Socio-Economic Planning Sciences xxx (xxxx) xxx Contents lists available at ScienceDirect Socio-Economic Planning Sciences journal homepage: www.elsevier.com/locate/seps Efficiency and quality in Colombian education: An application of the metafrontier Malmquist-Luenberger productivity index Alexei Arbona a, Víctor Giménez b, *, Sebastián López-Estrada a, b, Diego Prior b a Department of Economics, Pontificia Universidad Javeriana Cali, Colombia b Department of Business, Universitat Autònoma de Barcelona, Spain A R T I C L E I N F O A B S T R A C T JEL classification: This study uses the metafrontier Malmquist-Luenberger index to measure changes in the productivity of 4587 JEL schools in the Colombian education system. Public and private schools are differentiated and all the schools C61 participated in the Saber 11 standardized test between 2014 and 2017 in the subjects of mathematics, reading, H52 sciences, social and citizen sciences and English. This methodology is appropriate to measure productivity while I21 using good and bad outputs in the educational context. The general results indicate deterioration in both sectors; Keywords: this behavior is due to the change in best practices and the change in efficiency. Large gaps are also found Education Efficiency between departments. Metafrontier malmquist-luenberger index Inequality 1. Introduction Index of Educational Quality (ISCE) and the Management Index for Educational Quality (IGCE). The first focuses on the results of the Educational efficiency is a topic of intense political, social and aca­ educational process (progress, performance, efficiency and school demic debate [25] for various reasons. First, education is considered as environment), and the second, on the schools’ resources (efficiency in the main source of human capital accumulation of a country [39]. the infrastructure). The application of this index reflects the importance Second, education plays a fundamental role in the redistribution of that the government attributes to the efficient management of resources living conditions in society [37]. Finally, because educational budgets in education, showing that educational quality is being measured from are high and growing [28], it is important to ensure that educational different perspectives [68]. However, although the quality index aims to spending is carried out with a high level of efficiency. measure the efficiency in both components, it should be noted that there Different development plans around the world recognize education is no input-output logic in its construction. as a priority due to the social externalities it presents [52]. Colombia is The educational reform carried out with the implementation of the no exception, as shown by its efforts to close inequality gaps through Quality Index is relevant for the context in which it is developed and for education. At the local level, the Ministry of National Education (MEN) the objective it is intended to achieve; however, the methodology of the of Colombia is the institution responsible for managing resources in index is not robust. The main characteristics that can be improved and education, one of the focuses of which is to reduce gaps in access and that motivate the present study are: first, the components should not quality, thus improving the level of human capital and, in turn, pro­ have a priori weights within the indicators; second, the conceptualiza­ moting economic growth and development [5]. tion of efficiency must be underpinned on production theory [8]; and The main objective of recent educational policies has been to close finally, there must be a global vision that can have multiple benchmarks social gaps [40]; again, Colombia is no exception. To this end, the for comparison, and not a partial vision through two weighted indexes. Estimulos a la Calidad Educativa (Incentives to Educational Quality) It is worth pointing out that estimation of models based on data envel­ incentive system was designed under MEN Decree 501 of 2016, which opment analysis methods for the calculation of composite indicators has uses the Índice de Calidad (Quality Index) as a single measurement tool gained prominence [33], among other reasons, to avoid the subjectivity for the granting of stimuli. In turn, this index comprises the Synthetic of choosing the weights of the components a priori [61] and to have the * Corresponding author. E-mail address: victor.gimenez@uab.cat (V. Giménez). https://doi.org/10.1016/j.seps.2021.101122 Received 3 November 2020; Received in revised form 30 May 2021; Accepted 6 July 2021 Available online 7 July 2021 0038-0121/© 2021 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article as: Alexei Arbona, Socio-Economic Planning Sciences, https://doi.org/10.1016/j.seps.2021.101122 A. Arbona et al. Socio-Economic Planning Sciences xxx (xxxx) xxx possibility of estimating different groups under the same approach [6]. high-school students who participated in 2014 and 2017. This period is The main objective of the study is to evaluate the change in the relevant to analyze the evolution of productivity due to the change in the productivity of 4587 schools over time, differentiating between public1 regulations related to the Índice de Calidad, which proposes incentives to and private sectors. In addition, there are two specific objectives: first, to schools in matters of management and budget, and acts as a market propose a robust and integrative methodology of the concepts measured signal. partially by the incentive system Estimulos a la Calidad; and second, to The contribution of this work is threefold. First, it opens the way for analyze the change in the efficiency of the education system with the multiple applications in educational policy, since it is proposed as a introduction of the ISCE. The analysis takes into account the political­ benchmark for delivering incentives in the public sector, and it acts as a –administrative division of Colombia, which is divided into 32 de­ market signal in the private sector. Second, it responds to calls for partments and its capital district, to determine if there are differences. research related to educational efficiency which applies variables that This methodology also allows the analysis to consider three different take into account all the dimensions of standardized tests [41], and in­ orientations in the results, towards performance, towards inequality, or corporates both performance and inequality in the educational process both at the same time. [68]. Third, comparing the application employed in this study with The education sector in Colombia is a representative case of a other similar studies in the literature [7,31,32], the Saber11 database is developing economy with high social inequity, where there are large used instead of TIMSS (Trends in International Mathematics and Science gaps between public and private education [15]. Therefore, rigorous Study) to measure the outputs of the process, and the C600 (census of analysis of the differences between the sectors is important for devel­ schools in Colombia) provided by the DANE is used for the inputs. It is oping educational policies. The gaps between the public and private the first application of the metafrontier Malmquist Luenberger index for sectors are worrisome for two main reasons: first, there is high private a specific education system that applies partial frontiers for different spending by households that want to access a better education for their sectors. children, which generates strong pressure on their well-being [53]. The results show, on average, better performance change in the Second, there is evidence that the difference in resources between public than the private sector, although there is a general deterioration schools in the public and private sectors is one of the most relevant in the education system regardless of the orientation used. The public drivers of gaps [13], which depends mainly on the differences in their sector has a better performance change when there is an orientation funding. towards equality (bad outputs), driven mainly by the change in effi­ One of the main motivations of this study is to use efficiency mea­ ciency. Additionally, departments show different approaches to working sures with an input-output logic, and in turn, incorporate inequality in on educational performance, and present results that vary significantly. the education system by using undesirable or bad outputs, thus The study is organized into five sections. This introduction is fol­ improving the approach used by the government’s quality index. lowed by the literature review (Section 2). The methodological aspects Inequality and inequity in education are two problems of constant of the MML index and its decomposition are then described (Section 3), concern around the world and especially in developing countries. after which the databases used in the education system evaluation However, this study focuses on inequality as a bad output for various process are presented, the variables are explained, and the main results reasons. First, it is a relevant topic of constant debate that is attracting are reported (Section 4). Finally, the main conclusions are detailed increasing interest in the literature [46]. Second, it is a problem in which (Section 5). schools play a fundamental role, for example, by running additional classes or by grouping students according to levels [2]. And finally, the 2. Literature review effects of inequality are not only localized, but are also externalized and affect the whole economic system [16]. The study of efficiency of schools was strongly motivated by the Educational efficiency has gained relevance in the measurement of Coleman report, which highlighted the lack of participation of educa­ different problems [25], among which the temporal evolution of tional institutions in the struggle for equal opportunities in the United educational productivity is highlighted [3,27]. Only three studies have States [19]. This research line has been approached from different focused on measuring changes in educational productivity by incorpo­ perspectives [65]. In this study, we take the public economics perspec­ rating desirable and undesirable outputs [7,31,32]. Tsai et al. [68] state tive, where emphasis is placed on technical efficiency, generally through that the correct evaluation of the productivity of an education system a non-parametric approach, and the units evaluated are compared with must consider both outputs (good and bad) while controlling the inputs their peers according to the levels of inputs and outputs, mainly using and environmental variables. As far as we are aware, no evaluation has non-parametric frontier models. these characteristics, nor considers the differences between groups The non-parametric approach is highlighted for the major advan­ (public and private). As this paper offers an initial approach, it is tages it has over the other methods used in the literature. First, it is less therefore relevant to all contexts, but even more so for Colombia due to vulnerable to specification problems that affect econometric models the large inequality gaps and the country’s demonstrated interest in [62]. Second, it is not necessary to define assumptions about the dis­ education policy. tribution of errors and the production function [73]. And third, multiple To carry out this approach, we use the metafrontier Malmquist inputs and outputs can be used [66], which for the objective of this study Luenberger (MML) index developed by Oh [54] because it helps to is fundamental, since it takes into account performance and inequality at incorporate a temporal dimension in the analysis while considering the same time. In addition, in the field of educational efficiency, good and bad outputs in the process. Additionally, directional distance non-parametric methods have been the most frequently applied in the functions (DDF) are applied since they allow efficiency to be measured literature [25]. by improving the academic average while reducing the variance of the Within the line of public economics, the Malmquist index has been results. This paper analyzes the changes in the productivity of 4587 used to analyze temporary changes in productivity [31,60,64], which schools through the result in the standardized exams of senior can be split in two components: technological change and change in efficiency. Alternatively, the Hicks-Moorsteen index [4] has also been used with an approach based on performance and total factor produc­ 1 The National Administrative Department of Statistics (DANE) of Colombia tivity ratios. Studies addressing the evolution of productivity using the uses the terms ‘official’ and ‘unofficial’ to differentiate schools managed by the Malmquist index [14] have applied methodological complements that public administration from those that are not; however, in this article we refer enhance the scope; for example, by using directional distance functions to these schools as ‘public’ and ‘private’, respectively, as these terms will be (DDF) [18] efficiency can be measured with multiple approaches and more familiar to international readers. flexible orientations. Moreover, the use of metafrontiers [22,66], based 2 A. Arbona et al. Socio-Economic Planning Sciences xxx (xxxx) xxx on the approach of Battese et al. [6]; allows decompositions of different differences between the types of schools, and considering composite categories (Thieme et., 2013) and analyzes different groups or tech­ measures that integrate all areas evaluated and their standard deviation. nologies [24] in the same context. The present study is therefore the first approach of this type in this In this study, the metafrontier Malmquist Luenberger (MML) index research field. developed by Oh [54] is adopted, since it incorporates a temporal dimension in the analysis while considering the good and bad outputs of 3. Methodology the process, and in turn, it focuses on metafrontiers. To date, only three studies have measured changes in educational productivity considering The first part of this section explains the DDF, which are necessary to good and bad outputs [7,31,32], all of which prioritize the educational estimate the MML index. The second part presents the evolution of quality orientation. However, the authors are unaware of any other productivity measurement indices in general and explains the MML applications of the MML approach to the education sector, making this index model. study an innovation in this field. Although the MML has not been applied to the education sector, 3.1. Directional distance functions similar studies have been undertaken. In the first, Giménez et al. [31] used a global (non-radial) Malmquist index to measure the change in the The technology that models the set of production possibilities as­ productivity of 29 education systems from 2003 to 2012, considering as sumes K groups (k = 1,2 …, K) for T periods of time (t = 1, 2 …., T). A set variables the results of mathematics and reading and their standard of inputs and outputs is used in the process: the vector of inputs is x = deviation. Subsequently, Ben Yahia et al. [7] conducted research using (x1, …, xN) ∈ RN+ and the outputs are distinguished between desirable y DDF to work with bad outputs and non-discretionary inputs, based on a sample of Tunisian schools in 2012. Finally, Giménez et al. [32] used = (y1, …, yM) ∈ RM J + and undesirable b = (b1, …, bJ) ∈ R+ . The set of TIMSS data for 28 countries in an efficiency analysis from 2007 to 2011 production possibilities meets the following axioms [29]: with a global Malmquist-Luenberger model due to the presence of bad { ⃒ } P(X) = y, b ⃒ x can produce (y, b), x ∈ RN+ (1) outputs. These authors conclude that although there are large differ­ ences between countries, on average educational performance declined if (y, b) ∈ P(x) | y´< y, then (y,́b) ∈ P(x) (2) during those years. The quality of an education system does not depend exclusively on if (y, b) ∈ P(x) | b = 0, then y = 0 (3) academic performance. For this reason, the present study highlights not ( ) only educational achievement but also equality in the education process. 1 if (y, b) ∈ P(x) | 0 ≤ Ø ≤ 1, then y, Øb ∈ P(x) (4) Tsai et al. [68] frame two objectives as a golden rule in educational Ø policy: excellence (high performance) and equality (low variability in performance) in the results. The first objective has been thoroughly if (y, b) ∈ P(x) | (y´, b´) ≤ (y, b), then (y´, b´) ∈ P(x) (4.1) studied in the literature [45], for example, to find its determinants [41], The first axiom suggests that the set of outputs is strongly disposable, the role of environmental variables [49] and the differences between while the set of bad outputs is only weakly disposable; this means that education systems [56]. Additionally, most transnational studies that there are no additional costs to reduce the production of desirable out­ examine performance consider each area of interest separately [39]. puts, but the reduction of undesirable outputs can require more input However, it is desirable to have composite measures of academic per­ consumption or, alternatively, the reduction of good outputs. The sec­ formance to produce more reliable studies [41]. ond, known as ‘null-jointness’, indicates that decision making units The second objective, equality, has been analyzed in terms of the role (DMUs) cannot produce the desirable output without producing the of education systems in the standardization of opportunities [44]. undesirable output. The third axiom shows that bad outputs are weakly Equality in educational processes has often been measured through the disposable, indicating that minimizing this type of output is expensive. total variance of academic performance [43]. However, how to measure The fourth states that it is feasible to reduce the set of good outputs while and treat inequality among students within an educational system is a increasing the bad outputs proportionally by Ø; note that this axiom topic discussed in the literature [9], which has been debated in many must be contrasted with equation (4.1) since it allows a non- countries, with positions between a selective system (for example, proportional reduction of good and bad outputs. Germany, Hungary, Austria) or a comprehensive system (for example, To estimate the DDF used by the MML index, parametric and non- Japan, Canada, Norway) [38]. The main policies have focused on early parametric frontier models can be applied. In this study we chose the follow-up of students [26,39], grouping of skills and/or performance in non-parametric approach because it does not require the assumption of a the classroom [42] and individualized support [30]. In general, those in functional form, nor a specific distribution of the error term [66]. It also favor of homogenizing classes affirm an increase in efficiency, and those allows us to work with multiple sets of inputs and outputs without who are not in favor affirm that the level of low ability students is having to assume factor prices, which is appropriate and applicable to affected because of lowered expectations and self-esteem due, among the education sector, since they are unknown or difficult to estimate other reasons, to the peer group effect [9]. [25]. Based on the above, although there are different positions in the The main idea of the DDF, as can be seen in Fig. 1, is to maximize the literature regarding student inequality, the present study takes it into desirable outputs while minimizing the undesirable ones by maintaining account in terms of the standard deviation of students, mainly because it or reducing the level of inputs used. The mathematical expression of the is a measure that can be used for different types of standardized tests DDF is as follows: [57], and because it is a proxy of other measures used in previous studies { ⃒( ) } [43]. Also, the model jointly evaluates excellence and inequality, in D(x, y, b; g) = max β ⃒ y + β gy , b − β gb ∈ P(x) (5) pursuit of aspects of improvement based on the available inputs, following the quality education goal in the 2030 Agenda for Sustainable where g = (gy , gb ) is a directional vector, which indicates the direction of Development, which commits to “providing inclusive and equitable approach to the frontier. Following Chung, Färe and Grosskopf [18]; and quality education and promoting lifelong learning opportunities for all” if g = (y, b), the DDF can be rewritten as follows: [70]”. D(x, y, b; g) = max{ β : ((1 + β)y, (1 − β)b) ) ∈ P(x) } (6) This paper proposes an approximation through the MML index, which responds to calls in the literature by prioritizing both the per­ In the previous expression, the utility of the DDF can be analyzed, formance and the equality of the process, taking into account the where the objective is to increase the desirable outputs (y) in a β 3 A. Arbona et al. Socio-Economic Planning Sciences xxx (xxxx) xxx Fig. 2 shows the MML index and its components with two years and two groups. The intertemporal reference technology is the envelope (metafrontiers) of contemporary technology and the global reference technology is the envelope of the intertemporal reference technology. For the present study, the contemporary reference (F (oft)) is the closest reference against which each school in a specific group and year is compared (public schools in period t). The intertemporal reference technology (F (Of)) is the metafrontier that considers all the contem­ porary boundaries of a specific group (for example, public schools). Finally, the global reference technology (F(G)) is the metafrontier that includes all the groups and years under study (public and private schools in periods t and t + 4). In other words, to define the index and its decomposition, it must be considered that there are three types of possible references: technolog­ ical, intertemporal and global [69]. Oh [54] defines the MML produc­ tivity index as follows: ( ) MMLt;t+1 xt , yt , bt , xt+1 , yt+1 , bt+1 = G t t 1 + D (x , y , b ) t (7) = ( ) G t+1 1 + D x ,y ,b t+1 t+1 Fig. 1. Directional Distance Function. Source: [17,58]. 1 + Dt (xt , yt , bt ) = ( )× 1 + Dt+1 xt+1 , yt+1 , bt+1 ( / ) proportion while reducing the undesirable outputs (b) in the same 1 + DI (xt , yt , bt )) (1 + Dt (xt , yt , bt ) proportion. ×( ( ))/( ( ) )× 1 + DI xt+1 , yt+1 , bt+1 1 + Dt+1 xt+1 , yt+1 , bt+1 ( /( ) 3.2. Temporary productivity analysis ×( 1 + DG (xt , yt , bt )) 1 + DI (xt , yt , bt ) ( ))/( ( )) (8) 1 + DG xt+1 , yt+1 , bt+1 1 + DI xt+1 , yt+1 , bt+1 Changes in productivity have been quantified in the literature using ( ) the Malmquist Index, introduced by Caves et al. [14]; in a parametric MMLt;t+1 xt , yt , bt , xt+1 , yt+1 , bt+1 = framework. This index attributes the changes in productivity to two TE t+1 BPR t+1 TGR t+1 (9) = × × components: the change in efficiency (or catching up) and the techno­ TEt BPRt TGRt logical change. The first refers to how the units have approached or ( ) moved away from the contemporary production frontier during the MMLt;t+1 xt , yt , bt , xt+1 , yt+1 , bt+1 = (10) analyzed period, while the second quantifies how it has moved. Thus, = EC × BPC × TGC the first of the components is the effect attributable to the capacity for Following Oh’s [54] proposal, it should be taken into account that management, organization, and coordination, and the second is related the distances (D) shown in the previous equations (7) and (8) are the to the capacity for innovation [50]. DDF explained in the previous section, where a comparison is made with The MML index arose from the Malmquist Luenberger (ML) index a global (DG ), an intertemporal (DI ) and a contemporary (Dt ) frontier. [18] and Oh’s developments [54,55]. The ML evaluates changes in Additionally, Oh [55] proposes decomposing the MML index into three productivity when bad outputs are incorporated into the production components: the first is the change in efficiency (EC), the second is the function [29]. Later, based on the ML, Oh [55] proposed the global change in the best practices gap (BPC) and the third, the change in Malmquist Luenberger (GML), which focuses on comparing observa­ technology gap (TGC). If the MML index is greater than unity, it in­ tions against a single global frontier, and finally, Oh [54] adapted the dicates a positive change in productivity between periods t and t + 1; in GML to the use of metafrontiers [6]. Fig. 2. MML productivity index diagram. Source: [36]. 4 A. Arbona et al. Socio-Economic Planning Sciences xxx (xxxx) xxx other words, the distance to the global frontier is less in the period t + 1 output, one bad output, and five inputs. The good output is the sum of than in the period t. An MML index value less than unity is a deterio­ the generalized global score for the school [7,22,31,32,63,66]. The ration in the productivity of the units evaluated. global score is the weighted average2 of the individual scores of each of If EC is greater than unity, it shows an improvement in efficiency, the tests that students take in the exam, divided into the total weighting which is the measure of recovery of technical efficiency within a group (13) and multiplied by the number of tests (5). To generalize the variable during the added period, indicating how fast a DMU moves towards the at the school level, since the inputs are at this level, the sum of the global reference technology. Referring to the second component, BPC measures score is divided into the number of students who took the exam and the change between the best contemporary reference technology and multiplied by the total enrollment of the school. intertemporal technology: when BPC is greater than unity, it indicates The bad output, following the scale with which the global score is that the contemporary approaches the intertemporal frontier. Finally, built, is the standard deviation of the generalized global score for TGC is the technical gap between the frontier of intertemporal reference schools. To date, equality in educational processes has been measured technology in moment t and the global frontier reference in moment t + through the total variance of academic performance [43], whereas other 1. When TGC is greater than unity, it indicates a narrowing of the global approaches have used the percentage of students who do not reach technology gap from the frontier of a specific group. minimum scores on standardized tests [31,32]. However, standard de­ The DDF can be calculated from different types of frontier models, viation is used because it is a measure that can be applied for different →G w w w types of standardized tests [57], and according to the literature, having following different programming systems D (x , y , b ), →I w w w →w w w w homogeneous groups improves efficiency, which is the objective of the D (x , y , b ), D (x , y , b ) where w = t and t + 1. We chose to analysis of this study [9]. calculate the DDF with DEA type models because it avoids the need to The five inputs used are frequent in the literature [25]: the amount of assume a functional form for the production function a priori and it is electronic equipment, number of teachers working as managers, number possible to work with a combination of multiple inputs and outputs. To of teachers in classrooms, number of students enrolled, and the average calculate these distance functions using DEA type models, the following socioeconomic and cultural level of the school as an environmental optimization program must be solved [17,55]: variable. Electronic equipment is quantified as all desktops or laptops ( ) and tablets available and in use [1,51]. Dd xkẃ , ykẃ , bkẃ = maxβ (11) The next two inputs are the number of teachers in managerial roles s.t. and teachers in classrooms. The first one is used as a proxy for variables ∑ commonly used in the literature such as managers or administrative staff λk,w yk,w k,́w m ≥ (1 + β)ym , m = 1, …M. (12) [11,34,35], and the second is teachers in classrooms [3,21,48,67], whose main function is teaching. These two variables are included to ∑ λk,w bk,w k,́w j ≤ (1 − β)bj , j = 1, …J. (13) represent the human capital of the school, mainly due to the different functions of the two roles. The fourth input refers to the number of ∑ λk,w xk,w k,́w (14) students enrolled in the school [23,59]. n ≤ xn , n = 1, …N. Finally, there is a debate in the literature about how to include the socioeconomic index of the students in the estimations, where some λk,w ≥ 0 (15) authors use it as an input and others as an environmental variable. The superscript d in the objective function is intended to identify the However, despite this debate, most scholars use it as input [20,21,31, type of directional function since it can be contemporary, intertemporal 66]. On the other hand, the methodological relevance of the conditional or global. A vector λk,w appears in the restrictions; this vector shows the models is recognized as having opened up the possibility of adopting the intensity to construct the restriction, where k indicates the education socioeconomic index as an environmental variable; however, even so, sector. The estimation of the DDF through a DEA is an optimal solution some authors treat this variable as an input when using conditional for the calculation and decomposition of the index, where a β coefficient models [20]. In this article, as in other research, the socioeconomic is obtained, which is interpreted as the simultaneous increase (decrease) index is included as an input, since the evaluation is focused on the that can be achieved in the good (bad) outputs, given the input capacity of schools to make the most of the inputs [10,12,21]. In addi­ consumption. tion, because the databases do not offer the index, following Thieme The next section presents the empirical study, and describes the et al. [66]; this is defined as an estimated latent variable through a variables and sources used in the analysis. The main results and multiple correspondence analysis, considering the educational level of decomposition are also shown according to the multiple directions parents and the socioeconomic level of the household. All the goodness offered by the index. of fit statistics yield results according to the calculation of the indicator. Table 1 reports a summary of descriptive statistics for the variables 4. Empirical study: data and results used. The overall score shows an average decrease of 1.16%, and its standard deviation reflects a growth of 5.41%. It should be noted that This section describes the data used in the model. It is focus on the statistics show an inverse behavior to the ideal in the educational pro­ variables and sources used for the analysis. cess. The inputs have the following behavior: electronic equipment and teachers in classrooms increase on an average by 97.27% and 0.51%, 4.1. Sample and variables respectively. The total enrollment of students, the number of teachers in The study considers 4587 schools participating in Saber 11 (stan­ dardized test) in the years 2014 and 2017. These are the schools for 2 A weighted average is used (three points each for mathematics, reading, which complete information is available in all the variables during the social studies and natural sciences and one for English language) for two rea­ years of study. The sample is 75.5% from the public sector and 24.5% sons: first, the literature recommends using multiple areas of study [41] and not from the private sector. just math and/or language, as is common; and second, because it is the measure Based on the literature review, and in accordance with the proposal used for admission to higher education and for the design of educational pol­ in the methodological section, this study uses seven variables: one good icies in Colombia. In addition, the weights of the areas are defined by the ICFES and are therefore maintained, mainly because one of the largest gaps between public and private education in Colombia is in the area of English and the global score smooths it out. 5 A. Arbona et al. Socio-Economic Planning Sciences xxx (xxxx) xxx Table 1 an orientation towards good and bad outputs simultaneously, with a Descriptive statistics of inputs and outputs. coefficient of 0.1026, which means that good outputs can be increased Variables Year Average Standard Private Public by 10.26% while bad outputs are decreased in the same proportion deviation while maintaining the level of inputs. Output As we see in Table 2, the static analysis on average shows higher y1: global score 2014 202,410 174,972 157,323 217,182 levels of inefficiency in the public than in the private sector. Following 2017 200,059 172,283 159,337 213,277 the orientation towards good and bad output simultaneously, the public y2: standard deviation 2014 474.28 304.14 557 447 sector presents potential improvements of 0.2101 and 0.2079 in 2014 of the global score 2017 499.95 275.68 552 483 Input and 2017, respectively, and the private sector presents values of 0.1969 x1: electronic 2014 79.88 78.65 53.62 88.49 and 0.1562, showing better performance at 1.32% and 5.16%. The es­ equipment 2017 157.58 185.06 58.32 189.8 timate towards bad output shows inefficiencies of 0.6549 and 0.5715 in x2: enrollment of 2014 798.19 670.16 553.14 878.48 the private sector and 0.6754 and 0.6613 in the public sector for the school 2017 768.46 639.71 540.83 842.34 years 2014 and 2017, respectively; these results reveal gaps of 2.06% x3: teachers in 2014 2.93 1.95 3.33 2.8 management roles 2017 2.88 1.72 3.11 2.81 and 8.98%, respectively. Finally, the estimate with a good output x4: teachers 2014 31.64 22.90 27.57 32.97 orientation shows inefficiencies of 0.2121 and 0.1618 for the private 2017 31.8 22.63 27.84 33.08 sector and 0.2227 and 0.2169 for the public sector, highlighting gaps of x5: socio-economic 2014 5.04 0.85 3.97 5.39 1.05% and 5.51%. In summary, three results are evident in the educa­ and cultural index 2017 4.96 0.73 5.9 4.66 tion system. First, the private sector is more efficient in both 2014 and Source: self-devised. 2017. Second, the gaps between sectors widen in this period. And third when the focus is on performance (good output) and equality (bad management roles and the socioeconomic and cultural index show output) simultaneously, inefficiency values are lower. respective average decreases of 3.72%, 1.71%, and 1.59%. At the departmental level, there are considerable differences in the Due to the considerable heterogeneities in access to and quality of results. In the private sector with simultaneous orientation towards good education among Columbia’s 33 departments3 and educational sectors and bad outputs, the most efficient department (Casanare) has a value of (public/private), the results are presented with this disaggregation. The 0.1026 and the least efficient (Putumayo), a value of 0.4514, showing a average global score for the private sector shows a growth rate of 1.28%. 35% gap in 2014 compared to the 23% gap for 2017 in the public sector. Bad output shows positive behavior in the private sector and negative Following the orientation towards bad outputs and good outputs, the behavior in the private one. gaps in 2014 and 2017 are 39% and 41%, and 38% and 24%, respec­ The inputs show different behaviors according to education sector. tively. Finally, the Li test [47] is used to check whether there is a sig­ For example, electronic equipment had a growth rate of 114% in the nificant difference between the general distributions of the results (it is public sector and 8.77% in the private. The teaching staff presented a used to compare all the results calculated in this article). This relatively stable trend in both sectors, except for teachers in manage­ nonparametric statistical test compares two unknown distributions ment roles in the private sector (− 6.61%). Overall, the socioeconomic using kernel densities. Its main advantage is that unlike most statistical and cultural conditions of the students improved significantly: average tests, the Li test is not based on comparisons of means or medians, but growth for private schools was 48.61%, whereas these conditions compares two complete distributions against each other. deteriorated by 13.54% in the public schools. In next section, the inputs The results of this static analysis provide an initial overview of the and outputs are used to estimate the metafrontier Malmquist Luenberger efficiency of the Colombian education system in two moments, and show index. The results are presented by component and educational sector. the heterogeneity and the diverse patterns between the departments and types of orientation. 5. Results 5.2. Temporary analysis of educational efficiency The results section presents the analysis of educational efficiency4 for Colombia at the departmental level and by education sector from two Temporary analysis is carried out through a MML index described in points of view: first, a static analysis through a DEA to provide an initial the methodology section for the years 2014 and 2017, disaggregated by reference point. And second, a temporary analysis of productivity, education sector and type of orientation. Table 3 shows the results of the which focuses on the evolution over time. scores for the metafrontier Malmquist Luenberger index and its decomposition for all departments of Colombia according to education 5.1. Static efficiency analysis sector. Table 4 shows the summary of applying the MML with three orientations. First, following the orientation g = (y, b); that is, it aims to The static analysis5 was carried out through a DDF and DEA increase the desirable outputs (y) by percentage β and reduce the un­ described in the methodological section for the years 2014 and 2017, desirable outputs (b) in the same proportion. Second, with orientation g disaggregated by education sector and type of orientation. The analysis = (0, b); in this case, the objective is to minimize the bad outputs was performed for all departments of Colombia according to education sector and considering alternative orientations (good and bad output, Table 2 bad output and, finally, good output). The DEA estimation in the static Static analysis of educational efficiency with different orientations between analysis shows high variability in the results among the education sec­ 2014 and 2017. tors, types of orientation, and departments at different times. For Orientation Year Private Public example, in 2014 Casanare is the most efficient department when it has Good and bad outputs 2014 0,1969a 0,2101a 2017 0,1562a 0,2079a Bad outputs 2014 0,6549a 0,6754a 3 Descriptive statistics at the departmental level are available upon request. 2017 0,5715a 0,6613a 4 Good outputs 2014 0,2121a 0,2227a Before the estimation, a procedure was carried out to detect the extreme 2017 0,1618a 0,2169a values and outliers using, among other methods, super efficiency [72] and a multivariate method, bacon [71], after which the estimation was performed Source: self-devised. without these extreme values. . 5 a Departmental results are available upon request. 1%. 6 A. Arbona et al. Socio-Economic Planning Sciences xxx (xxxx) xxx Table 3 Educational improvement (orientation towards good and bad output) between 2014 and 2017. Department Private Public EC BPC TGC MML EC BPC TGC MML Amazonas 1.1378 0.9129 0.9766 1.0145 0.9859 1.0111 1.0072 1.0040 Antioquia 1.0153 0.9566 0.9580 0.9304 0.9984 0.9881 0.9924 0.9790 Arauca 1.0507 0.9395 0.9897 0.9769 1.0366 0.9682 0.9928 0.9964 Archipiélago de San Andrés 0.9776 1.0595 0.9666 1.0011 0.9469 1.0302 0.9983 0.9739 Atlántico 1.0403 0.9508 0.9809 0.9703 1.0129 0.9720 0.9976 0.9823 Bogotá, D.C 1.0281 0.9649 0.9611 0.9534 1.0051 0.9682 0.9815 0.9551 Bolívar 1.0222 0.9774 0.9686 0.9677 0.9953 0.9964 0.9964 0.9882 Boyacá 1.0499 0.9582 0.9517 0.9575 1.0143 1.0287 0.9795 1.0220 Caldas 1.0352 0.9753 0.9537 0.9629 0.9811 1.0104 0.9934 0.9847 Caquetá 1.0265 1.0185 0.9771 1.0216 0.9882 1.0273 0.9933 1.0084 Casanare 1.0436 0.9912 0.9804 1.0141 1.0044 1.0106 0.9967 1.0117 Cauca 1.0849 0.8755 1.0200 0.9688 0.9385 1.0402 1.0091 0.9851 Cesar 1.0521 0.9184 0.9861 0.9529 1.0194 0.9743 0.9964 0.9896 Chocó 0.9615 1.0308 0.9973 0.9885 Córdoba 1.0808 0.9698 0.9607 1.0070 0.9895 1.0089 0.9912 0.9894 Cundinamarca 1.0352 0.9589 0.9720 0.9649 1.0080 0.9931 0.9879 0.9889 Guainía 1.0332 0.9590 0.9990 0.9899 Guaviare 1.0056 0.9473 1.0128 0.9648 0.9286 0.9630 0.9987 0.8931 Huila 1.0668 0.9552 0.9449 0.9628 1.0175 1.0258 0.9849 1.0279 La Guajira 0.9838 1.0268 0.9786 0.9886 0.9541 1.0376 0.9986 0.9886 Magdalena 1.0156 0.9544 0.9703 0.9405 0.9861 1.0033 0.9898 0.9793 Meta 1.0453 0.9401 0.9522 0.9356 0.9886 0.9911 1.0046 0.9843 Nariño 1.0299 0.9603 0.9653 0.9547 1.0050 1.0167 0.9745 0.9957 Norte de Santander 1.0695 0.9066 0.9798 0.9500 1.0328 0.9710 0.9909 0.9938 Putumayo 1.3821 0.7326 1.0089 1.0216 1.0181 1.0175 0.9859 1.0213 Quindío 1.0805 0.9194 0.9629 0.9566 0.9833 1.0034 0.9922 0.9790 Risaralda 1.0457 0.9386 0.9535 0.9360 0.9875 0.9950 0.9950 0.9777 Santander 1.0491 0.9482 0.9506 0.9457 1.0070 0.9968 0.9856 0.9893 Sucre 1.0348 0.9676 0.9702 0.9714 0.9826 0.9952 0.9936 0.9715 Tolima 1.0464 0.9691 0.9587 0.9722 0.9781 1.0184 0.9919 0.9880 Valle del Cauca 1.0462 0.9436 0.9954 0.9827 0.9932 0.9777 0.9984 0.9695 Vaupés 1.0065 0.9984 0.9987 1.0035 Vichada 0.9921 1.0205 0.9963 1.0087 Total 1.0374b 0.9546a 0.9680b 0.9586a 0.9977b 0.9958a 0.9909b 0.9845a MML: Metafrontier Malmquist Luenberger. EC: Efficiency Change. BPC: Best Practices Change. TGC: Technical Change Gap. Source: self-devised. a 5%. b 1%. without considering the desirable outputs. Finally, the results with the Table 4 orientation g = (y, 0), where the objective is to maximize good outputs, Summary of educational improvement (three orientations) between 2014 and and bad outputs are not considered. 2017. Table 3 shows the results of the estimation of the MML index and its Orientation Component Private Public components with a simultaneous orientation towards good output and Good and bad outputs EC 1.0374b 0.9977b bad output. Changes in productivity are positive when the coefficients BPC 0.9546a 0.9958a are greater than unity and negative when lower values are obtained. The TGC 0.9680b 0.9909b aggregate results offer an overview of the whole system, although the MML 0.9586a 0.9845a Bad outputs EC 1.0302b 1.0005b analysis is broken down at the departmental level for a more detailed BPC 0.9583 0.9973 analysis. The education system, in general, shows a decrease in pro­ TGC 0.9655b 0.9908b ductivity of 4.14% (column 4) in the private sector and 1.55% (column MML 0.9680b 0.9909b 8) in the public sector, the latter showing a better performance by Good outputs EC 1.0396b 0.9985b BPC 0.8911b 1.0080b 2.59%. The behavior in the private sector is mainly driven by a decrease TGC 0.9024b 0.9258b in the BPC and TGC of 4.54% (column 2) and 3.20% (column 3) MML 0.8360b 0.9318b respectively, offset by an EC improvement of 3.74% (column 1). In the MML: Metafrontier Malmquist Luenberger. public sector, the behavior of the TGC (0.91%), the BPC (0.42%) and the EC: Efficiency Change. EC (0.23 deteriorates in all cases. BPC: Best Practices Change. The EC is the component with the best performance in the private TGC: Technical Change Gap. sector as behavior improves in 84% of the departments. In the BPC Source: self-devised. (Archipelago of San Andrés, Caquetá, La Guajira) and TGC (Cauca, a 5%. Guaviare, Putumayo), only 9% of the departments show positive b 1%. behavior. In the public sector, the component with the highest number of departments with positive behavior is the BPC (51%), followed by the EC (41%) and the TGC (9%). The gaps between departments are different for each of the 7 A. Arbona et al. Socio-Economic Planning Sciences xxx (xxxx) xxx components, although the difference between the best and worst output and bad output). The results confirm that the public sector per­ department evaluated in each component is always greater in the private forms better than the private sector, regardless of the orientation. The than in the public sector. The gaps are 40.5%, 32.7% and 7.5% in the superior productivity of the public sector is in line with the study by private sector and 10.8%, 8.1% and 3.5% in the public sector for the EC, Mancebón et al. [51]; who find better levels of performance in public BPC, and TGC, respectively, and the MML index presents a 9.1% gap for schools than in their private counterparts in the Spanish education the private and 13.5% for the public sector. The results of the estimation system. of the index oriented to good output and bad output simultaneously The components of the index show great differences in behavior show a better change in public sector performance except for the depending on the sector analyzed. The private sector is mainly driven by component directly related to the change in efficiency. the change in efficiency (EC), and to a greater extent when the perfor­ Table 4 6 presents the summary of the results of the estimation of the mance orientation is followed (3.96%). At the departmental level and MML index and its components with all orientations. On average, for the considering each of the components, the results are as follows: (i) high orientation towards bad output, the public sector has a better change in variability among departments and, in general, their level of produc­ performance in both the MML index and the BPC and TGC components tivity is correlated with the efficiency in the initial situation; (ii) less than the private sector. The differences in favor of the public sector dispersion of the results in the public than in the private sector; (iii) considering the orientation to bad output are 3.90%, 2.53% and 2.30% greater variance in the index when focusing on performance. for the components BPC, TGC, and MML, respectively, and the differ­ The results of the educational tests, in general, are analyzed directly, ence in EC in favor of the private sector is 2.97%. When the orientation without considering either bad outputs or the available resources. This is to good output is followed, the differences in average increase to one of the reasons why the findings of the present study do not coincide 11.69%, 2.34% and 9.58% for the BPC, TGC and MML components, and with current discussions surrounding academic results of public and the EC has a difference in favor of the private sector of 4.11%. The private schools. From a public policy point of view, 75% of the schools orientation towards bad outputs focuses on equality in the education evaluated are public; therefore, there should be a concern for access to a system, showing differences between sectors and highlighting the gaps. more egalitarian education without sacrificing good performance. In The results reveal that for both sectors the greatest participation in conclusion, a simple analysis must be distinguished in absolute terms as increasing productivity is provided by the EC. in the present study, in which the focus is on the productivity and The results of estimates with a static and dynamic approach equality of the education system. considering multiple orientations reveal a difference between the ap­ The MEN Estímulos a la Calidad program in Colombia, introduced in proaches and orientations; four common patterns emerge. First, overall 2016 to evaluate and encourage academic quality, is one of the moti­ performance deteriorates in the private sector, and to a greater extent vators of this study. The results confirm that, on average, the program when it is oriented towards performance. Second, the EC is the had no positive effects on the evolution of productivity of schools in the component that drives positive changes in productivity. Third, the least education system. That is, school productivity decreased from 2014 to efficient departments (Amazonas, Putumayo) at the initial moment, 2017. This decrease may be due to various reasons, of which we iden­ considering the static approach (DEA), are those that are marginally tified two. First, the program applies a blanket educational policy to the more easily able to improve their productivity. Finally, large gaps are whole country, regardless of the context. The second reason is related to evident between the departments. the Estímulos a la Calidad program and the new measurement of schools’ quality. One component of the measurement is the students that pass the 6. Conclusions academic year (promotion in the index), which can incentivize teachers to raise students’ grades, which may not guarantee the minimum The present study evaluates the evolution in the productivity of the learning standards. Colombian education system between 2014 and 2017, taking into ac­ There are several implications for education policy. First, the Estí­ count students’ results in the Saber 11 test and the endowments of mulos a la Calidad program and its adaptability should be restructured educational institutions, and differentiating between orientations to­ according to the context. Additionally, variables that cannot be biased wards the types of results (performance and equality) and the education by the MEN’s design of incentives must be taken into account. Finally, sectors. public policy must focus on all students in the education system. For this The methodological approach is based on the metafrontier Malm­ reason, maximum attention should be paid to guaranteeing minimum quist Luenberger index (MML), and to our knowledge, this study is the learning standards for all students and not simply to the results of first application in the education sector. This is a relevant tool to analyze standardized tests. the educational context since it allows bad outputs to be incorporated Although the study met its objectives and the results are relevant to into the process while considering both performance and different the context in which it was developed, three limitations should be groups in the evaluation. From an empirical point of view, the study addressed in future research. First, the availability of a restricted period aimed to analyze the change in the productivity of the Colombian ed­ in the databases; expanding this period would provide a better overview ucation system and the similarities and differences between the sectors, of the approach. Second, this study, like many others, carries out ana­ and to discover the performance of the different units in a context of lyses with the schools as its unit of analysis; however, there is a tendency quality incentives. It also seeks to offer a robust tool with greater scope towards considering the student as the unit of analysis. Additionally, for measuring efficiency than the Quality Index explained in the intro­ taking into account different academic courses, instead of standardized ductory section. tests, could yield results with greater scope. Third, the effects of The results of the MML index according to the different orientations departmental characteristics should be analyzed, since, in an environ­ show, on average, deterioration in the private sector of 16.4% (good ment such as Colombia, institutions and location can play an important output), 3.20% (bad output) and 4.14% (good and bad output), role in educational productivity. depending on their orientation. For the public sector, there is a deteri­ oration of 6.82% (good output), 0.91% (bad output) and 1.55% (good Funding Víctor Giménez, Sebastian López-Estrada and Diego Prior acknowl­ 6 As the focus is on components and sectors, results at the departmental level edge the financial support of the Spanish Ministerio de Economía y for the second and third orientation are not presented here, but are available Competitividad (ECO2017-88241-R). upon request. This decision was made at the suggestion of one of the reviewers and to reduce space. 8 A. Arbona et al. Socio-Economic Planning Sciences xxx (xxxx) xxx References [33] Greco S, Ishizaka A, Tasiou M, Torrisi G. On the methodological Framework of composite indices: a review of the Issues of weighting, aggregation, and Robustness. Soc Indicat Res 2019;141(1):61–94. [1] Agasisti T. How competition affects schools’ performances: does specification [34] Grosskopf S, Moutray C. Evaluating performance in Chicago public high schools in matter? Econ Lett 2011;110(3):259–61. the wake of decentralization. Econ Educ Rev 2001;20(1):1–14. [2] Agasisti T, Falzetti P. Between-classes sorting within schools and test scores an [35] Haelermans C, Ruggiero J. Estimating technical and allocative efficiency in the empirical analysis of the Italian junior secondary schools. Int Rev Econ 2017;64: public sector: a nonparametric analysis of Dutch schools. Eur J Oper Res 2013;227 1–45. (1):174–81. [3] Aparicio J, Crespo-Cebada E, Pedraja-Chaparro F, Santín D. Comparing school [36] Han H, Zhong Z, Wen C, Sun H. Agricultural environmental total factor ownership performance using a pseudo-panel database : a Malmquist-type index productivity in China under technological heterogeneity: characteristics and approach. Eur J Oper Res 2017;256(2):533–42. determinants. Environ Sci Pollut Control Ser 2018;25(32):3209–11. [4] Aparicio J, López-Torres L, Santín D. Economic crisis and public education. A [37] Hanushek EA. The economics of schooling: production and efficiency and public productivity analysis using a Hicks-Moorsteen index. Econ Modell 2018;71:34–44. schools. J Econ Lit 1986;24(3):1141–77. [5] Badunenko O, Henderson DJ, Russell RR. Polarization of the worldwide [38] Hanushek EA, Ludger W. Does educational tracking affect performance and distribution of productivity. J Prod Anal 2013;40(2):153–71. inequality? differences-in-differences evidence across countries. Econ J 2006;116: [6] Battese GE, Prasada Rao DS, O’Donnell CJ. A metafrontier production function for 63–76. estimation of technical efficiencies and technology gaps for firms operating under [39] Hanushek EA, Woessmann L. The role of Cognitive skills in economic development. different technologies. J Prod Anal 2004;21(1):91–103. J Econ Lit 2008;46(3):607–68. [7] Ben Yahia F, Essid H, Rebai S. Do dropout and environmental factors matter? A [40] Hanushek EA, Kimko DD. Schooling, Labor Force quality, and the growth of directional distance function assessment of tunisian education efficiency. Int J Educ Nations. Am Econ Rev 2000;90(5):1184–208. Dev 2018;60:120–7. [41] Hauser RM. ‘On ‘Quality and Equity in the Performance of Students and schools.’ [8] Bessent A, Bessent W. Determining the comparative efficiency of schools through Center for Demography and Ecology working. 2009. Paper No. 2009–06. data envelopment analysis. Educ Adm Q 1980;16(2):57–75. [42] Hindriks J, Verschelde M, Glenn R, Schoors K. “Ability Tracking, social Segregation [9] Betts JR, Shkolnik JL. The effects of ability grouping on student achievement and and educational opportunity: evidence from Bel- gium.” CORE discussion paper. resource allocation in secondary schools. Econ Educ Rev 2000;19:1–15. Center for Operations Research and Econometrics, Université Catholique de [10] Bradley S, Johnes G, Millington J. The effect of competition on the efficiency of Louvain-la-Neuve; 2010. secondary schools in England. Eur J Oper Res 2001;135(3):545–68. [43] Huang MH. Classroom homogeneity and the distribution of student math [11] Brennan S, Haelermans C, Ruggiero J. Nonparametric estimation of education performance: a country-level fixed-effects analysis. Soc Sci Res 2009;38(4):781–91. productivity incorporating nondiscretionary inputs with an application to Dutch [44] Jackson M, Jonsson JO, Rudolphi F. Ethnic inequality in choice-driven education schools. Eur J Oper Res 2014;234(3):809–18. systems: a longitudinal study of performance and choice in england and Sweden. [12] Camanho AS, Portela MC, Vaz CB. Efficiency analysis accounting for internal and Sociol Educ 2012;85(2):158–78. external non-discretionary factors. Comput Oper Res 2009;36(5):1591–601. [45] Johnes G, Johnes J. The international Handbook on the economics of education. [13] Castro Aristizabal G. ¿ Por qué los colegios privados en Colombia obtienen mejores Cheltenham: Edward Elgar; 2004. p. 613–742. resultados académicos ? Lumen Gentium 2019;3:9–31. [46] Josa I, Aguado A. Measuring Unidimensional inequality: Practical Framework for [14] Caves DW, Christensen LR, Diewert W. The economic theory of index numbers and the choice of an appropriate measure. Soc Indicat Res 2020;149:541–70. the measurement of input, output, and productivity. Econometrica 1982;50(6): [47] Li Q. Nonparametric testing of closeness between two unknown distribution 1393–414. functions. Econom Rev 1996;15(3):261–74. [15] Celis MT, Jiménez ÓA, Jaramillo JF. ¿Cuál Es La brecha de La Calidad Educativa en [48] López-Torres L, Nicolini R, Prior D. Does strategic interaction affect demand for Colombia en La Educación básica Y en La superior? ICFES 2012;3:67–98. school places? A conditional efficiency approach. Reg Sci Urban Econ 2017;65: [16] CEPAL (Comisión Económica para América Latina y el Caribe). Balance Preliminar 89–103. de las Economías de América Latina y el Caribe, 2017 (LC/PUB.2017/28-P). 2018. [49] López-Torres L, Prior D. Centralized allocation of human resources. An application Santiago, 2018. to public schools. Comput Oper Res 2016;73:104–14. [17] Choi Y, Oh D Hyun, Zhang N. Environmentally sensitive productivity growth and [50] Luo F, Ma W, Sun C, Tian M, Zhou Y. Analysis of the performance of government its decompositions in China: a metafrontier Malmquist–Luenberger productivity Investment and Interventions to Promote small-scale Irrigation works in China’s index approach. Empir Econ 2015;49(3):1017–43. major grain-producing areas-based on the two-step model of Malmquist-FGLS. [18] Chung YH, Färe R, Grosskopf S. Productivity and undesirable outputs: a directional Irrigat Drain 2017;66(3):293–304. distance function approach. J Environ Manag 1997;51(3):229–40. [51] Mancebón MJ, Calero J, Choi Á, Ximénez-de-Embún DP. The efficiency of public [19] Coleman J, Campbell E, Hobson E, McPartland J, Mood A, Weinfield F, York R. and publicly subsidized high schools in Spain: evidence from PISA-2006. J Oper Equality of educational opportunity. Washington, DC: US Government Printing Res Soc 2012;63(11):1516–33. Office; 1966. [52] McMahon W. "The social and external Benefits of education", ch. 6 in: international [20] Cordero JM, Polo C, Santín D, Simancas R. Efficiency measurement and cross- Handbook on the economics of education. Edward Elgar Publishing; 2004. country differences among schools: a robust conditional nonparametric analysis. [53] OECD. Education in Colombia. OECD Publishing; 2016. Econ Modell 2018;74:45–60. [54] Oh D. A global Malmquist-Luenberger productivity index. J Prod Anal 2010;34 [21] Cordero JM, Santín D, Simancas R. Assessing European primary school (164):183–97. performance through a conditional nonparametric model. J Oper Res Soc 2017;68 [55] Oh D. A metafrontier approach for measuring an environmentally sensitive (4):364–76. productivity growth index. Energy Econ 2010;32(1):146–57. [22] Cordero JM, Prior D, Simancas R. A comparison of public and private schools in [56] Parteka A, Wolszczak-Derlacz J. Dynamics of productivity in higher education: Spain using robust nonparametric frontier methods. Cent Eur J Oper Res 2016;24 cross-european evidence based on bootstrapped Malmquist indices. J Prod Anal (3):659–80. 2013;40(1):67–82. [23] Crespo-Cebada E, Pedraja-Chaparro F, Santín D. Does school ownership matter? An [57] Phillips M, Chin T. School inequality: what do we know? In: Neckerman KM, unbiased efficiency comparison for regions of Spain. J Prod Anal 2014;41(1): editor. Social inequality (467–520). Russell Sage Foundation; 2004. 153–72. [58] Picazo-Tadeo AJ, Prior D. Environmental externalities and efficiency [24] De la Torre EM, Gómez-Sancho JM, Perez-Esparrells C. Comparing university measurement. J Environ Manag 2009;90(11):3332–9. performance by legal status: a Malmquist-type index approach for the case of the [59] Podinovski VV, Ismail I, Bouzdine-Chameeva T, Zhang W. Combining the Spanish higher education system. Tert Educ Manag 2017;23(3):206–21. assumptions of variable and constant returns to scale in the efficiency evaluation of [25] De Witte K, López-Torres L. Efficiency in education: a review of literature and a secondary schools. Eur J Oper Res 2014;239(2):504–13. way forward. J Oper Res Soc 2017;68(4):339–63. [60] Portela MC, Camanho AS, Keshvari A. Assessing the evolution of school [26] Dupriez V, Dumay X, Vause A. How do school systems manage pupils’ performance and value-added: trends over four years. J Prod Anal 2013;39(1): heterogeneity? Comp Educ Rev 2008;52(2):245–73. 1–14. [27] Essid H, Ouellette P, Vigeant S. Productivity, efficiency, and technical change of [61] Ray AK. Measurement of social development: an international comparison. Soc Tunisian schools: a bootstrapped Malmquist approach with quasi-fixed inputs. Indicat Res 2008;86(1):1–46. Omega 2014;42(1):88–97. [62] Sinuany-Stern Z, Mehrez A, Barboy A. Academic departments efficiency via DEA. [28] Eurostat. GDP per Capita, consumption per Capita and price level indices. Eurostat Comput Oper Res 1994;21(5):543–56. statistics explained. 2014. http://ec.europa.eu/eurostat/statisticsxplained/index. [63] Tavana M, Ebrahimnejad A, Santos-Arteaga FJ, Mansourzadeh SM, Matin RK. php/GDP_per_capita,_consumption_per_capita_and_price_level_indices. A hybrid DEA-MOLP model for public school assessment and closure decision in [29] Färe R, Grosskopf S, Noh DW, Weber W. Characteristics of a polluting technology: the City of Philadelphia. Soc Econ Plann Sci 2018;61(1):70–89. theory and practice. J Econom 2005;126(2):469–92. [64] Thanassoulis E, Kortelainen M, Johnes G, Johnes J. Costs and efficiency of higher [30] Ferrer-Esteban G. Trade-off between Effectiveness and Equity ? An analysis of education institutions in England: a DEA analysis. J Oper Res Soc 2011;62(7): social sorting between classrooms and between schools. Comp Educ Rev 2016;60 1282–97. (1):151–83. [65] Thieme C, Prior D, Giménez V, Tortosa-Ausina E. Desempeño de los centros [31] Giménez V, Thieme C, Prior D, Tortosa-Ausina E. An international comparison of educativos: ¿Un problema de recursos o capacidades organizativas? Hacienda educational systems: a temporal analysis in presence of bad outputs. J Prod Anal Publica Espanola 2011;199(4):81–118. 2017;47(1):83–101. [66] Thieme C, Prior D, Tortosa-Ausina E. A multilevel decomposition of school [32] Giménez V, Thieme C, Prior D, Tortosa-Ausina E. Comparing the performance of performance using robust nonparametric frontier techniques. Econ Educ Rev 2013; National educational systems: inequality versus achievement? Soc Indicat Res 32(1):104–21. 2019;141(2):581–609. 9 A. Arbona et al. Socio-Economic Planning Sciences xxx (xxxx) xxx [67] Tran CD, Villano RA. Measuring efficiency of Vietnamese public colleges: an research papers have been most of them published in international journals such as Socio- application of the DEA-based dynamic network approach. Int Trans Oper Res 2018; Economic Planning Sciences, Health Care Management Science, Omega, Journal of Pro­ 25(2):683–703. ductivity Analysis, Economics of Education Review, Fiscal Studies or the European Journal [68] Tsai SL, Smith ML, Hauser RM. Families, schools, and student achievement of Operational Research among others. Dr. Giménez has been consultant for the Inter- inequality: a multilevel MIMIC model approach. Sociol Educ 2017;90(1):64–88. American Development Bank among other companies and institutions. He often serves [69] Tulkens H, Vanden Eeckaut P. Non-parametric efficiency, progress and regress as a reviewer for prestigious journals, worked on several research projects with external measures for panel data: methodological aspects. Eur J Oper Res 1995;80(3): founding. 474–99. [70] UNDP, UNESCO. Education 2030: Incheon Declaration and Framework for action Sebastian López-Estrada, PhD(c), has researched in the fields of Education eco­ for the implementation of sustainable development. In Report, http://uis.unesco. nomics, social policy and Cultural economics. He is a researcher at the Universitat org/sites/default/files/documents/education-2030-incheon-framework-for-actio Autònoma de Barcelona (Spain) and he was professor at Pontificia Universidad Javeriana, nimplementation-of-sdg4-2016-en_2.pdf; 2015. Cali (Colombia). [71] Weber S. Bacon: an effective way to detect outliers in multivariate data using Stata (and Mata). STATA J 2010;10(3):331–8. [72] Wilson PW. Deterministic with nonparametric outputs multiple. J Bus Econ Stat Diego Prior Jiménez, Ph.D., Full Professor in the Department of Business at the 1993;11(3):319–23. Universitat Autonoma de Barcelona (Spain). Member of the Editorial Committee of five [73] Worthington AC. An empirical survey of frontier efficiency measurement academic journals (Editor-in-Chief in one of them) related with accounting control, effi­ techniques in education. Educ Econ 2001;9(3):245–68. ciency and financial analysis. Member of the Scientific Committee of fourteen national and international conferences. My research interests, published in international journals, are the efficiency analysis of organizations and firms’ financial analysis. In the efficiency Alexei Arbona, PhD, is associate professor of the Department of Economics at the analysis field, my research is oriented towards the design of models to assess the efficiency Pontificia Universidad Javeriana, Cali (Colombia). His research has been in the fields of of organizations, both public and private, and to design programs to improve their per­ social policy and the interaction between the public and private sectors for the provision of formance. In the financial analysis, my research is oriented towards the comparative public goods. benchmark of performance and risk of the European firms in the framework of global competition. I have worked on 36 research projects with external financing, appearing as Dr. Víctor Giménez is Associate Professor of the Business Department at the Uni­ principal researcher in 27 of them. I have published 82 articles, 66 of them in peer- versitat Autònoma de Barcelona (Spain). His research interests are the measurement of reviewed Spanish and international academic journals. I have written 14 books and efficiency in organizations, stock management as well as project management. His 47 books chapters. I have supervised 16 doctoral dissertations. 10