An estimation of technical efficiency for Florida public elementary schools

Transcription

1 Economics of Education Review ] (]]]]) ]]] ]]] An estimation of technical efficiency for Florida public elementary schools Stephen J. Conroy a,, Nestor M. Arguea b,1 a School of Business Administration, University of San Diego, 5998 Alcala Park, San Diego, CA , USA b Marketing and Economics Department, University of West Florida, University Parkway, Pensacola, FL 32514, USA Received 6 September 2005; accepted 30 August 2007 Abstract We use a frontier production function estimation technique to analyze whether elementary schools in Florida are operating at an efficient level and to explain any inefficiencies. A motivation for this analysis comes from recent state and federal level educational initiatives designed to improve school accountability and reduce class sizes. Results presented here indicate that while Florida elementary schools are not operating at efficient levels (with regional mean inefficiency estimates in the % range), they compare favorably to published results for other states. One factor associated with higher inefficiency is student promotion rates something which does lie within the purview of school administrators and may have important policy implications. However, other factors associated with higher inefficiency (percent free-lunch eligible, higher crime and violence, higher suspension rates and not having a parent teacher organization) are indicators of conditions that lie largely beyond the direct control of public schools, casting doubt on the effectiveness of recent accountability measures to improve efficiency. r 2007 Elsevier Ltd. All rights reserved. JEL classification: I21; I28 Keywords: Technical efficiency; Economics of education; Efficiency; Productivity; Frontier production function; Florida elementary schools 1. Introduction Questions about the productivity of the educational system often arise among policymakers and the general public. Recent policy initiatives at both the federal (No Child Left Behind) and state (e.g., the Corresponding author. Tel.: ; fax: addresses: sconroy@sandiego.edu (S.J. Conroy), narguea@uwf.edu (N.M. Arguea). 1 Tel.: ; fax: A-Plus Plan in Florida) level that emphasize accountability of schools are one example of this concern. What is not clear is the extent to which schools are working within their potential, based on their allocated resources. In other words, what is the degree of technical (in)efficiency in the school system, and what can be done in an effort to improve public education? The main objective of this current endeavor is to provide evidence in support of possible alternatives to improve the performance of public schools, in particular those schools that are not satisfying the standards set in Florida /$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi: /j.econedurev

2 2 ARTICLE IN PRESS S.J. Conroy, N.M. Arguea / Economics of Education Review ] (]]]]) ]]] ]]] This study will address the following issues: (1) Are Florida elementary schools operating at a level considered less than fully efficient? (2) What is the extent of the inefficiency? (3) What are the main determinants of technical inefficiency? and (4) What corrective mechanisms are recommended? 2. Background A common thread among recent initiatives at both the state and federal level has been school accountability. For example, the No Child Left Behind (NCLB) initiative at the federal level provides state and school district report cards as a way of providing information to parents and communities about school progress. Just prior to this federal initiative, the state of Florida s A-Plus plan for education included provisions to (set) high standards and provide adequate funding, and then hold schools and educators accountable for the performance of the students they are entrusted to educate (from Plan for Education on At the heart of these initiatives are two assumptions: (a) schools are not operating at their frontier of technical efficiency and (b) policymakers know how to correct any inefficiencies. However, these remain largely assumptions since heretofore only a few published reports have attempted to estimate the inefficiencies and provide direction for correcting them, and none of these was for Florida. We attempt to remedy this dearth of information by estimating the level of inefficiency for elementary schools in Florida the fourth most-populous state in the Union and one which includes a large, ethnically diverse population which may present their own unique challenges. Prior attempts to measure technical efficiency using a stochastic frontier approach include work by Cooper and Cohn (1997), who estimate the mean inefficiency for South Carolina schools ranges from 21.1% to 4.9% and Ruggiero and Vitaliano (1999), who find mean cost inefficiency for New York State schools of 14% and a weak relationship between expenditure and pupil performance. Public school funding can be viewed as a scarce resource which, if not used wisely, can affect funding availability for other programs (e.g., health care, housing, income assistance, etc.). Thus, even if local public school administrators may not be concerned about maximizing efficiency, society as a whole should be. Second, this type of technical efficiency analysis offers specific insights into which factors (e.g., free lunch eligibility, crime rate, promotion rates, etc.) are relevant in reducing inefficiencies. As such, policymakers can assess the efficacy of specific initiatives and programs in terms of their ability to improve efficiency. 3. Productive efficiency Efficiency in production is an empirical concept that involves the comparison of the maximum potential output, given a combination of inputs, with the observed output value. The closer (more distant) the observed output value from the maximum potential output, the higher the level of efficiency (inefficiency). This definition is referred to as technical efficiency, as opposed to economic efficiency where prices of inputs and output are involved. By concentrating on output measures, Debreu and Farrell (Debreu, 1951; Farrell, 1957) suggested an empirical measure of technical efficiency based on current output and the potential output that current input utilization should generate. Our goal in this paper is to concentrate on estimating technical efficiency to provide estimates that allow us to compare schools in terms of a distance measure from a theoretical boundary representing the best a school can do with given resources. A well established methodology called stochastic frontier production models (Aigner, Lovell, & Schmidt, 1977; Meeusen & van den Broeck, 1977), discussed in Section 3.1, will allow us to obtain these estimates. Research by Ruggiero and Vitaliano (1999) found that this theoretical construct provided similar results (0.86 rank order correlation) to the data envelopment analysis approach Stochastic frontier production functions A stochastic frontier production function measures the maximum feasible output. 3 The linear stochastic unobserved frontier, denoted as y *, can be represented as y i ¼ x 0 i b þ v i, (1) where v i is assumed to be N(0, s 2 ) and independent of x i. The model is completed assuming that the 2 For a review of data envelope analysis, see, for example, Lovell (1993). 3 For a different approach using mathematical programming, see, for instance, Lovell (1993).

3 S.J. Conroy, N.M. Arguea / Economics of Education Review ] (]]]]) ]]] ]]] 3 actual level, y, is y i ¼ y i u i, (2) and therefore the frontier function becomes y i ¼ x 0 i b þ v i u i, (3) where now there is an error term that is composed of two elements: a random error (v i ) and a onesided, non-negative error (u i ). The v i are random variates with mean zero, and are assumed to be independent of u i and x 0 i. This error is the typical random term of regression models. The u i are nonnegative random variates, which are independent of v i and x 0 i. The values of u i constitute a measure of technical inefficiency. In this paper, we use a half-normal distribution and we treat v i and u i as iid errors. Ignoring heteroscedasticity in both error terms may lead to bias in the estimates. Given that our focus is on the estimation of inefficiencies and considering that the literature on stochastic frontiers indicates that the size of the bias is small when no heteroscedasticity is included in either error term (the biases tend to cancel out as they work in opposite direction), we proceeded without including heteroscedasticity. If there were heteroscedasticity, then we would expect only small variations in the ranking of schools based on inefficiencies and in the direction of change of inefficiencies in response to the exogenous factors. With the purpose of confirming this conjecture, we estimated a model with the size of the school as a cause for heteroscedasticity, computed the inefficiency term and compared it with the model without heteroscedasticity. The comparison showed a correlation coefficient of 0.95, which resulted in a negligible change in the ranking of schools. 4 Once the parameters are estimated, we obtain estimates of technical inefficiency ( ^u i ) following Jondrow, Lovell, Meterov, and Schmidt (1982) who provided a method for that estimation. We then proceed to estimate a model to explain the level of inefficiency as a function of some exogenous variables, as presented in Section These results are available from the authors upon request. Rather than impose a correction for heteroscedasticity when we do not know the precise form of it and worsen the specification of the model, we decided to accept a small bias that based on the correlation shown above does not present a big problem in the identification of the most inefficient schools. For a more detailed and technical discussion see Kumbhakar and Lovell (2000). 5 The recognition that the level of inefficiency is a function of exogenous variables is evident in recent work by Wang and Schmidt (2003), and previous work by Simar, Lovell, and Vanden 3.2. Data The data on school outcomes, characteristics and resources used in this study were collected from different sources and cover the academic year School characteristics data were obtained from the Florida Department of Education (FDOE, 2000) report on school indicators. Data on race information, pupil teacher ratios and free-lunch eligibility and number of students were obtained from a national data set referred to as the Common Core of Data (CCD) produced by the National Center for Educational Statistics at the US Department of Education. The CCD data set provides information on public schools and school districts in the United States. The output variable is a proxy for school performance approximated by the mean Florida Comprehensive Assessment Test (FCAT) score for the school (as it is standard in studies of this type). Demographic variables and school characteristics are included along with measures related to funding, e.g., expenditures per student and teacher experience. We considered the quality of the student population an important determinant of academic student achievement. For this reason, we included the percentage of students enrolled in programs for the gifted. As a consequence, the number of observations with non-missing values for gifted was reduced from 1434 to 1256 schools in Florida. 4. Results The analysis of inefficiency in the Florida school system using a stochastic frontier model proceeded in two stages. In the first stage, we estimated a stochastic frontier function to obtain inefficiency scores for all the schools in our sample. In the second stage, we estimated a model to explain the behavior of the inefficiency scores. Table 1 provides some basic descriptive statistics on the input variables for the 1256 elementary schools included in the estimations. We included both the mathematics and the reading portion of the (footnote continued) Eeckaut (1994). They suggest incorporating that information into the frontier model and the estimation of the model using either non-linear least squares or maximum likelihood. Several attempts on our part to use maximum likelihood did not succeed because of problematic numerical optimization and the corresponding non-convergence. Instead, we use a two-step more modest approach.

4 4 ARTICLE IN PRESS S.J. Conroy, N.M. Arguea / Economics of Education Review ] (]]]]) ]]] ]]] Table 1 Descriptive statistics Variable Mean S.D. Minimum Maximum N School-level characteristics FCAT scores: mathematics FCAT scores: reading Teachers average years of experience Interaction: per pupil expenditures free lunch eligibility Student-level characteristics Proportion of black students Proportion of students with limited English proficiency Students with disabilities (%) Mobility of students Interaction: mobility of students absenteeism rate Crime-violence: total Ratio of gifted students Rate of suspensions or expulsions Rate of promotion of students Ratio of free-lunch eligible students Parental background Parent Teacher Organization indicator 0, Interaction: PTA or PTO free lunch eligibility Regional dummies Northwest region indicator 0, Northern region indicator 0, Central region indicator 0, Southern region indicator 0, FCAT for our analysis. Their sample means were and , respectively. We have divided the variables into four general areas: school-level characteristics, student-level characteristics, parental background and regional dummies. For schoollevel characteristics, the average number of years of teachers experience was years, ranging from 1.6 to We included an interaction term of expenditure per pupil and free lunch eligibility, which had a mean of Among the student-level characteristics, the mean proportion of Black students was slightly more than one in four (0.27). Approximately 86% of schools had some students with limited English proficiency and 15.25% of students were disabled or in special education. The mean mobility and crime and violence rates were 36.73% and 26.56%, respectively. The mean gifted percentage was 4.44% and the rate of promotion of students 96.08%. The mean free-lunch eligibility percentage was 44.4%. We included two measures of parental involvement. The Parent Teacher Organization (PTO) indicator variable is a dummy variable for whether the school had a parent teacher organization (a school-level organization for parents and teachers, not affiliated with the PTA, though similar in many respects). Approximately 26% of schools had a PTO. We also included an interaction term of PTO or PTA (Parent Teacher Association an official, nationwide organization with local memberships) with free-lunch eligibility. The mean value was Having a PTA or PTO has been demonstrated elsewhere to be associated with higher mean achievement (Arguea & Conroy, 2003). For the regional dummies, approximately 43% of the schools were located in the central region of Florida, with 31% in southern Florida, followed by 15% in the north and 10% in the northwest. Before proceeding with the estimation of the stochastic frontier model we first applied a test suggested by Coelli (1995) with the objective of checking the skewness of the residuals from a least squares regression. The test is based on the third moments of the OLS residuals. Given the nature of the inefficiency component of the residuals (u i ), existence of skewness is consistent with an inefficiency component in the error term. A negative third sample moment will indicate the presence of

5 S.J. Conroy, N.M. Arguea / Economics of Education Review ] (]]]]) ]]] ]]] 5 technical inefficiency. With OLS residuals, the null hypothesis of non-negativity of the third moment was rejected at the 1% level. Even though the main goal in the estimation of stochastic frontier functions is to produce inefficiency scores, we provide a brief account of the main effects observed in Table 2. We estimate two variations of the stochastic frontier model using math scores and reading scores, and as a consequence we estimate two inefficiency scores for each school. The specification follows a Cobb Douglas functional form. Therefore, all variables, except dummy indicators, are measured in natural logs. According to the results presented in Table 2, the two school-level characteristics included in this estimation were significant at the 1% level and had the expected sign. Since math and reading scores were similar (except for limited English proficiency, as noted below), we discuss only the math scores here. Higher levels of teacher experience exert a positive effect on mean student achievement. A 100% increase in average experience of teachers is associated with a 1.7% increase in student achievement. This is similar to some previous findings, though the prior evidence for a positive teacher experience effect is by no means conclusive (see Hanushek, 1986). The effect of expenditures per student is not statistically significant when included by itself (these results are not shown here). However, the interaction of expenditures per student and the ratio of free-lunch eligible students seems to have a negative effect on performance. The free-lunch eligibility ratio exerted a negative influence on achievement when interacted with expenditures and also with the PTO or PTA dummy variable. Free-lunch eligibility ratios are likely to be proxies for lower wealth and education of parents, which has been found to be negatively associated with educational outcomes. As noted in Hanushek (1986, p. 1163), Virtually regardless of how measured, more educated and more wealthy parents have children who perform better on average. In terms of the student-level characteristics, percent Black and student mobility both exerted a negative influence on achievement. The Black effect may be a proxy for socioeconomic status (SES) effects not controlled for in the freelunch eligibility variable. Indeed, the effect is not large, with a 100% increase in the proportion of Blacks associated with about a 0.6% decline in test scores. These results are not surprising given previous findings linking poverty and educational attainment (Mayer, 1997), but have a very important policy implication discussed below. The mobility effect is larger, with a 100% increase in student mobility associated with a 2.6% reduction in achievement. Prior investigations have found that student mobility disrupts learning (Alexander, Entwisle, & Dauber, 1996; Kerbow, 1996; Rumberger, 2003) and imposes a negative externality on the school Table 2 Maximum likelihood regressions Variable Log math scores Log reading scores Estimate z-ratio Estimate z-ratio Constant *** *** Teachers average years of experience *** *** 5.87 Interaction: per pupil expenditures free lunch eligibility *** *** 7.73 Proportion of black students *** *** 9.37 Proportion of students with limited English proficiency *** 3.53 Mobility of students *** *** 8.49 Ratio of gifted students *** *** Interaction: PTA or PTO free lunch eligibility *** *** Southern region indicator 0, *** *** Variance parameters for compound error (standard errors in parentheses) ^l *** (0.0040) *** (0.0047) ^s *** (0.0002) *** (0.0003) Log likelihood ^s v *** (0.0015) *** (0.0018) ^s u *** (0.0027) *** (0.0031) Note: ***, **, *, denote significance at the 0.01, 0.05 and 0.10 levels, respectively. Logs used for all continuous variables.

6 6 ARTICLE IN PRESS S.J. Conroy, N.M. Arguea / Economics of Education Review ] (]]]]) ]]] ]]] (Hanushek, Kain, & Rivkin, 2004; Rumberger, Larson, Ream, & Palardy, 1999). A 100% increase in the percent of gifted students was associated with a 1.1% increase in achievement. This variable likely serves as a proxy for the proportion of students with higher innate ability (see Cooper & Cohn (1997) for a discussion), who would require less formal academic instruction in order to produce the same level of achievement. Schools located in the southern part of the state are associated with about a 2.1% lower achievement. Since South Florida schools are more likely to be in larger, urban and more ethnically diverse areas than the rest of the state, perhaps there are other big city factors at work that are not being controlled for in these models that are associated with lower scores. The limited English proficiency result differs for the math and reading achievement estimations. While the effect is not significant at even the 10% level for math, it is negative and significant at the 1% level for reading. We suspect that limited English proficiency translates into negative reading effects more readily than in mathematics, since reading ability requires a more thorough command of the English language. 5. The inefficiency component of the disturbance In addition to the skewness test presented in Section 4 that justifies the estimation of a stochastic frontier model, we also confirmed the existence of technical inefficiency by performing a likelihood ratio test on s 2 u, by rejecting the null hypothesis of no technical inefficiency in both empirical models for math and reading as shown in Table 2. In the estimation of the inefficiency component of the disturbance term, we follow the procedure suggested by Jondrow et al. (1982) (see also Greene (1993)). Two common assumptions about the distribution of u i include the half-normal and exponential distribution. We use the most common, half-normal distribution. 6 The conditional expectation of the inefficiency component has the following form: E½u i j i Š¼ sl fð i l=sþ ð1 þ l 2 Þ Fð i l=sþ il, (4) s 6 The estimates of the stochastic frontier production function and the inefficiency component were obtained with the software Stata/SE 9.1. where f(.) is the density of the standard normal distribution (PDF), F(.) is the standard normal cumulative distribution function (CDF), s ¼ ðs 2 u þ s2 v Þ1=2, and l ¼ s u /s v. The parameter l is a different parameterization of the inefficiency component of the half-normally distributed inefficiency term u i. It measures the relative contribution of s u and s v in the composite error term in Eq. (3). In our estimations ^l is greater than one and it is statistically significant for both the math and reading frontier models indicating a dominance of the one-sided error component u i, and therefore demonstrating the existence of inefficiencies in the Florida public elementary school system. In Section 5.1 we try to measure the extent of inefficiency A model for the inefficiency component of the disturbance In Table 3 we present the stratified average inefficiencies by region within the state. We have labeled the inefficiency scores for the math and reading production functions as u m and u r, respectively. The mean inefficiencies for math vary from for the central region to in the southern region of Florida. In other words, the central region has the lowest average level of inefficiency at 4.1%, while the south has the highest average inefficiency at 4.5% (though the difference is not statistically significant). In terms of reading, the lowest mean inefficiency score occurred in the northern region (4.1%), while the highest inefficiency was in the central Florida region (5.1%) (and the difference in this case is statistically significant). Following Greene (1997), 7 we have attempted to explain the behavior of the inefficiency measures by providing estimates of a model that include those measures as a function of some of the variables that we believe might be important in explaining those variations. Due to the truncation of the inefficiency measures (u i X0) we present estimates of a Tobit model for each of the two measures (Table 4). The explanatory variables, except for crime-violence: total, are measured in their original units. Positive estimates indicate that the inefficiency increases with that particular variable. Negative terms indicate 7 After the estimation of parameters, and the computation of the inefficiency measures, this seems to be as Greene calls it a natural third step in the analysis of the stochastic frontier model y See Greene (1997, p. 109).

7 S.J. Conroy, N.M. Arguea / Economics of Education Review ] (]]]]) ]]] ]]] 7 Table 3 Average inefficiencies for mean math and reading scores by region Region Type N Mean S.D. Minimum Maximum South Florida ^u m ^u r Central Florida ^u m ^u r North Florida ^u m ^u r Northwest Florida ^u m ^u r Total ^u m ^u r Table 4 Tobit maximum likelihood regressions for mean math and reading scores Variable Inefficiency for math scores (^u m ) Inefficiency for reading scores (^u r ) Estimate t-ratio Estimate t-ratio Constant *** 2.67 Crime-violence: total ** Students with disabilities (%) * Rate of suspensions or expulsions *** *** 4.17 Rate of promotion of students *** *** 3.95 Ratio of free-lunch eligible students *** *** 8.10 Parent Teacher Organization indicator 0, ** 2.36 Southern region indicator 0, ^s *** *** Note: ***, **, *, denote significance at the 0.01, 0.05 and 0.10 levels, respectively. Log used for the crime and violence variable. that inefficiency is negatively (or efficiency is positively) correlated with those factors. In both Tobit estimations (for math and reading scores), the coefficients for rate of suspensions or expulsions, rate of promotion of students and freelunch eligibility ratio are positive and significant, suggesting that they contribute to inefficiencies. There are two ways to interpret this result. First, from a policy perspective, one might expect that higher suspension and expulsion rates would improve the learning environment for students (at least for those not directly involved in the suspensions and expulsions). However, a more likely interpretation is that higher suspension and expulsion rates is a proxy for schools that have high proportion of students with discipline problems, resulting in more chaotic and disruptive learning environments. Higher student promotion rates are also associated with higher inefficiencies. One explanation could be that schools which promote students to higher grades at higher rates (instead of holding back the lowest-performing students) may create inefficiencies in learning in the form of negative peer effects. Teachers must then spend more time in remedial learning, instead of teaching at grade level. Importantly, this is one effect that could be the result of administrative decisions and, as such, within the control of school administrators. As noted above, higher proportions of free-lunch eligible students proxies for a lower average SES of students, which suggests that these students have fewer resources (including parental education) from which to draw. Thus, schools must devote more time to teaching things that may be learned outside of the classroom by students of higher SES. Crime and disability are both positive and significant, though only for the math score and disability is marginally significant. It appears that there may be

8 8 ARTICLE IN PRESS S.J. Conroy, N.M. Arguea / Economics of Education Review ] (]]]]) ]]] ]]] some inefficiencies (e.g., through diversion of resources from learning to discipline or disability students) associated with higher crime and violence and disability, though it is unclear why this effect does not exist for the reading estimations. Similarly, the coefficient for the PTO dummy variable was negative and significant, though only for the reading estimation. Perhaps, parent teacher organizations assist schools directly with learning (e.g., by providing parent volunteers, funding for programs, etc.), thereby reducing inefficiencies. On the other hand, the existence of PTOs may be an indicator for greater parental interest and involvement in the home (see Arguea & Conroy (2003) for a discussion). To compare our results with State of Florida assessments (i.e., school grades ), we compiled a list of schools for a specific county (Escambia) to test whether the lowest-performing schools (those with scores of D or F ) were at the bottom (i.e., with the highest inefficiency estimates). Indeed, the only two schools in Escambia County to receive failing grades for two consecutive years, Spencer Bibbs and A.A. Dixon, were in the bottom grouping (i.e., beyond two standard deviations from the mean) for math, and at the very bottom of our list for reading, with inefficiency estimates of 15.3% and 16.4%, respectively (results available from authors upon request). At the same time, A.K. Suter and Pensacola Beach, two perennial A schools, ended up at the top of our list for math and reading, respectively. 6. Discussion This paper used fourth and fifth grade FCAT scores among public school students in Florida to test whether Florida schools were operating at efficient levels and, if not, to estimate the extent of inefficiency. We used a standard stochastic production frontier model introduced by Aigner et al. (1977) that measured the level of technical efficiency for school production, i.e., learning, as measured by mean student achievement. Motivation for these estimations came about as a result of recent state and federal initiatives including school accountability measures and the class-size reduction amendment in Florida that have assumed that (a) some schools are inefficient and (b) administrators know how to correct these inefficiencies. Results presented here are suggestive that (a) schools were not operating at efficient levels and (b) the mean levels of inefficiency varied by region from a mean of 4.1% for reading in North Florida and math in Central Florida, to 5.1% in Central Florida for reading, with South Florida close behind (5.0%). Since the peninsular region of Florida has a much larger immigrant population, perhaps the higher level of inefficiency for reading achievement is due to increased resources that must be spent on teaching English to non-native speakers, and the direct relationship between English language proficiency and reading ability. This would also explain why these regions were more efficient in math relative to reading. While much of this effect should have been captured in the limited English proficiency control variable, this is an imperfect measure. Overall, these levels of inefficiency compare favorably to those reported for other states (e.g., 14% in New York and % in South Carolina), suggesting that on average Florida schools are performing more efficiently. However, there is still room for improvement and it is important to note that the intra-regional range included some schools with as high as 23.7% inefficiency for reading and 19.0% for math (both in Central Florida). The Tobit estimations presented in Table 4 indicated that several inputs were significant predictors of inefficiencies. For example, higher free lunch eligibility ratios, suspensions/expulsions and student promotion rates were associated with increased inefficiency. An indicator for parent teacher organizations (though only significant for reading) was associated with reduced inefficiency. With one exception (i.e., promotion rates of students), what is perhaps most noteworthy and potentially vexing to current incentive-based, accountability policies is that educational efficiency appears to be moderated largely by socioeconomic and parental factors which lie largely beyond the control of local public schools. An interesting policy implication is that spending on school-based initiatives (e.g., class size reduction amendments) may channel scarce resources away from other areas that could be more effective (e.g., poverty reduction, higher education, etc.). However, we urge caution in making specific policy recommendations from these results for the following reasons. First, we have not been able to control for every potential input to the educational process. We would like to have more robust parental involvement variables such as time spent with children on homework. Second, our outcome

9 S.J. Conroy, N.M. Arguea / Economics of Education Review ] (]]]]) ]]] ]]] 9 measure for technical efficiency (test scores) does not include other potentially important outcomes. While using test scores as a proxy for output is standard practice in this area of research (see, for example, Cooper & Cohn, 1997; Ruggiero & Vitaliano, 1999), we acknowledge that it is merely a proxy for student learning more broadly defined. Other research endeavors into this area may wish to consider including other important student outcomes such as enhanced creativity and imagination, leadership skills and character. Further, this current analysis has been limited to (a) public elementary schools within the state of Florida and (b) students in fourth and fifth grades. Future research endeavors should attempt to address these matters. References Aigner, D. J., Lovell, C. A. K., & Schmidt, P. (1977). Formulation and estimation of stochastic frontier production function models. Journal of Econometrics, 6, Alexander, K. L., Entwisle, D. R., & Dauber, S. L. (1996). Children in motion: School transfers and elementary school performance. The Journal of Educational Research, 90(1), Arguea, N. M., & Conroy, S. J. (2003). The effect of parent involvement in parent teacher groups on student achievement. School Community Journal, 13(2), Coelli, T. J. (1995). Estimators and hypothesis test for a stochastic frontier function: A Monte Carlo analysis. Journal of Productivity Analysis, 6(4), Cooper, S. T., & Cohn, E. (1997). Estimation of a frontier production function for the South Carolina educational process. Economics of Education Review, 16(3), Debreu, G. (1951). The coefficient of resource utilization. Econometrica, 19(3), Farrell, M. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society, Series A, General, 120(Part 3), Florida Department of Education (2000) Funding for Florida school districts. Statistical Report, EIAS Series , September. Greene, W. H. (1993). The econometric approach to efficiency analysis. In H. O. Fried, C. A. K. Lovell, & S. S. Schmidt (Eds.), The measurement of productive efficiency (pp ). New York: Oxford University Press. Greene, W. H. (1997). Frontier production functions. In M. Hashem Pesaran, & P. Schmidt (Eds.), Handbook of applied econometrics: Microeconomics, Vol. II. Massachusetts: Blackwell Publishers. Hanushek, E. A. (1986). The economics of schooling: Production and efficiency in public schools. Journal of Economic Literature, 24(September), Hanushek, E. A., Kain, J. J., & Rivkin, S. G. (2004). Disruption versus Tiebout improvement: The costs and benefits of switching schools. Journal of Public Economics, 88, Jondrow, J., Lovell, C. A. K., Meterov, I. S., & Schmidt, P. (1982). On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics, 19, Kerbow, D. (1996). Patterns of urban student mobility and local school reform. Journal of Education for Students Placed at Risk, 1(2), Kumbhakar, S. C., & Lovell, C. A. K. (2000). Stochastic frontier analysis. New York: Cambridge University Press. Lovell, C. A. K. (1993). Production frontiers and productive efficiency. In H. O. Fried, C. A. K. Lovell, & S. S. Schmidt (Eds.), The measurement of productive efficiency (pp. 3 67). New York: Oxford University Press. Mayer, S. E. (1997). What money can t buy. Cambridge, MA: Harvard University Press. Meeusen, W., & van den Broeck, J. (1977). Efficiency estimation from Cobb Douglas production functions with composed error. International Economic Review, 18, Ruggiero, J., & Vitaliano, D. F. (1999). Assessing the efficiency of public schools using data envelopment analysis and frontier regression. Contemporary Economic Policy, 17(3), Rumberger, R. W. (2003). The causes and consequences of student mobility. Journal of Negro Education, 72(1), Rumberger, R. W., Larson, K. A., Ream, R. K., & Palardy, G. J. (1999). The educational consequences of mobility for California students and schools. Berkeley, CA: Policy Analysis for California Education. Simar, L., Lovell, C. A. K., & Vanden Eeckaut, P. (1994). Stochastic frontiers incorporating exogenous influences on efficiency. Discussion Paper No Institut de Statistique, Universite Catholique de Louvain. Wang, H., & Schmidt, P. (2003). One-step and two-step estimation of the effects of exogenous variables on technical efficiency levels. Journal of Productivity Analysis, 18,