School competition and efficiency in elementary schools in Mexico

International Journal of Educational Development 46 (2016) 23–34

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International Journal of Educational Development journal homepage: www.elsevier.com/locate/ijedudev

School competition and efficiency in elementary schools in Mexico Rocio Garcia-Diaz a,*, Ernesto del Castillo b, Rene´ Cabral c a

Department of Economics, Tecnolo´gico de Monterrey, Campus Monterrey Av. Garza Sada 2501, Monterrey, NL, Mexico, C.P. 64849 EGAP Government and Public Policy, Tecnolo´gico de Monterrey, Campus Monterrey Ave. Rufino Tamayo, Garza Garcı´a, NL, Mexico, C.P. 66269 c EGADE Business School, Tecnolo´gico de Monterrey, Campus Monterrey Eugenio Garza Laguera & Rufino Tamayo, Garza Garcı´a, NL, Mexico, C.P. 66269 b

A R T I C L E I N F O

A B S T R A C T

Article history: Received 7 May 2015 Accepted 29 September 2015

This paper examines how competition impacts the technical efficiency of schools. We model competition between schools using Geographical Information System (GIS) tools in order to develop a Herfindahl– Hirschman market concentration index (HHI) and then follow a stochastic frontier analysis with alternative specifications that enable us to obtain the best unbiased efficient estimators. We find three important results. First, a higher degree of competition from public and private schools significantly increases elementary school efficiency in Mexico, as measured by the outcomes in a nationwide standardized test. Second, we find a positive, though small, association between expenditure on education and school outputs. Third, private schools perform significantly better due to the differential incentives they face in terms of competition. ß 2015 Elsevier Ltd. All rights reserved.

JEL classification: I21 I22 D24 R12 Keywords: Competition Stochastic production frontier Efficiency Spatial analysis Panel data

1. Introduction The improvement of educational outcomes is an important part of the governments’ agenda in developing countries. Mexico increased its education expenditure by 24% during 2000–2010, which represents an additional 1.2% of the GDP. This expenditure represents, during that period, the highest increment in educational expenditure among OECD members. Despite these expenditure efforts, the outcomes in education, as measured by the Program for International Student Assessment (PISA) in mathematics, science and reading, are almost unchanged and statistically significant below the OECD average. The meager results achieved by Mexican students in the PISA test suggest two possibilities. The first is that expenditure on education does not have a significant impact on school achievement. The second is that, even if additional expenditure on education has a positive impact on education outcomes, the inefficient use of resources hinders any possible improvements in terms of school achievements.

* Corresponding author. Tel.: +52 81 8358 2000x4305; fax: +52 81 8358 2000x4351. E-mail addresses: [email protected] (R. Garcia-Diaz), [email protected] (E. del Castillo), [email protected] (R. Cabral). http://dx.doi.org/10.1016/j.ijedudev.2015.09.015 0738-0593/ß 2015 Elsevier Ltd. All rights reserved.

In addition to a more careful treatment of technical efficient production, in which inputs are associated to the best possible outcome, market environment variables such as competition play an important role in explaining production outcomes. This relationship is not clear in the case of education markets, since education is considered to be a public good. Recent advances in analyzing policies aimed at enhancing parental choice to improve competition have found mixed results. For instance, Dee (1998) found that competition from private schools does have a positive and statistically significant impact on high school educational outcomes. Rouse and Barrow (2008), reviewing the empirical evidence on the impact of education voucher programs, observe a positive association between competition and educational outcomes, but find small achievement gains for students offered education vouchers. Likewise, the evidence presented by Greene and Kang (2004) finds significantly positive effects of private competition for some school outputs (such as dropout rates and standardized tests), but little, if any, effect on measures such as the percentage of students receiving high quality diplomas in New York high schools. Also, Gibbons et al. (2008) distinguish between the effects of competition on achievements and school performance. Using data from elementary schools in England they find little evidence of a link between choice and achievement, but observe a positive association between competition and school

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R. Garcia-Diaz et al. / International Journal of Educational Development 46 (2016) 23–34

performance. Indeed, inefficiency arises from a lack of incentives in schools to behave efficiently and competition can be an incentive mechanism to promote a better use of resources in schools (Grosskopf et al., 2001). It seems that in terms of competition, schools should not be evaluated by their outcomes, but by the efficient use of their inputs. This paper contributes to the debate on competition and school efficiency. We use data from the Mexican educational system to test the relationship between urban school market concentration (lack of competition) and school technical efficiency (the percentage that represents observed school production out of the maximum school production, given a set of inputs). The estimation process is not straightforward since neither efficiency nor competition is directly observable. In order to measure school efficiency we use the stochastic frontier analysis method (SFA) which allows us to calculate technical efficiency, controlling for the inputs and exogenous factors in which schools operate. Given the lack of consensus on the SFA estimation process, we use a series of estimators following different econometric specifications for panel data, including both fixed and random effects. In addition, we follow Hoxby (2000) to measure competition in a school market by using the Herfindahl–Hirschman market concentration index (HHI) as a proxy of the competition level every school market faces. The index we estimate employs the application of Geographical Information System (GIS) tools to incorporate the degree of competition each school faces from peer schools spatially located within a radius of 1 km in their local market area. The development of the literature on production outcomes and efficiency is scattered among several approaches which are restricted to specific interests. This analysis follows two strands in the literature. One, related to the estimation procedure using panel data to isolate the effect of competition on the efficiency of public school districts (Millimet and Collier, 2008; Kirjavainen, 2012). The other analyzes the effect of competition on public schools’ efficiency following the stochastic frontier analysis and using the geographic information system approach to measure the degree of competition (Misra et al., 2012). We apply the stochastic frontier analysis, but use a single equation estimation procedure in order to avoid the potential biases a two-stage estimation procedure may cause. Our panel consists of 27,068 schools which represent 63% of urban elementary schools in Mexico during the 2009–2011 academic periods 2009–2011. We focus our analysis in urban elementary education markets since they are less concentrated than rural elementary education markets, where in many localities there is public education monopoly. On the other hand, there is low quality of data for rural elementary education associated to an under-representation of indigenous schools due to a very low participation in public evaluation programs. The panel covers both private and public schools; at national level private schools represent 8.68% of the elementary school market. The private school ratio varies considerably across the country, for instance, private schools are mostly concentrated in urban areas where the ratio average is 19.1% while in some states, such as Aguascalientes, can be as high as 36%. The data was obtained from Mexican Ministry of Education through the national system of statistic information on education (SNIEE, for its Spanish acronym). The educational output we use is a standardized performance test known as ENLACE, which is a partial measure of an education process but is the best available proxy that enables us to achieve objective, transparent comparisons. We found three main results in our analysis. The first consists of showing that the lack of competition is positively and significantly related to schools’ technical inefficiency. We found that school markets that are more competitive are more efficient in their use of school inputs, for both public and private schools. These results go in tandem with previous findings observed by Grosskopf et al.

(2001) and Millimet and Collier (2008) for other countries. The results suggest that promoting competition between schools, either public or private, can improve schools’ technical efficiency, which may translate into better educational outcomes. The second result is that public expenditure per student has a small and positive impact on the ENLACE test in Spanish and mathematics. This result suggests that increasing expenditure on education per student would increase the outcomes in the ENLACE test, ceteris paribus. However, the increment in expenditure would have to be very high for the improvement in outcomes to become evident. This result is similar to Barrow and Rouse’s (2005) findings which also conclude that expenditure on education is significant yet higher in the case of public schools. The third result shows that the most relevant factor to explain the outcomes in the ENLACE tests is the type of school (public or private) considered. On average, private schools obtain better results in the ENLACE test outcomes than their public counterparts. Other studies using Mexican data have found similar results (Dı´az Gutie´rrez and Flores Va´zquez, 2010; Blanco, 2011). It is highly evident that the incentive structure private schools face is a key determinant of their efficient performance. The rest of paper is structured as follows. Section 2 briefly discusses the theoretical basis of frontier efficiency and the measurement of school competition. Section 3 introduces the data and the empirical estimation strategy. The results of the estimations appear in Section 4. Finally section 5 concludes the article. 2. Competition in elementary schools in Mexico School competition has been widely studied in the literature. It is a complex issue and the evidence found in empirical estimations is sometimes contradictory. In this study we focus our analysis on 42,552 Mexican urban elementary schools, 82.6% of which are public and 17.43% private. Moreover, we stress two issues regarding methodologies and implementations of the competition concept: the delimitation of a school market and the way of measuring competition within a school market. We employ three factors for the delimitation of the school market: the geographic factor, the school type (public or private) and the school shift. The geographic factor delimits the school markets through specific geographic zones: regions, counties, school districts, etc. The idea is that parents look for a place to live in relation to the places where the best schools for their children are located and that schools are concentrated in specific school districts. Several authors have delimited the school market according to geographic zones and county sizes (Barrow and Rouse, 2005; Millimet and Collier, 2008), while others have used the school district markets in metropolitan areas (Hoxby, 2000; Greene and Kang, 2004). Nevertheless, Misra et al. (2012) pointed out that geographic delimitation is subject to an aggregation bias given that sometimes the real competition is not captured. In order to avoid this, they proposed a novel method to delimit the market influence zone of every school, using geographic information system (GIS) tools to create a school competition index for the state of Mississippi. In order to define different school market sizes, the authors drew circles, such as a 5, 15 and 25 mile radius, around each school. The availability of a geo-referenced database of Mexican elementary schools allows us to use the GIS tools proposed by Misra et al. (2012), with the differences that result from the educational market structure in Mexico. Given the geographic concentration, we draw a circle of a 1 km radius around each public or private school. This choice may seem arbitrary, yet it is supported by several sensitivity tests carried out using GIS tools.

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Fig. 1. School market delimitation.

The nearest neighbor statistical test revealed that the expected median distance between close urban schools is 0.64 km considering the entire Mexican territory. From which, we consider delimiting a 1 km radius to distinguish educational markets a reasonable compromise. The second factor that delimits the school market is the types of producers that participate in the market. There is substantial evidence that distinguishes two types of educational markets: public and private schools. In our analysis, we follow the same distinction which implies that public schools compete with other public schools and private schools compete with other private schools. It is clear that private schools have economic incentives to compete against other private schools in order to attract and retain students. This situation is different from public schools which lack from this incentive, because public funding is usually independent of the number of students schools attract. In addition, public schools have such a high occupancy that they do not need to look for or attract new students. Under those circumstances, the hypothesis that public schools compete against other public schools seems to have no support in theory. Nevertheless, we follow different specifications of the model to test whether this hypothesis is sustainable in the empirical application. The third factor that delimits a scholar market corresponds to shifts1. Delimitation is based on the assumption that parents choose their children’s shift (morning or afternoon) according to their needs. Therefore, we suppose that schools compete against other schools in the same shift. Given that nearly all private schools only have a morning shift, we recognize that this assumption may not be valid for private schools. Public schools’ decision to offer an afternoon shift is not necessarily driven by market structure per se, but rather by the need to use the school infrastructure to cover the demand for education services. Children who do not obtain a place in the morning shift are usually placed in the afternoon shift in public schools. Hence, the distinction between the morning and afternoon shifts is only relevant for the case of public schools.

Considering the above-mentioned factors, we classify three types of school markets: markets without competition (market 0); markets with different types of competitors (market 1); and markets with the same type of competitors (market 2). Fig. 1 summarizes all the criteria in the delimitation of a market from which we have defined three types of school markets. We now turn to the way in which we measure competition in a school market. The degree of competition a producer faces in a given market is a non-directly observable characteristic, thus neither is directly quantifiable. Under these circumstances, there are some proxies of the degrees of competition schools face in a given market. One commonly used is enrollment size at private schools as a proxy to school competition (Couch et al., 1993; Geller et al., 2006). Marlow (1997), for instance, uses the number of schools for every 1000 students in a school district. Misra et al. (2012) proposes a gravity-based index. Nevertheless, the indicator most used in the literature, and the one followed in this paper, is the Herfindahl–Hirschman (HHI) index. The HHI is a concentration measure that assesses the size of the producers in relation to the size of the market. The index is defined as the sum of the squared market shares of all the schools in the market. The HHI index is given by, HHI ¼

n X ðsi Þ2

(1)

i¼1

where si is the market share of school i. Since the index may take values between 0 and 10,000, we use the normalized version of the index which is bounded between 0 and 1. Higher (lower) values of the index imply a higher (lower) market concentration and a lower (higher) degree of competition. Misra et al. (2012) pointed out some of the limitations of the index: it does not consider the distance between competitors and it also overlooks the size of the competitors in relation to the installed capacity. 3. The model and the estimation strategies

1 Most schools in Mexico have two shifts or ‘‘turnos". Students might attend the morning shift or the afternoon shift depending on parents’ preference and the schools’ place availability.

We depart from a model in which schools i = 1, . . ., n are maximizing their production of output in a period t using N inputs to produce a single output. A stochastic production function model

R. Garcia-Diaz et al. / International Journal of Educational Development 46 (2016) 23–34

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can be written as: yit ¼ f ðxit ; bÞTEit

(2)

where yit is the output of school i in period t, 0 < TEit ðyit ; xit Þ < 1 is the technical efficiency component, xit is a vector of inputs which may be time variant. The production model will usually be linear in the logs of variables, hence: lnðyit Þ ¼ lnf ðxi ; bÞ þ lnðTEi Þ

(3)

Now assuming that f(xit,b) takes the usual log-linear Cobb– Douglas production function, the empirical specification is the following: 0

yit ¼ a þ b xit m0 zi þ vit þ uit

i ¼ 1; . . .; N;

t ¼ 1; . . .; T

(4)

where a is the intercept in period t, b0 xit is an input function (defined in a vector x) which is time variant. The term m0 zi represents the observed heterogeneity (defined in a vector zi) that is not related to production structure (inputs), but that captures specific effects of each producer. The term vit represents the idiosyncratic error term of each producer in each period. The term, uit represents technical inefficiency, which must be positive since it is limited by the optimal production frontier. The compound error term according to Aigner et al. (1977) is defined as the sum of the idiosyncratic error in period t (vit) and the absolute value of the technical inefficiency in period t (uit),

eit ¼ vit uit ¼ yit b0 xit m0 zi

(5)   where both terms follow the assumptions vit ¼ N 0; s 2v and   uit ¼ jU it j where U it  N 0; s 2 . That is, assuming both the idiosyncratic error in period t (vit) and the technical inefficiency in period t (uit) follow a normal distribution2. The technical inefficiency in period t, uit, is a non-observable characteristic and cannot be measured directly. In order to overcome this limitation, Jondrow et al. (1982) proposed a conditional technical inefficiency (uit ) under normal distribution ˆ assumption. The technical efficiency (TE) of each unit they use can be calculated as: TEjlms ¼ expðEðsuit jeÞÞ

(6)

From this equation it is possible to assess technical efficiency (TE) for every unit i at time t. The TE estimator can take values between 0 and 1, where 0 corresponds to a completely inefficient unit and 1 corresponds to a completely efficient unit. That is, if TE = 0.80 means that, while having the same inputs and resources, a given producer manages to produce 80% of what a fully efficient producer produces. Several estimation techniques for stochastic frontier analysis are employed. The series of tests conducted for all the models can be found in Appendix 1. We depart from the simplest methodology, ordinary least squares (OLS), in order to estimate the parameter of the production function. We ran a series of test for the OLS where we found problems of heteroscedasticity that traditionally are not thought to be serious because even when it is present, the estimators are unbiased and consistent (yet inefficient). Although this may be true, the presence of heteroscedasticity in the context of stochastic frontier analysis can be critical because of the bias it can cause in the technical efficiency estimators. 2 Other authors have proposed alternative distributions that allow more flexible estimations of the technical efficiency. Stevenson (1980) developed a stochastic frontier model with truncated normal distribution, while Green (1980) proposed the use of gamma and exponential distributions. Nevertheless, in empirical applications of the stochastic frontier models the normal distributions are still the most employed (Kumbhakar and Lovell, 2000).

Recent literature provides evidence that the use of panel data improves stochastic frontier estimation when compared to crosssectional data (Schmidt and Sickles, 1984; Kumbhakar and Lovell, 2000; Greene, 2005a). The use of panel data also allows the relaxation of assumptions about the inefficiency term distribution, thus avoiding the assumption of no correlation between effects and regressors. This offers the possibility of using several estimation methods depending on the assumptions made with respect to the distribution of the inefficiency term. Panel data models with random effects (RE) and fixed effects (FE) allow us to reduce the bias that characterizes OLS estimators resulting from the individual heterogeneity of producers. We performed a series of tests for the RE and FE estimators and found that both estimators are more appropriate than LS. There is no evidence of serial autocorrelation, although we found persistence in the heteroscedasticity in the estimation3. Regarding the selection of the appropriate model, the Hausman test suggests that the FE model is preferable to the RE model. This result is unexpected since it is well known that the FE model specification causes a bias due to the proliferation of parameters in the model. In the context of stochastic frontier analysis with panel data, efficiency is important and RE estimators are more efficient than FE estimators (Kennedy, 2008). Greene (2008) also argues in favor of the RE model given that the omission of time-invariant variables in the FE model would later appear masked in the form of inefficiency, which would create ambiguity in the interpretation of frontier coefficients. Given these considerations, we employ three different specifications of the stochastic frontier model following a RE panel data specification: Time-invariant stochastic frontier, time-variant stochastic frontier and time-variant stochastic frontier with exogenous factors. All the models presented here require assumptions about the distribution of the inefficiency term (ui) defined in (5). Nevertheless, panel data utilization enables us to relax the assumption about the distribution of ui and in this paper we follow a truncated-normal distribution because it gives us a flexible representation of the efficiency pattern in the data (Kumbhakar and Lovell, 2000, p. 83). The time-invariant stochastic frontier by Battese and Coelli (1988), referred to as BC88, was the original theory of stochastic frontier production, which assumed inefficiency as an intrinsic characteristic of each producer. That is, it assumed that the technical inefficiency was time-invariant. Battese and Coelli (1988) recognize that the time-invariant assumption behind this model cannot be accomplished in the long term, for two reasons. The first reason is that competitive markets eventually punished inefficient producers, forcing them to improve their efficiency or leave the market. The second reason refers to the adaptation process a producer faces in the long run. That is, when producers notice they are inefficient, they are expected to re-organize their inputs to become more efficient in the long run. This adaptation process makes it difficult to sustain the time-invariant assumption. Nevertheless, the assumption is not particularly problematic in the case of short panel data like ours (T = 3). Battese and Coelli (1995) proposed a method that allows us to estimate producers’ inefficiency variation in a period of time (we call this method, BC95). The major advantage of this innovation is that with sufficiently long panel data, the BC95 estimator would allow us to follow producers’ performance during a given period of time, in terms of the relative efficiency of each producer in the industry. The inefficiency estimators are more robust, which is 3 The Breusch-Pagan test is based on the Lagrange multiplier. The test consists of contrasting residuals from LS against those obtained from RE or FE. The null hypothesis is that there are no random effects, which implies that LS estimators are appropriate to estimate the model.

R. Garcia-Diaz et al. / International Journal of Educational Development 46 (2016) 23–34

highly relevant for the consistent estimation of the technical efficiency of each producer. Greene (2008) finds that with this specification technical efficiency estimators are robust to the choice of alternative error distributions, contrary to the case of time-invariant assumptions where the estimators are very sensitive to changes in the error distribution. Our last stochastic frontier specification consists of incorporating exogenous factors that explain the variation within the technical inefficiency term. It is important to make a distinction in the type of exogenous factors that impact the technical inefficiency term. For instance, exogenous factors such as location, competition, weather, etc., play a role in the producer’s efficiency, but should not be considered as inputs. Given that these factors are not inputs, they cannot be incorporated directly into the stochastic frontier production function. The true random effects (TRE) model proposed by Greene (2005b) simultaneously solves three issues related to the specification of stochastic frontier models. First, because it considers technical efficiency to be a time variant, it increases the inefficiency estimation strength. Second, it enables us to model the heteroscedasticity in the term ui which generates unbiased estimators of the individual technical efficiency of each producer. Third, it allows us to incorporate exogenous factors in a single stage, hence avoiding the model bias of a two-stage process. It is important to note that the latter advantage of TRE models turned out to be crucial for our estimation purposes, since we were able to incorporate competition as an exogenous factor that determines the inefficiency terms of each school. Given that there is no precise definition about where to locate the exogenous variables when correctly modeling individual heterogeneity, Greene (2004) suggests distinguishing the exogenous variables in two types. The first type comprises the exogenous variables related to industry-specific characteristics (such as competition, technology, etc.) and should be associated with the technical efficiency term (ui). The second type are the exogenous variables that characterize the producer’s environment, on which producers have no decision power (e.g., weather, geography, etc.) and that should be associated with the traditional error term (vi). This formulation makes it possible to model with greater precision the heteroscedasticity found in the LS model and that persists through RE and FE panel data estimations. 4. Data The panel data we use for this paper was obtained from the Mexican Ministry of Public Education. It consists of a sample of 27,068 urban elementary schools observed annually during the 2009–2011 period. All observations in the panel correspond to the sixth grade, which is the last year of elementary school. This allows us to capture the terminal efficiency of elementary schools. The panel structure presents a typical configuration, that is, there are many units (N = 27,068) observed over a short period of time (T = 3). This provides us with a total of 81,204 observations. The panel is balanced since we do not have any non-observed data problems in the sample. It covers 63% of the urban elementary schools in Mexico, which represents 48% of sixth-grade students and 35.9% of the schools evaluated by the ENLACE test in 2012. The construction of the panel is based on two public information sources: data from the Mexican Ministry of Public Education and data from the Mexican National Population Council. In particular, we use two databases: the SNIIE 2009–2011 education database4, which covers school characteristics, ENLACE test outcomes (2010– 2012)5 and the CEMABE database, which is a census of schools,

teachers and students in elementary education6. The CEMABE comprises a more detailed information about school characteristics and teachers. Finally, we obtained the urban marginalization index for each AGEB from the National Population Council7. Ideally, we would like to have a database that is as representative and heterogeneous as possible. However, we cannot dismiss the potential bias caused by the school self-selection process. In this case, some schools would have to choose not to be part of the evaluation process and not to participate in the ENLACE test and to avoid submitting information to the SNIEE, etc. We found particular self-selection problems in 4 out of 32 states: Guerrero, Michoaca´n, Chiapas and Oaxaca. Oaxaca is an extreme case where according to CEMABE database there is no information reported on urban elementary schools on ENLACE test results. We will now describe the variables that capture inputs and outputs in the school production function. Table 1 presents the descriptive statistics of the variables used in our analysis. Most empirical applications employ standardized test results as a proxy of the school production output and in this paper, we use the standardized ENLACE test for this purpose. In particular, we focus on the mathematics and Spanish tests corresponding to 6th grade students enrolled in elementary education. The test results we are using correspond to the average points obtained by each school in the test; these results are calculated and published annually. There is no disaggregated information regarding education expenditure per school. Considering that the OECD (2012) recently pointed out that wages and salaries represent 91% of the total education expenditure in Mexico, we constructed a proxy variable that represents the total number of employees per student. In order to do that, we divide the total number of school employees (teachers, school principal, administrative and maintenance personnel) that work in the school and divide it by the number of students in the school. Quality in education regarding teaching experience is an important variable that has no data that fully characterizes this factor in our sample. However, we use a proxy variable of teaching quality in the schools. The variable we use is a proportion of teachers that work in the school and participate in the National Teaching Career Program (PNCM, for its Spanish acronym). The program promotes teachers’ professionalization in education through seminars, course updates and continuous training programs. The PNCM creates incentives (monetary and professional) so the teachers enrolled in the program can achieve five levels of teaching career updates. We recognize that this variable has important limitations since it does not capture teachers’ experience, although it can reveal the teachers’ own interest in acquiring more knowledge on educational practice. School size is a traditionally important variable in the literature on educational production function. We define the school size variable as the total number of students in the school. There is no information available regarding school infrastructure facilities and teaching material. Nevertheless, we have information about the schools that have internet access for its students in classes. The student’s internet access variable is represented with a dichotomous variable with a value of 1 if the school has internet access for its students and 0 if not. The characteristics of the school principal are important determinants in the school production function. In the absence of information on experience, we use a dichotomous variable that reveals whether the position of school principal is a full-time job or he/she has other teaching activities in the school. We expect that, 6

4

We made a public request to the SNIEE database, although there is a shorter version available at http://www.snie.sep.gob.mx/geosepv2/. 5 The ENLACE test results are available at http://www.enlace.sep.gob.mx/.

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The ENLACE test results are available at http://www.enlace.sep.gob.mx/. An urban AGEB is a geographic area that comprises a group of household blocks delimited by streets, avenues or any other characteristic for which the land use is for housing, industrial, services and commercial. The AGEB are part of urban localities. 7

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Table 1 Descriptive statistics. Variable name

Average

Min

Max

Std. deviation

Variance

ENLACE Spanish ENLACE mathematics Expenditure per student Teacher ratio in PNCM School Size Marginalization Index Student’s internet access School principal without class HHI Private Morning school N = 81,204

557.0788 564.0807 2.8229 0.4193 5.4492 2.1227 0.4384 0.8981 0.2572 0.1743 0.7052

300 267 5.6021 0.0001 1.3863 0.5730 0 0 0 0 0

858 903 0.5306 2 7.5011 8.1115 1 1 1 1 1

63.5567 72.5164 0.4784 0.3311 0.7251 0.9003 0.4962 0.3025 0.3402 0.3794 0.4559

4039.4580 5258.6250 0.2289 0.1096 0.5258 0.8105 0.2462 0.0915 0.1158 0.1439 0.2079

Source: Authors’ own estimations from SNIIE and CEMABE databases from 2010–2012.

Fig. 2. Enlace Test results.

on average, full-time school principals would perform their administrative tasks better than those who also teach and need to divide their time between both activities. The dichotomous variable has a value of 1 if the school principal is not teaching and 0 if he/she also has to fulfill other administrative tasks. According to their source of funding, schools are classified as public or private. The distinction is relevant since, on average, private schools obtained better results than public schools on the ENLACE tests. Fig. 2 presents the ENLACE test results in Spanish and Mathematics for public and private schools. The figure depicts a clear statement about the incentives private schools face in order to capture the resources families privately invest in their children’s education. We can see that the outcomes in the ENLACE test results in both, Spanish and Mathematics are on average superior for the case of private schools. The urban marginalization index is a multidimensional index that measures levels of poverty and vulnerability in Mexican localities. The index is estimated by the Mexican National Population Council using the 2010 Mexican Population Census. The marginalization index presents AGEB as a unit of analysis and

ranges from 2 to 6 points, with larger values indicating a higher level of marginalization for a given locality. In order to avoid negative values we transformed the original data so the range of values varies between 0 and 108. 5. Results The main results of the paper are presented in Tables 2 and 3, for the test results in the subjects Spanish and mathematics, respectively. The columns in these tables correspond to the estimation of Eq. (4) under LS, FE and RE techniques. The results reveal that nearly all the estimated parameters are consistent and have the expected sign. We found that the FE estimator differs in magnitude and significance from the rest of the estimators; then again, this may be due to the incidental parameter bias of the model. The original fitting criterion, that follows the clustered LS estimators, shows a R2 value of 0.28 for the Spanish test and 8 The method for completing these transformations and the correlation values between both indices for validation are available from the authors upon request.

Panel

Stochastic frontier

LSE Stacked Ln expenditure per student Ln teacher ratio in PNCM Ln total students enrolled in school School Principal without class Private Morning Students with internet access Marginalization Index HHI—Market 1 Constant

FE

0.0289 (19.09)** 0.0036 (19.01)** 0.0293 (32.66)** 0.0124 0.1263 0.0364 0.0104 0.0218 0.0218 6.2310

(8.50)** (62.79)** (42.77)** (14.48)** (50.08)** (50.08)** (1,623.78)**

RE 0.0090 (2.33)* 0.0005 (1.5300) 0.0055 (1.2000) 0.0021 1.1800

6.3108 (313.61)**

BC 88 0.0282 (16.64)** 0.0028 (14.44)** 0.0297 (28.57)** 0.0081 0.1201 0.0357 0.0107 0.0218 0.0029 6.2306

Sigma (u) HHI—Market 1 Constant

(6.02)** (53.54)** (32.41)** (11.05)** (38.12)** (1.9300) (1496.3)**

BC 95 0.0262 (4.78)** 0.0030 (3.59)** 0.0179 (4.12)** 0.0075 0.1156 0.0340 0.0107 0.0202 0.0005 6.3553

3.4738 (58.80)**

Sigma (v) Margination index Constant Adjusted R2 Log likelihood Number of observations Number of schools

(2.72)** (14.10)** (12.71)** (3.13)** (6.17)** (0.1200) (347.13)**

0.2801 149,176 81,204 27,068

222,244 81,204 27,068

Notes. Robust standard errors in parentheses. Log likelihood simulated for TRE model. * Significant at a confidence level of 95%. ** Significant at a confidence level of 99%.

. 81,204 27,068

157,217 81,204 27,068

TRE 0.0434 (8.13)** 0.0040 (4.37)** 0.0347 (7.10)** 0.0120 0.1312 0.0321 0.0138

(3.10)** (15.54)** (13.32)** (4.59)**

6.2318 (263.12)**

0.0426 (9.16)** 0.0030 (4.46)** 0.0315 (6.64)** 0.0071 0.1221 0.0305 0.0149

(2.68)** (17.05)** (11.75)** (4.63)**

6.2584 (295.14)**

0.4751 (4.09)** 5.0867 (55.33)**

0.7351 (7.35)** 6.6559 (53.21)**

0.1501 (4.51)** 5.0867 (55.33)**

0.2644 (5.16)** 6.0888 (48.58)**

147,388 81,204 27,068

160,181 81,204 27,068

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Table 2 Enlace Spanish results.

29

30

Panel

Stochastic frontier

LSE stacked Ln expenditure per student Ln teacher ratio in PNCM Ln total students enrolled in school School Principal without class Private Morning Students with internet access Marginalization Index HHI—MARKET 1 Constant

0.0312 0.0056 0.0256 0.0126 0.1102 0.0276 0.0092 0.0135 0.0047 6.2646

FE (16.73)** (23.96)** (22.89)** (6.58)** (45.10)** (26.07)** (10.54)** (23.75)** (3.14)** (1317.82)**

RE 0.0476 0.0003 0.0076 0.0039

(9.66)** 0.9200 1.3100 1.8300

6.4176 (256.58)**

BC 88 0.0353 0.0038 0.0271 0.0090 0.0932 0.0278 0.0095 0.0135 0.0042 6.2692

Sigma (u) HHI—MARKET 1 constant

(15.71)** (15.00)** (18.88)** (5.21)** (32.43)** (20.07)** (7.90)** (18.52)** (2.23)* (1146.17)**

BC 95 0.0353 0.0038 0.0271 0.0090 0.0932 0.0278 0.0095 0.0135 0.0042 6.7467

4.3035 (81.39)**

Sigma (v) Margination index Constant Adjusted R2 Log likelihood Number of observations Number of schools

(4.41)** (7.97)**** (4.43)** (2.75)** (12.01)** (7.17)** (2.56)* (3.30)** 0.7900 (270.32)**

0.1204 116,490 81,204 27,068

193,567 81,204 27,068

Notes. Robust standard errors in parentheses. Log likelihood simulated for TRE model. * Significant at a confidence level of 95%. ** Significant at a confidence level of 99%.

. 81,204 27,068

129,219 81,204 27,068

TRE 0.0426 0.0058 0.0325 0.0120 0.1127 0.0239 0.0119

(5.27)** (7.77)** (5.69)** (2.26)* (13.01)** (5.93)** (2.92)**

6.2406 (295.84)**

0.0416 0.0039 0.0299 0.0082 0.0948 0.0239 0.0114

(5.33)** (7.86)** (4.87)** (2.47)* (12.25)** (5.81)** (2.79)**

6.2580 (265.62)**

0.9176 (2.97)** 5.4216

2.0107 (2.89)** 9.7672 (10.21)**

0.1415 (5.13)** 4.5780 (75.77)**

0.1845 (5.20)** 5.2532 (74.90)**

115,889 81,204 27,068

125,725 81,204 27,068

R. Garcia-Diaz et al. / International Journal of Educational Development 46 (2016) 23–34

Table 3 Enlace mathematics results.

R. Garcia-Diaz et al. / International Journal of Educational Development 46 (2016) 23–34

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Table 4 Technical efficiency descriptive statistics. Panel A: Spanish Stochastic frontier model

Average

Min.

Max.

Std. deviation

Variance

Time invariant (BC88) Time variant (BC95) Time variant and exogenous factors (TRE)

0.9284 0.9686 0.9612

0.6810 0.6701 0.5962

0.9937 0.9888 0.9892

0.0405 0.0131 0.0216

0.0016 0.0002 0.0005

Panel B: Mathematics Time invariant (BC88) Time variant (BC95) Time variant and exogenous factors (TRE)

0.6082 0.8781 0.9456

0.4659 0.4194 0.6543

0.8157 0.9998 0.9654

0.0389 0.0478 0.0127

0.0015 0.0023 0.0002

Note. N = 81,204.

0.12 for the mathematics test. These values are low, yet not surprising given that previous results in the literature find that less than 30% of test result variance is attributable to the schools (Blanco, 2011; Ferna´ndez and Blanco, 2004). Regardless of the estimated model and the subject test, we found that the expenditure per student is positive and significant. Still, the magnitude of the elasticity is small. This implies that an increase in public expenditure on education would marginally improve the results on the ENLACE test, ceteris paribus. These results support the evidence found by the OECD (2013) that in the last ten years Mexico has increased education expenditure per student by 14%, while changes in the PISA test, for the same period, were just marginal. The qualifications of the teachers in the analysis, proxied by the enrollment of teachers in the National Teaching Career Program, show positive and mostly significant effects, except for the case of fixed effect model. Given that the impact factors are close to zero, in practical terms, it can be ruled out that being enrolled in the National Teaching Career Program would have positive effects on the test outcomes. This is certainly a discouraging result and could only be understood in the light of some policy arrangements inside the structure of the Education Workers Labor Union (SNTE for its Spanish acronym). Reyes (2011) documents that there are perverse instructional arrangements in the SNTE, which cause the National Teaching Career Program to be more of a control mechanism than a tool to improve the quality of education. Our results are consistent with Reyes (2011) findings. In a similar fashion, morning schools, school size, and the number of classes attended per school principal, all show the expected signs and are significant, yet the elasticities are close to zero. The dichotomous variable students with internet access that refers to school quality infrastructure is consistently significant with a small positive impact on students’ outcomes through all the models. The marginalization index was associated with negative results in the test for all the estimated models, except BC95 and TRE for which the interpretation is different. The result implies that the more marginalized the school, the lower the outcomes in the tests. These results support the strong negative relationship between marginalization and school achievement (Ferna´ndez and Blanco, ˜ oz and Villarreal, 2009; Blanco, 2011). On the contrary, 2004; Mun private schools present positive and significant coefficients in all the models estimated. They also observe the higher coefficients in each model for both test subjects. These results suggest that private elementary schools, on average, obtain better results than their public counterparts. The interpretation of the marginalization index in models BC95 and TRE is different from the rest of the models. In these cases, the marginalization index helps to model the environment in which the schools operate. Therefore, it is considered an exogenous factor associated with the traditional error variance (vi) that is useful for controlling the heteroscedasticity found in the stochastic frontier

estimations. In both models, the marginalization index coefficient should be interpreted as a lineal relationship between the traditional error term variance (sv) and the marginalization index. We found this relationship positive and significant in both models for the two subject tests. From these results we can say that the higher the marginalization index the more dispersed the error term in the frontier. We now focus on the effects of competition on the type 1 school market. The interpretation of the coefficient is similar to the previous cases. The difference is that, in this case, market concentration is used to model the technical inefficiency term in the production frontier (su). We find positive and significant coefficients which indicate that higher concentration levels (less competition within the market) increase the school technical inefficiency dispersion. In other words, we observe that less competition implies higher technical inefficiency. These results are supported by the economic theory and are similar to other authors’ findings for the United States (Marlow, 1997; Millimet and Collier, 2008) and United Kingdom (Bradley et al., 2001). One of the main aims of this paper is to measure the efficiency parameters. For this purpose, we are going to use the stochastic frontier results to calculate the technical efficiency of each producing unit with the following Eq. (4). In addition, we argue that a reduction in technical inefficiency (ui) is equivalent to an increase in technical efficiency (TEi). Table 4 panels A and B present the descriptive statistic results of the schools’ technical efficiency in the analysis for Spanish and Mathematics tests, respectively. Table 4 Panel A shows the variation in the average technical efficiency between the models. The highest dispersion in technical efficiency is found in the BC95 model, while the lowest dispersion is obtained in the BC95 model. The former model presents a variation range of 13%, while in the TRE it is 21% and 40% in BC95. The technical efficiency for the mathematics test presented in Table 4 Panel B is similar to that obtained for the Spanish test. Nevertheless, the variance of the TRE model is now the lowest. The results obtained here are consistent with those of Greene (2008) Table 5 Technical efficiency Spearman correlation. Panel A: Spanish Stochastic frontier model

BC88

BC95

TRE

Time invariant (BC88) Time variant (BC95) Time variant and exogenous factors (TRE)

1.0000 0.6827* 0.3447*

1.0000 0.8534*

1.0000

1.0000 0.6420* 0.5717*

1.0000 0.6916*

1.0000

Panel B: Mathematics Time invariant (BC88) Time variant (BC95) Time variant and exogenous factors (TRE)

Note. N = 178,845. * Significant at a confidence level of 95%.

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R. Garcia-Diaz et al. / International Journal of Educational Development 46 (2016) 23–34

Fig. 3. Technical efficiency statistics using the TRE model.

and Kirjavainen (2012), with both authors reporting that the TRE model reduces the magnitude of variance in technical efficiency. Greene (2008) explains that this is due to the modeling of the individual heterogeneity within the frontier. In this case, the inefficiency term (ui) is cleared from time-invariant individual effects, which in other cases are captured by the technical inefficiency term. Following Kirjavainen (2012), in Table 5, we run the Spearman correlation test in order to verify if the estimators of technical efficiency are consistent. We use the Spearman test because the monotonic relationship between efficiency estimators violates the assumptions associated with the Pearson test. The test results confirm a difference between TRE technical efficiency estimators from the other alternatives, due to the purge from individual effects. There is in principle no single answer to the question of how the selection of frontier stochastic models should be. As a pragmatic choice, the TRE model appears to be a reasonable compromise given that the estimator is more robust to the presence of heteroscedasticity, enabling us to model its potential sources. The second reason is that it allows us to purge the technical inefficiency term from individual effects, thus obtaining more balanced efficiency estimations. The third reason is that we obtain technical efficiency estimations that vary in time, from which we can follow the evolution over time of each school’s efficiency. Fig. 3 is a box plot graph that shows the technical efficiency statistics using the TRE model9. This graphic allow us to see the mean of the data and its variability. Fig. 3 consists of two panels, one for the Spanish test results and the other for the mathematics test result. The figure distinguishes eight cases when it furthers 9 The box plot graph desegregates data variability in quartiles. In addition, it shows minimum, maximum, mean and atypical values (outliers).

disaggregates according to the type of school (public and private) and level of concentration in the market (concentrated, C, and nonconcentrated, NC). In a first general inspection, we can see that the technical efficiency variance reduction affects the variability range for the result in the Spanish and mathematics tests. In general, public schools tend to have a lower technical efficiency than private schools (for both the Spanish and mathematics test outcomes). Now, considering the market concentration, we can observe that public schools in concentrated markets (cases 1 and 5) are the least efficient. While the public schools in non-concentrated markets (cases 2 and 6) improve with respect to the previous case. However, public schools are still somehow below the results obtained in terms of efficiency by the private schools, which is the case for private schools in concentrated markets (cases 3 and 7) and the private schools in non-concentrated markets (cases 4 and 8). In general, the evidence suggests private school markets are more competitive and more efficient.

6. Conclusions This paper presents a stochastic frontier analysis to study the effect of competition on elementary schools’ educational outcomes in Mexico. We found two main results: First, there is evidence of a positive association between competition and school efficiency in elementary schools. Second, there is a small but significant effect of expenditure on education per student in terms of educational outputs, as measured by the nationwide standardized academic test ENLACE. In relation to school characteristics we found that the type of school is a relevant factor to explain school outputs, since private schools obtain on average better outputs than public schools. This may suggest that the incentive structure in private schools is an important determinant in school outputs.

R. Garcia-Diaz et al. / International Journal of Educational Development 46 (2016) 23–34

We propose several models to define the best consistent and unbiased technical inefficiency estimator. The models discussed here have consistent and unbiased estimators with persistent heteroscedasticity problems in both the fixed effect and random effect panel data models. The heteroscedasticity problems pose serious limitations in the estimation of school technical efficiency given that it represents a component of the error term. We find the True Random Effects presents a reasonable compromise since it enables us to model heteroscedasticity controlling for exogenous factors in both the random noise component and the technical inefficiency component. We observe that schools’ technical efficiency is positively associated with the urban environment and negatively associated with the localities’ marginalization index. We also find that controllable school inputs show a low correlation (less than 15%) with schools’ technical efficiency.

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There are, of course, a number of other issues that could usefully be studied here. For example, we do not consider competition from private schools independently and separately from public schools (largely for no convergence in the estimation process); nor can we study directly school expenditure per student, teacher experience, investment in infrastructure, for all these variables we proxied. Also, for future research we could use instrumental variables to treat potential endogeneity problems in the estimation process, as proposed by Hoxby (2000) and Gibbons et al. (2008).

Appendix A. Series of tests on the models Table A1

Table A1 Tests performed on the all the models in the analysis. Test

Subject

Performed tests to the OLS Model Global significance Spanish (F test) Global significance Mathematics (F test) White test Spanish

Statistic value

p-Value

Rejection criterion

Judment

Conclusion

3743.76

0.00

0.05

Null hypothesis rejected

1418.86

0.00

0.05

Null hypothesis rejected

5054.37

0.00

0.05

Null hypothesis rejected

4501.21

0.00

0.05

Null hypothesis rejected

jointly, the coefficients are significant jointly, the coefficients are significant The residual variance is non homoscedastic The residual variance is non homoscedastic The residuals are not normally distributed The residuals are not normally distributed There is no evidence of multicollinearity There is no evidence of multicollinearity

White test

Mathematics

Shapiro–Wilk W test

Spanish

0.9557

0.00

0.05

Null hypothesis rejected

Shapiro–Wilk W test

Mathematics

0.9734

0.00

0.05

Null hypothesis rejected

Variance Inflation Factor (vif) Variance Inflation Factor (vif)

Spanish

2.30

Mathematics

2.30

Performed tests to the RE model Wald test Spanish

Average vif < 2.5 Average vif < 2.5

17,673.22

0.0000

0.05

Null hypothesis rejected

6548.52

0.0000

0.05

Null hypothesis rejected

Wald test

Mathematics

Wooldridge test

Spanish

6.724

0.0095

0.05

Null hypothesis rejected

Wooldridge test

Mathematics

2.727

0.0987

0.05

Breusch–Pagan test

Spanish

12,178.64

0.0000

0.05

Null hypothesis non rejected Null hypothesis rejected

Breusch–Pagan test

Mathematics

14,012.48

0.0000

0.05

Null hypothesis rejected

Hausman test

Spanish

195.29

0.0000

0.05

Null hypothesis rejected

Hausman test

Mathematics

307.24

0.0000

0.05

Null hypothesis rejected

2.89

0.0210

0.05

Null hypothesis rejected

40.53

0.0000

0.05

Null hypothesis rejected

6.724

0.0095

0.05

Null hypothesis non rejected

Performed tests to the FE model Global significance Spanish (F test) Global significance Mathematics (F test) Wooldridge test Spanish Wooldridge test

Mathematics

2.727

0.0987

0.05

Null hypothesis rejected

Wald heteroskedasticity Wald heteroskedasticity

Spanish

1.10E + 10

0.0000

0.05

Null hypothesis rejected

Mathematics

1.00E + 10

0.0000

0.05

Null hypothesis rejected

jointly, the coefficients are significant jointly, the coefficients are significant There is evidence of first order autocorrelation There is no evidence of first order autocorrelation RE estimator is preferred over OLS RE estimator is preferred over OLS The difference between RE and FE is systemic. FE is preferred The difference between RE and FE is systemic. FE is preferred

Jointly, the coefficients are significant Jointly, the coefficients are significant There is evidence of first order autocorrelation There is no evidence of first order autocorrelation The residual variance is non homoscedastic The residual variance is non homoscedastic

R. Garcia-Diaz et al. / International Journal of Educational Development 46 (2016) 23–34

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