probability that other alternatives are selected. The mixed logit model, however, allows for variation in the parameters of the utility function across individuals, and does not require the IIA assumption. To show robustness to the assumption of IIA, I respecify the utility function as

Ui, j = αiq j + βidi, j + i, j, (4)

where αi ∼ f (α, θ α)andβi ∼ f (β, θ β ). This logit speciﬁcation assumes that the utility function is identical across students. When students submit rankings of multiple schools, the rank-ordered logit model can also be used to estimate a student-speciﬁc value of school quality. Rather than estimating the parameters of the utility function from the full set of rankings, the parameters are estimated for each student, so that each regression is computed with data from the student’s three rankings. The likelihood function associated with student i is

Ui,i Ui,i Ui,i3 e 1 e 2 e Li = . (5) eUi,n eUi,n eUi,n n∈N n∈N\i1 n∈N\i1,i2

6. DATA Data were available for 4,313 high school applications for the 2014–15 school year. A data observation consisted of a ZIP code, a first choice, and usually a second and third choice. Individual student characteristics such as race, income, and gender were unavailable. These students are primarily ninth graders (approximately fourteen years old). Several students ranked the same school twice, or submitted a request for a school that did not exist. These students were removed from the data. Approximately 550 of these applications were incomplete, meaning the students did not rank three schools. These students remained in the sample, except when the model was estimated without the first-ranked school. The rankings of the students are presented in table 2. Rankings suggest that students do respond to school quality, particularly with their first selection. Naaman Forest and South Garland, which have lower TAKS scores than other schools, received approximately half as many first-place rankings as other schools.

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Table 2. Number of Students Submitting School Rankings

School First Choice Second Choice Third Choice Any Choice

Sasche 725 652 382 1,786 Garland 703 731 774 2,208 Rowlett 696 544 524 1,764 Lakeview Centennial 681 484 492 1,657 North Garland 619 489 507 1,615 Naaman Forest 455 667 747 1,869 South Garland 434 286 311 1,031

Table 3. First Choice by ZIP Code

ZIP Code

School 75040 75041 75042 75043 75044 75048 75088 75089 75098

Garland 288 118 113 78 51 10 13 32 0 South Garland 31 220 8 166 30240 Naaman Forest 168 7 60 12 187 18 0 3 0 North Garland 122 44 246 58 115 8 9 17 0 Lakeview Centennial 62 102 13 433 14 10 13 32 2 Rowlett 122 26 4 52 4 0 248 233 1 Sasche 175 9 104 15 22 196 8 178 18 Total 968 526 548 820 396 242 293 499 21 Area (Sq. Miles) 15.7 7.3 6.7 19.0 9.3 8.0 10.9 11.2 4.0

Note: Bolded numbers indicate the ZIP code in which a school is located.

Sasche and Rowlett, which are the highest-scoring schools, were among the most often ranked schools. Garland, the oldest school in the district, is consistently ranked among students’ preferences, even though it is slightly below the average level of school quality in the district. This suggests that there is some degree of non-quality-related characters associated with school preferences. Student home location is available at the ZIP-code level. Students generally rank a school that is geographically close as their first choice. The distribution of first choices by ZIP code of student residence is presented in table 3. The bolded entry in each row indicates the ZIP code in which a school is located (e.g., Garland High School is located in ZIP code 75040 and South Garland High School is in ZIP code 75043). As is shown, schools receive the greatest portion of their first- place rankings from students who reside in the same ZIP code in which the school is located. The travel distance was computed between the centroid of each U.S. Census block and each school using Google Maps driving directions. Travel distances were computed in both miles and minutes of driving. If a Census block intersected a school’s busing zone, then the travel distance and time were treated as zero because transportation costs are assumed by the school district rather than the household. Census blocks were geographically overlaid onto ZIP Code Tabulation Areas (a spatial representation of ZIP codes). A map of the overlaid Census blocks is presented in figure 2. The travel distance between each ZIP code and each school is calculated as the population-weighted

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Figure 2. Census Block Map.

average between Census blocks in the ZIP code and the school. Travel distances range from 1.6 miles to 13.6 miles, and travel times range from 3.9 minutes to 20.2 minutes. The travel cost was computed as a function of distance and time. Following the U.S. Department of Transportation, the cost of driving one mile at the time of this study is 59.2 cents to account for gasoline, wear and tear, and so forth. Zamparini and Reggiani (2007) provide a recent meta-analysis of 90 studies on the value of travel time and, using that study’s average value of time spent commuting, the opportunity cost of time is calculated at 55 percent of the wage rate, which averages $23 per hour in Garland, Texas. One-way travel costs, then, range from $1.41 to $10.20, and each ZIP code has a relatively low-cost option—in every case there is a school with a travel cost below $2.50. Each student was assumed to take two trips per day and attend school 180 days per year for a total of 360 one-way trips per year. Although students can reenter the school choice process each year, students can continue at their assigned school without re- entering the choice process, thus the annual number of trips is multiplied by the four years a student will attend the high school. For ease of computation, travel costs were expressed in hundreds of dollars.

7. RESULTS The parameters α and β of the utility function are estimated via maximum likelihood for the full sample, as well as the subsamples and sensitivity analyses. In each case, the models are estimated twice: once measuring school quality with tenth-grade TAKS scores and once using eleventh-grade TAKS scores. Estimation results are presented in tables 4 and 5. Both school quality and travel cost have statistically signiﬁcant eﬀects

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Table 4. Value of School Quality Estimation Results

Full Sample No Same ZIP School No Closest School

10th Grade 11th Grade 10th Grade 11th Grade 10th Grade 11th Grade

Travel cost −0.024*** −0.023*** −0.023*** −0.023*** −0.022*** −0.0212*** (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) School quality 0.018*** 0.0259*** 0.015*** 0.019*** 0.014*** 0.015*** (0.001) (0.002) (0.001) (0.003) (0.001) (0.003) Value of school quality (dollars) 76.49 110.24 65.67 83.15 65.79 68.98 Number of students 4,313 4,313 4,083 4,083 4,068 4,068 Number of decisions 11,928 11,928 8,781 8,781 8,613 8,613 AIC 35,498 35,684 25,190 25,191 25,330 25,446

Notes: AIC = Akaike’s Information Criterion. Standard errors in parentheses. ***p < 0.01.

Table 5. Value of School Quality Estimation Results

First Choice Only No First Choice

10th Grade 11th Grade 10th Grade 11th Grade

Travel cost −0.029*** −0.029*** −0.020*** −0.023*** (0.001) (0.001) (0.000) (0.000) School quality 0.024*** 0.024 0.0143*** (0.025)*** (0.002) (0.004) (0.001) (0.003) Value of school quality (dollars) 82.97 85.62 70.07 124.49 Number of students 4,313 4,313 3,871 3,871 Number of decisions 4,313 4,313 7,615 7,615 AIC 12,538 12,676 22,783 22,836

Notes: AIC = Akaike’s Information Criterion. Standard errors in parentheses. ***p < 0.01.

on utility in each model, and in every case the sign of the coeﬃcient corresponds with theory. The value of school quality is the ratio of the marginal utility of school quality divided by the marginal utility of decreasing expenditure: −α/β. For the full sample, the average value of a unit of school quality over four years of high school education is $76 and $110, when school quality is measured with the tenth- and eleventh-grade TAKS scores, respectively. The total value of a high school education can be computed by multiplying these estimates by the level of school quality. The tenth-grade estimates ﬁnd that the total value of a high school education ranges between approximately $3,800 and $5,800, and the eleventh-grade estimates correspond to substantially higher values, between $8,900 and $10,200 for four years of education. These numbers can be interpreted as the tuition parents would be willing to pay for the GISD schools if they were private and provided transportation services. In both cases the increase in value from the worst school to the best school is around $2,000. In order to facilitate comparisons to other estimates of the value of school quality, I also note the value for a standard deviation of school quality. The standard deviation of school quality across schools in the GISD is 10.3 and 4.5 percentage points for the tenth- and eleventh-grade TAKS scores, respectively, so the corresponding value of a

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standard deviation of school quality is $783 and $495. These results are presented in table 4. After removing the neighborhood schools from students’ choice sets, estimates of school quality fall slightly. The value of school quality using tenth-grade TAKS scores falls approximately $10 to $65 per TAKS point. School quality values fall more substantially when the eleventh-grade TAKS scores are used to measure school quality—to $83 or $69 per unit of school quality, depending on whether neighborhood schools are measured based on shared ZIP codes or distance. These results suggest that, to the ex- tent that removing schools within the same ZIP code or the closest school captures the effect of sorting toward a neighborhood school, sorting behavior is not leading to downward bias in the value of school quality. It is not clear what is driving the lower values for school quality of nonlocal schools. One potential explanation is that school quality is, in fact, negatively correlated with the local amenities that households value. For example, if households have preferences for low housing prices and short commutes to Dallas, these attributes would generally be negatively correlated with high-quality local schools. Alternatively, the measurement of local schools is quite coarse and misspeci- fication in the definition of a local school may lead to lower estimates of the value of school quality. The results are also economically comparable when the value of school quality is estimated without using the first choice. These values are presented in table 5. Estimates increase by only $7 to a total of $83 when tenth-grade scores are used to measure school quality, and decline by $25 to $85 when school quality is measured with eleventh-grade scores. These values move in opposite directions because the relative quality of schools differs between the two measurements of school quality. Similarly, the estimated value of school quality is also economically comparable when the parameters are estimated using only the students’ first choice. As expected, the computed value of school quality declines for tenth-grade TAKS scores and increases for eleventh-grade TAKS scores—to $70 and $124, respectively. Finally, the mixed logit model is estimated via simulated maximum likelihood and results are presented in table 6. Because the estimated parameters are random variables, the expected willingness to pay for a one-unit increase in the value of school quality is computed as E(WTP) = E(α)/E(β). The results are comparable—but slightly lower—than those obtained from the logit model. Although the standard deviation of the estimated parameter is statistically significant, the standard deviation of the parameter is not statistically different from zero. The similarity between the computed value of school quality in the logit and mixed logit models suggests it is unlikely that viola- tions of the IIA assumptions are biasing parameter estimates. Utility parameters are also estimated via maximum likelihood for each student in the sample. A summary of these estimates is presented in table 7. The median estimate corresponds with a value of school quality of approximately $58 per one-point increase in tenth-grade TAKS scores, or $144 for each one-point increase in eleventh-grade TAKS scores. The individual estimates of the value of school quality vary dramatically across individuals, and the value of school quality appears to be negative for many students. Indeed, the mean estimate is negative for both tenth- and eleventh-grade TAKS scores. Moreover, the standard errors of the estimated coefficients of the utility function

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Table 6. Mixed Logit Estimation Results

Tenth Grade Eleventh Grade

Mean travel cost −0.028*** −0.028*** (0.000) (0.000) Mean school quality 0.021*** 0.032*** (0.000) (0.003) Std. dev. travel cost 0.015*** 0.015*** (0.001) (0.001) Std. dev. school quality 0.000 0.000 (0.003) (0.008) Value of school quality (dollars) 74.21 115.38 Number of students 3,744 3,744 Number of decisions 11,232 11,232 AIC 35,131 35,324

Notes: AIC = Akaike’s Information Criterion.Standard errors in parentheses. ***p < 0.01.

Table 7. Summary of Student-Specific Value of School Quality Estimation Results

Tenth Grade Eleventh Grade

Minimum −68,515.59 −188,719.80 Mean −106.78 5,914.88 Median 73.40 104.99 Maximum 20,508.13 19,302,390 Failed to converge 615 816

are, in general, quite large because each utility function is estimated with only three observations. Although the estimates are imprecise because students submit only three rankings, it is worth briefly describing why student-specific estimates of the value of school quality could be useful. The majority of the school choice literature has focused on the incentive-compatibility and fairness attributes of assignment mechanisms, but student- specific estimates of the value of school quality allow for an ex-post assessment of the efficiency of an assignment scheme. In this case, an assessment of allocation efficiency was not possible because student assignment data were unavailable. These values cannot, however, be used as an assignment mechanism, because such a mechanism would not induce students to honestly reveal their preferences. A student could increase her probability of getting her first choice by falsely identifying a distant, high-quality school as her second choice and increasing her perceived value of school quality. It is also worth noting that the estimated value of school quality depends on student location—a student who lives near the best school cannot signal as high of a value of school quality as a student who lives far away. Such an assignment mechanism would doubtless be challenged on the grounds of fairness. Further, although most school improvement schemes are targeted at an entire school or school district, some interventions can impact a single student or a small set of students. In justifying such interventions, it is important to understand the value

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of the school quality improvement to the student who will experience it. This approach allows for this type of student-speciﬁc valuations.

8. COMPARISON TO HEDONIC RESULTS Nguyen-Hoang and Yinger (2011) review hedonic estimates of the value of school quality and find, on average, a one-standard deviation increase in school quality will result in a 4 percent increase in home values. Average home values in Garland, Texas, are approximately $200,000, which would correspond to a value of school quality of approximately $8,000 per standard deviation if the travel cost model gave results equivalent to a 4 percent change in home values. Meanwhile, the travel cost model suggests that a standard deviation of school quality for over four years is worth between $495 and $783. Although the coarse data on student location may be responsible for some of the discrepancy, it is important to note that the value of school quality measured with the travel cost model is not the same as the value of school quality estimated in hedonic models. First, hedonic models capture the ancillary benefits of high-quality schools, such as decreased neighborhood crime. In the travel cost model, however, ancillary benefits of high-quality schools are not included because the characteristics of a student’s neighborhood remain unchanged no matter the student’s school assignment. Further, homeownership conveys not only access to high-quality schools, but also the ability to resell that access in the future. Hedonic estimates can, therefore, be thought of as the value of school quality in perpetuity, and the travel cost model values school quality for a single student. If the value of school quality estimated in this paper were received in perpetuity, each standard deviation of school quality would be worth approximately $4,100 to $6,500, assuming a 3 percent discount rate. This places the estimates derived in the travel cost model closer to hedonic estimates of the value of school quality, although they remain smaller. This differential can be interpreted as the value of ancillary benefits that are associated with living near good schools.

9. CONCLUSION Although a great deal of economic interest has been placed on valuing school quality, the majority of the literature estimates the value of owning a home, which conveys access to a high-quality school. Living near a quality school provides access to a high-quality education, but there are avenues—other than school attendance—through which homeowners may value living near a good school. This paper presents an approach, using the travel cost that households are willing to bear to reach a school, that estimates only the value of school attendance, rather than ancillary benefits associated with quality schools. Moreover, the travel cost approach avoids the threat of unobserved, spatially varying neighborhood attributes that confounds hedonic techniques. Because the value of school quality can be estimated using any desired subsets of students, the travel cost approach is well suited to estimating differences in the value of school quality between students, or across demographic groups although, because of data limitations, this paper did not estimate heterogeneous values of school quality. The resulting estimate of the value of school quality is substantially lower than the values suggested from hedonic models. This difference could, in part, be explained

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by differences in measuring permanent access to quality schools relative to temporary access. Another possible explanation for the difference is the removal of the ancillary benefits from the estimate. This result is robust to several assumptions about student effort in ranking. This suggests that the travel cost approach can be used to estimate the value of school quality any time that students can elect to attend a nonlocal school, regardless of whether multiple, ranked preferences are submitted. The assumptions linking wages to travel cost may also lead to an underestimation of the value of school quality, particularly if adults value their travel time at more than 55 percent of the wage rate or if the opportunity cost of students’ time is nonzero.4 The coarse nature of the student location data may also lead to bias. For example, if students who live near a ZIP code boundary choose to attend a school that is just on the other side of the ZIP code, the value of proximity will be underestimated because those students are treated as leaving from the population-weighted centroid of the ZIP code. This will cause the estimated value of school quality to be overestimated. The estimated value of school quality is robust to the number and type of preferences that are used to estimate the value of school quality. Although there is concern that sorting behavior could bias the estimate of school quality downward, the results are robust to several approaches to control for such sorting behavior. Finally, this paper also provides a method for estimating student-specific values of school quality. Although the estimates are noisy for this dataset, this approach will be- come more accurate if the number of schools that a student is required to rank increases. Further, although characteristics of the individuals were unavailable in this case, this approach lends itself for estimating differential effects of school quality based on observable heterogeneity, such as race and income. Estimating the value of school quality to a particular student is important, both because it can be used to justify student-specific policy interventions and also because it can be used to assess the economic efficiency of an assignment mechanism. It is not possible, however, to devise an economically efficient, incentive-compatible assignment mechanism using the travel cost model, thus the advantage of this approach is limited to assessing the efficiency of other assignment mechanisms. This paper adds a new tool—and a new question—to the debate over the value of school quality. Because public goods like school quality are not openly purchased, there is no price signal associated with school quality. The travel cost, however, places an implicit cost on school quality, allowing the value of good public schools to be assessed as if school quality were a private good.

ACKNOWLEDGMENTS I thank Alexandra Graddy-Reed, Timothy Hamilton, Steve Sexton, and Laura Taylor for helpful comments, and I thank Babetta Hemphill at the Garland Independent School District for help in obtaining the data. I also thank two anonymous referees for their helpful comments. All remaining errors are my own.

4. Inclusion of the opportunity cost of student travel time at 55 percent of the minimum wage increases the implied value of school quality by only a few dollars per unit of school quality.

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APPENDIX A: DATA DESCRIPTION The school choice data come from the Garland Independent School District (GISD), and were provided in response to my request. A unit of observation comprises a unique identiﬁcation number, a ZIP code, and up to three schools representing the students’ ﬁrst, second, and third choices. The ZIP code corresponds to the location of each student’s home. A map of school attendance zones is available from the GISD Web sites. Using ArcGIS, these maps were overlaid onto a map of U.S. Census blocks. The two layers were intersected, providing a list of Census blocks that were eligible for busing to each school. For all block–school combinations that were not eligible for busing, travel distance was calculated using Google Maps. In particular, the centroid of each Census block was calculated in latitude–longitude space, and the fastest driving distance between that point and the school was calculated. Congestion time was ignored. The transportation- eligible block–school combinations were assigned a travel distance of zero. Texas Assessment of Knowledge and Skills (TAKS) data are available from the Texas Education Agency (TEA). TAKS scores are available from the TEA at http://tea.texas.gov /student.assessment/taks/. Although TAKS scores are available at a subject-test level, the results were computed using aggregate TAKS scores for a particular grade–school combination. An aggregated TAKS score corresponds to the percentage of students who have passed all required TAKS exams in a particular grade level at a school.

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APPENDIX B: ALTERNATIVE SPECIFIC CONSTANT REGRESSION RESULTS

Table B.1. Alternative Specific Constant Regression: Stage 1

Travel cost −0.025*** (0.000) Garland 0.287*** (0.015) South Garland −0.692*** (0.026) Naaman Forest 0.328*** (0.017) North Garland 0.032* (0.018) Lakeview Centennial 0.402*** (0.021) Rowlett 0.373*** (0.018) Sasche 0.267*** (0.021) Number of students 4,313 Number of decisions 11,928 AIC 34,733

Notes: AIC = Akaike’s Information Criterion. Standard errors in parentheses. *p < 0.1; ***p < 0.01.

Table B.2. Alternative Specific Constant Regression: Stage 2

Tenth Grade Eleventh Grade

Intercept school quality −1.299 −2.633 (0.898) (3.056) 0.022 0.032 (0.013) (0.035) Value of school quality 88.84 129.73

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