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Factors that affect students’ performance in Science: An application using Gini-BMA methodology on PISA 2015 dataset

Existing theoretical and empirical evidence on the determinants of students’ performance reveals a direct link between pre-primary education and achievement test scores in primary school. Relying on the first-of-its-kind 2015 wave data from the Programme of International Student Assessment (PISA), the present study analyses the associations between students’ performance in science and a broad set of variables, including regressors that proxy pre-primary education. Employing a Gini Regression Bayesian Model Averaging (BMA) approach to account for model uncertainty, it is found that non-attendance in pre-primary education is a robust determinant with a negative impact on students’ performance in science. This result is confirmed both under Gini-BMA and OLS-BMA methodology.


Introduction
Formal education is without any doubt one of the major concerns for policy makers since it determines an individual's income and amplifies inequalities of economic and social opportunities. A study by Carniero and Heckman (2003) points out that investments in human capital have dynamic complementarities and that skills obtained in the child's lifetime expedite the development of additional future skills. So, "learning begets learning" and generates important benefits in terms of medium and long-term schooling and socio-economic outcomes, including the promotion of productivity and economic efficiency. Among all educational stages, there is an emerging consensus that early childhood education interventions provide a cognitively simulating environment that enhance school readiness, academic performance, social integration, and longterm skill development (Myers (1995), Entwisle and Alexander (1993), Waldfogel (2002), Brooks-Gunn (2003), Carniero and Heckman (2003)). Becker (1964) is of the view that early childhood investments bring higher returns compared to future investments because recipients have a longer time to enjoy the benefits. Along the same lines, a study by Heckman (2006) reveals that preprimary education generates the highest possible annual return that gradually fades at higher levels of education program.
This study investigates the associations between students' performance in science and a set of variables that are classified into 14 categories, including attendance and non-attendance in preprimary education. These indicators are relative to students', families' and schools' characteristics. To test this, a large cross-national dataset, the 2015 round of the Programme for International Student Assessment (PISA) is applied. Due to the limited number of observations, Principal Component Analysis is applied to reduce the dimensionality of the dataset, while retaining as much as possible of the variation present in it. Therefore, motivated by the proliferation of possible explanatory variables in explaining students' performance in science and the relative absence of guidance from economic theory, 43 variables are taken simultaneously into consideration. To ensure a comprehensive search, Gini-BMA methodology and OLS-BMA methodology are employed.
Bayesian Model Averaging (BMA) approach constructs estimates that do not depend on a particular model specification, but rather they are conditional on the model space. Thus, inference is averaged over models, forming a weighted average of model specific estimates where the weights are given by the posterior model probabilities. This framework permits to consider a wider range of possible explanatory variables and to end up with those that can effectively explain the relationship. To estimate the coefficients, Gini regression methodology is incorporated into the BMA. The Gini methodology is a rank-based methodology that takes into consideration both the variate values and their ranks and it is based on the Gini Mean Difference (GMD) as a measure of dispersion. Among the two types of Gini regression coefficients that can be attributed to GMD, the focus has been on the semi-parametric approach. The semi-parametric nature of the Gini regression coefficient is justified because it does not rely on the linearity assumption nor on any distributional assumptions and the regression coefficients can simply be interpreted as weighted average of slopes. Even though both the OLS and Gini share an underlining linear structure, they differ in that the estimated marginal responses (i.e., the BMA unconditional posterior means (PSE) as presented in Table 5) are generated differently and in the case of the Gini in a much more robust fashion. This fits well in using a BMA approach that relies on the uncertainty that surrounds the estimates of beta coefficients (i.e., the BMA unconditional posterior means (PSE) as presented in Table 5). Using alternative semi-parametric methods based on local smoothers (Pagan and Ullan (1999)) would not lend themselves directly to the use of BMA since the estimation of marginal effects would not be expressed in a single coefficient as in the case of both OLS and Gini.
A comparison between the Gini-MA results and the OLS-MA ones suggests that the determinants that are important under Gini analysis are not necessarily similar to the ones that are important under OLS analysis. Among the most important outcomes found in this study is the effect of pre-primary schooling on students' test scores. In particular, the results grounded in both Gini and OLS coefficients suggest that the "percentage of students who had not attend pre-primary education" has a negative impact on science test scores. Also, the regressor "attendance in preprimary education" was found to be strongly robust with a surprisingly negative effect on students' performance and only under the Gini-MA analysis. As was argued, that apparent negative result may conceal the significant variation in years of pre-primary education and the potential benefits and costs between too many or too few years of pre-primary education. In other words, the apparent negative result of the "attendance in pre-primary education" variable reflects the significant trade -offs that may exist between entering pre-primary education at a very early age and missing out on parental care at these very early ages, while it is clear that some pre-primary education is crucial and important, but not at all costs. (Entwisle and Alexander (1993), Myers (1995), Waldfogel (2002), Brooks-Gunn (2003), Carniero and Heckman (2003), Velez et.al (1993), Wößmann (2005), Bjorklund and Salvanes (2011), Waldfogel and Washbrook (2011, and press b)).
Children tend to reap the greatest benefits if preschool programs are of high quality . Most of the economic research on this topic recognizes that pre-primary education of exclusive quality is a high-yield investment with longstanding benefits , Heckman and Lochner (2000), Reynolds et. al. (2011)). Also, along with the early childhood interventions, many studies have found that home conditions are another crucial determinant of child's educational achievement (Bjorklund and Salvanes,2011). Both Velez et.al (1993) and Wößmann (2005) agree that, apart from preschool attendance, parental involvement and family features are key components in students' performance. Becker (1981Becker ( , 1985 and Becker and Tomes (1986) embrace the theory of family to provide a reasonable justification for the failure of preschool education.
Apart from pre-primary education, there are also other factors that stand out for their influence on students' performance. In a recent study, Helal et. al. (2019) identify three classes of factors that lead to lower academic performance: the socio-demographic factor, which involves all students from indigent socio-economic background and those with special entry requirements, the academic one which includes all students with limited access to the course resources and forum and the course assessment factor, which refers to students with low level contributions to the course level activities or to students who study-off campus or part-time. According to Tinajero et. al (2012), Brazilian university students' academic achievement is significantly enhanced by cognitive style and learning strategies. Hanushek and Wößmann (2006) take into consideration institutional differences by splitting schools into differing and non-differing ability systems and examine their impact on students' mean performance.
The contribution of this paper to the current literature is threefold. First, it is a contribution to the narrow literature that focuses on the factors that affect students' performance in science. For this purpose, 43 variables, classified into 14 categories, are taken simultaneously into consideration. Most importantly, it attempts to shed a light on the following question: Does pre-throughout the schooling years up to age 16. Similarly, using data from the Early Childhood Longitudinal Study, Manguson et. al. (2007) show that pre-school enrolment in the United States is associated with higher reading and mathematics skills at the time of entry into the first grade, but these effects dissipate for most children by the end of the first grade.
There are several reasons that justify the diminishing trend of/in gains from early childhood interventions. Esping-Andersen et. al. (2012) and Reynolds (1993 state that children at risk due to family's low-income, poverty and other related factors cannot secure a continuous development if there is no a coherent, continual and adequate support provided by government funded preschool and primary grade intervention programs. Specifically, Zigler and Berman (1983) mention that a one-year intervention cannot "inoculate a child against continuing disadvantage" (p.898). Barnett (2011) mentions that interventions are not compelling when graduates from the early educational intervention programs attend public schools with limited efficiency. Further, Schulman et. al. (1999) and Barnett et.al (2004) acknowledge that although most of the states, across the United States, have established prekindergarten curriculum standards, they differ in terms of quality, accessibility, and availability of resources. Most importantly, few of them have established mechanisms to implement these comprehensive standards/prekindergarten initiatives.
Along with the early childhood interventions, many studies have found that home conditions are another crucial determinant of child's educational achievement (Bjorklund and Salvanes,2011). Both Velez et.al (1993) and Wößmann (2005) agree that, apart from preschool attendance, parental involvement and family features are key components in students' performance. A child's development begins within the family and depends on the parents' educational and cultural levels (Wößmann, 2005). Waldfogel and Washbrook (2011, and press b) support that parents that are educated and receive high income, spend more time to prepare their children's reading skills. In contrast, parents with lower income and less education have more possibilities to engage in harsh and incompatible parenting teaching behaviours that may negatively affect child's progress. Becker (1981,1985) and Becker and Tomes (1986) embrace the theory of family to provide a reasonable justification for the failure of preschool education. Many authors correlate family's income with the quality of pre-school education too (Bainbridge et al., 2005;Magnuson and Waldfogel, 2005;Meyers et al.,2004). Low-income families are less likely to enrol their children to pre-school care, and if they do, they are most likely to be characterized by low-quality. In contrast, children from prosperous families are more likely to be registered in high-quality pre-schools. Attending systematically poorer quality pre-schools is an additional reason why gains from pre-school may eventually fade (Esping-Andersen et. al.,2012).
Expanding pre-primary education is an effective instrument to improve school progression and raise average achievement for less advantaged children. Extensive research has been conducted both on the short-run and long-run effects (see among others Barnett (1992), Barnett (1995), Review of Economic Analysis forthcoming 13 (2021) www.RofEA.org Danziger and Waldfogel (2000), Currie (2001), Blau and Currie (2006), Ludwig and Miller (2007)). Dumas and Lefranc (2010) find that extending pre-school enrolment in France is beneficial in terms of schooling outcomes, including test scores, for children from disadvantaged households. Heckman et. al. (2013) evaluate the results of the early childhood education Perry Preschool program that targeted to children from economically disadvantaged families. Outcomes reveal that children who participated in this program tended to create improvements in personality skills and enhance academic motivation. In particular, there is a boost in the long-term achievement test scores, with the effect being stronger for girls than for boys. Research suggests that disadvantaged children take the greatest advantage if, these special programs are of high quality , Heckman and Lochner (2000), Neuman, Kamerman, Waldfogel, and Brooks-Gunn (2003), Reynolds et al. (2011), Waldfogel (2006). Although there is ample empirical evidence that early childhood intervention programs have significant positive effects on the results of children from disadvantaged or minority background, it is not clear whether such pre-school programs influence the outcomes of children in the population as a whole. As typical preschool or prekindergarten programs vary in the quality of learning environments they provide and in the availability of financial resources, little is known about whether universal intervention can promote children's cognitive and academic outcomes (Gilliam and Zigler (2001)).
Many recent papers document the effects of universal preschool enrolment on the education of children in the entire population in a variety of other counties. Estimates obtained for developing counties testify positive and in some cases long-lasting effects of preschool attendance. Exploiting the information given by the Uruguayan Household Survey, Berlinski et. al. (2008) notice that attendance in pre-primary education reduces the probability for grade retention, grade failure and early drop-out during the primary and secondary schooling years. Aguilar and Tansini (2012) recognize that early exposure to pre-primary education has a positive effect on children's performance in the first year at public schools in Montevideo, Uruguay, and this effect remains positive but weakens after six years. Berlinski et.al. (2009) study the effects of Argentina's expansion of universal pre-primary schooling and find that pre-primary education positively affects third grade standardized Spanish and Mathematics test scores as well as students' behavioral skills. Taiwo and Tyolo (2002) notice that first grade Botswana students with pre-school experience achieve higher scores in English language, mathematics and science compared to students without such an experience. Using data for Thailand obtained from the Programme of International Student Assessment (PISA) for the years 2009 and 2012, Pholphirul (2017) reveals that pre-schooling attributes positively on cognitive skills in reading, mathematics and science with the mother's education attainment being a decisive factor on child's enrollment to preschool. According to this information, early exposure to pre-primary schooling appears as a successful and cost-effective policy to prevent late entry, early drop-out rates and early grade failure in poor countries, where Review of Economic Analysis forthcoming 13 (2021) www.RofEA.org large share of young population is excluded from compulsory education already at an early age (UNESCO, 2005).
There is considerable evidence for the impact of universal early childhood schooling in developed countries too. Using Census data, Cascio (2009) examines the long-run results of an expansion in universal kindergarten in the late 60s and early 70s across the United States. She reports no effect on the labour market outcomes and regarding the educational ones, the only positive influence is the reduction in grade retention. Goux and Maurin (2008) apply a differencein-difference approach and find that one additional year in pre-elementary school in French has no important effect on children's subsequent educational skills. Baker et. al. (2005) show that the establishment of full-time and highly-subsidized kindergartens in the Canadian province of Quebec in the late 1990s, corresponds to an increase in the labour supply by married women and a decline in children's outcomes. Similarly, Dickson (2012) displays that the extension of free early education in the UK to all three -year-olds does not have any impact on reading, writing and mathematics when children reach the age seven. Only for deprived Local Education Authorities the results turn to be positive. In contrast  find that Oklahoma's universal pre-school program contributes positively to cognitive scores. In Japan, the expansion of both kindergartens and nursery schools is associated with higher achievement rates both in high school and college (Akabayashi and Tanaka (2013)).

Literature Review for Bayesian Model Averaging (BMA) approach
Classical Statistical Analysis disregards the theory and specification uncertainty, which jointly refer to as model uncertainty. As indicated by Leamer (1983), whimsical decisions about choice of functional forms and control variables leads to fragile inferences based on economic data. Bayesian Model Averaging (BMA) has successfully addressed model uncertainty in the model selection process, providing a comprehensible mechanism to embody ambiguity into conclusions about parameters. To construct estimates, it does not condition on a specific set of theories and covariates, but rather extracts information from a universe of candidate models. The result is a weighted average of model specific estimates, where posterior model probabilities are employed as weights.
The initial approach to deal with model averaging dates back to Roberts (1965) 1 , who proposed a marginal distribution for outcomes of any unobserved sample, the so-called "predictive distribution". This distribution is defined as the weighted averaged of posterior probabilities of two models. Building up to this idea, Leamer (1978Leamer ( ,1983 presented Extreme Bound analysis and set the fundamentals for the BMA methodology 2 . This technique was further studied by Raftery (1988Raftery ( , 1993, George and McCulloch (1993), Madigan and Raftery (1994), Drapper (1995), Kass and Raftery (1995), Kass and Wasserman (1995), Raftery, Madigan and Hoeting (1997), Hoeting et all (1999) among others.
BMA has proven a valuable tool in empirical settings with alternative theories, transmission mechanisms, a massive number of covariates and a limited number of observations. Empirical growth theory is a characteristic example. Theory uncertainty appears extensively in growth regressions since different theories, which use a specific subset of regressors, cannot exclude other compatible and interrelated theories from also being suitable. Brock and Durlauf (2001) refer to this by theory open-endedness, implying that existing growth theories needs to be considered when the effect of a specific growth theory on growth is to be analyzed. Apart from the aforementioned, Fernandez, Ley and Steel (2001a), Doppelhofer, Miller and Sala-i-Martin (2004), Durlauf, Kourtellos and Tan (2008), Durlauf, Kourtellos and Tan (2012), Massanjala and Papageorgiou (2008), Eicher, Papageorgiou and Raftery (2011), Ley and Steel (2007) are among those who endorse the use of this methodology to generate robust growth determinants.
Economic forecasting is another field which is affected by model uncertainty, since forecasts often depend on the model selected. Insightful contributions of forecast uncertainty can be found in Garratt, Lee, Pesaran and Shin (2003), Min and Zellner (1993), Raftery, Madigan and Hoeting (1997), Fernandez, Ley and Steel (2001 a, b), Ley and Steel (2009). In the context of forecasting inflation, Koop and Korobolis (2012) apply dynamic model averaging and selection strategies in state-space models to allow for both the coefficients in each model and the forecasting model to evolve over time. An alternative approach to predict inflation is used in Eklund and Karlsson (2007) who replace the standard marginal likelihood with the so-called forecast weights.

Bayesian Model Averaging (BMA) methodology
BMA provides a probabilistic framework to simultaneously deal with model and parameter uncertainty. To describe the relationships between all the unknown parameters and the data, a joint probability distribution is needed. To construct estimates, instead of conditioning on a single model, a model space M= {M1,…,Mk} is taken into consideration, whose elements cover all the implementation of this methodology, alternative solutions have been suggested (Leamer (1985), Granger and Uhlig (1990), Sala-i-Martin (1997)). Considering the case of normal linear regression models, model uncertainty occurs from the selection of the "best'' model, or alternatively, from the selection of the explanatory variables to include in the right-hand side:

Review of Economic
where Y and ε are nx1 vectors of the dependent variable and the error term respectively, X is a nxq matrix of candidate regressors that may or may not be included in the model, B is an nxq matrix with the parameters to be estimated and n is the total number of observations. If some of the elements of the parameter vector, β= (β1, β2,….βq) equal zero, there are 2 q candidate models in total to be estimated, indexed by Mk for k= 1,….,2 q . Each of these models offers to explain the data and represents a distinct subset of the candidate regressors. For instance, model Mk takes the form: where Y is the dependent variable, ε is the normal error term, 1 ( ) ,…, ( ) is a subset of X1,..Xq and β = ( β1 (k) ,.…., βqk (k) ) is a vector of regression coefficients to be estimated. The vector θk= (β0, β (k) , σ) summarizes the parameters for the given model Mk.
The implementation of a pure Bayesian approach addresses model uncertainty, but its implementation rests firmly on solving the common challenge of specifying the priors over models in the model space and the prior distribution for the parameters of each model. Elicitation of prior parameters can be extremely critical for the outcome and any differences in the results can be attributed to the use of alternative prior assumptions. It is acknowledged (Fernandez, Ley and Steel (2001b), Kass and Raftery (1995), George (1999a) that posterior model probabilities, which are employed as weights for averaging estimates across models, are sensitive to the specification of priors over model-specific parameters. More detailed discussion about model and parameter priors can be found in Steel (2020) as well as in the Appendix. Motivated by the proliferation of possible explanatory variables in explaining students' performance in science and the relative absence of guidance from economic theory, 43 variables are taken simultaneously into consideration. These quality indicators are relative to students', families' and schools' characteristics and are classified into 14 categories: Educational outcomes, Participation in Education, Fields of education, Student evaluation and assessment, Classroom environment/school climate, Students' engagement drive and beliefs, After-school activities, Access to ICT, Performance and socio-economic status, Performance and diversity, Resources for education, Professional development of teachers, School evaluation and Governance. More detailed description is provided in Table 4. BMA approach is employed by calculating the weighted average of model specific estimates using Gini and OLS estimates. The weights attached to each estimate is identical to posterior model probabilities. Being totally agnostic about whether any of these regressors is included in the true model, a prior probability of 0.5 is attached to each one, implying a uniform model prior. Regarding the parameter space, the unit information prior (UIP) is adopted, where the integrated likelihood is proxied by the Schwarz Information Criterion (SIC). According to Eicher, Papageorgiou and Raftery (2011), the combination of the unit information prior with a uniform prior over the model space generally outperforms competing priors. Considering an extremely large set of possible models, by allowing for any subset of up to 43 regressors to be included into the model, the Bayesian framework undertakes the specification uncertainty in a straightforward and formal way. It deals with a set of 2 43 = 8.8x10 12 (over eight trillion) different models. To determine this numerical problem and, following Madigan and York (1995) and Lee and Steel (2007), Markov Chain Monte Carlo (MCMC) techniques are practiced, using the so-called Markov chain Monte Carlo Model Composition ( 3 ) sampler, with 5x10 6 recorded draws after a burn-in of 1 million draws.

Gini Regression coefficient
Although Least Squares methodology ranks as one of the most popular practices for estimating the relationship between a set of regressors on the conditional expected value of the dependent variable, it relies on certain assumptions, whose violations might result in non-robust estimates. The Gini regression, introduced by Olkin and Yitzhaki (1992), is proposed as an alternative. Its utilization is justified whenever the investigator wants to relax the traditional assumptions, such as the convenient world of normality and the linearity of the model. The Gini methodology is a rankbased methodology that takes into consideration both the variate values and their ranks and it is based on the Gini Mean Difference (GMD) as a measure of dispersion 4 .
Between the at least 14 distinct presentations that exist for GMD, the focus has been on the formula that relies on covariances (Lerman and Yitzhaki (1984)) 5 . That is, if F(X) is the cumulative distribution function which is uniformly distributed on [0,1], the GMD is expressed as: 4 The GMD as a measure of spread/variability was first initiated by Corrado Gini (1912).

= 4 { ( ( ) − [ ( )])} = 4 [ , ( )]
( 3) which is four times the covariance between a random variable X and its cumulative distribution function F(X). Equivalently, the above can be rewritten as: which equals the one third of the slope of the OLS regression curve of the dependent variable, as a function of the observation's positions in the array, F(X), having arrayed the observations in ascending order. Alternatively, it is the weighted average of the slopes defined between two adjacent observations.
There are two types of Gini regressions related to GMD. The first one, known as the Rregression (Hettmansperger (1984)), is based on the minimization of the GMD of the residuals. The second one, known as the semi-parametric approach (Olkin and Yitzhaki (1992)), imitates the OLS methodology by replacing the variance-based expressions by the equivalent GMD terms 6 . The combination of these two allows one to identify and test whether the implicit assumptions about the underlying distributions are supported by the data or not. Apart from that, the superiority of the Gini-based analyses also lies on the fact that Gini estimators are robust to the existence of extreme values or measurement errors and to the asymmetry of the distribution. Under that case, heavy tailed distributions can be used (Serfling (2010)). In addition, only the first moment conditions are needed for the Gini methodology to be implemented (Stuart and Ord (1987, p.58)). Focusing on the second approach, and assuming a simple regression, the population semiparametric Gini regression coefficient is based on the covariance presentation of the GMD and is obtained by replacing the covariance expressions in the OLS regression coefficient by the corresponding Gini covariances: where F(X)=R(X) represents the regressor's rank 7 . For the case of multiple Gini regression coefficients, a set of linear equations composed of simple Gini regression coefficients must be solved. Starting from the multiple regression model, expressed in population parameters: where the index 0 illustrates the dependent variable, and are the regression coefficients in the simple regressions of Xk on R(Xk) and 0 are the semi-parametric Gini regression coefficients as given in presentation (5). Rearranging, defining the following column vectors β0= (β01 , β02,…, β0Κ ), = ( 1 , 2 ,.., ) and provided that the rank of the matrix that consists of the βk's coefficients is K, it comes: where A -1 is a K x K matrix while the β's are K x 1 vectors. Imposing the set of restrictions, known as "orthogonality conditions", described by: the identities of (10) turn into: The previous expression shows the Gini estimator is a function of slope coefficients of semiparametric simple Gini regressions β0. Consequently, it is a semi-parametric Gini estimator. Since, most of the concepts and parameters in the Gini framework are parallel in structure to the OLS framework, it is natural to view presentation (5) as an OLS instrumental variable (IV), where F(X)=R(X) serves as an instrument 8 . According to Olkin and Yitzhaki (1992), when the model is given by: where Y is the dependent variable (N x 1 vector), X ≡ [ ] is the matrix of regressors (N x K with the first column of ones), is the vector of parameters to be estimated (K x 1) and ε is the vector of errors (N x1), the semi-parametric Gini regression yields an estimator of , where Rx = R(X).

Analyzing PISA DATA
PISA (or alternatively, Programme of International Student Assessment) is created by the Organization for Economic Co-operation and Development (OECD) to test the 15-year-old students' skills and knowledge in reading, mathematics and science. Through this procedure vital 8 Although the Gini regression framework can imitate the OLS concepts, they differ in interpretations and properties. Under the Gini IV analysis, the concept of IV is applied twice. As a first step, the sample's empirical cumulative distribution function replaces the variable, without questioning the validity of the rank to serve as an IV. As a next step, another variable that satisfies all the conditions from an IV perspective is required (Yitzhaki and Schechtman (2004)).
information is collected regarding students' ability to compete globally, to collaborate for problem solving, to think critically and creatively. What students know and can do, where all students can succeed and what school life means for students' lives are the three questions that constitute the core of this triennial international education survey. Seventy-nine countries and economies and a sample of 600,000 students among 32 million in total, participate in a two-hour test carried out every three years 9 . Only students between the age of 15 years and 3 months and 16 years and 2 months can engage in this survey, while the choice of schools and students is as broad as possible so that a wider range of different educational backgrounds and abilities to be covered. The PISA test's content is based on the curricula found across the world, without promoting or imposing anyone of those and neither there is a need to find similar characteristics between them. The goal is to assess countries' performance by providing scores for each subject area while the mean score in each subject can be used to rank the participating economies. Since the test is not designed to evaluate individual students, a considerable number of alternative test forms, covering all aspects of test material, exists and gives the opportunity to obtain a much greater coverage of the content 10 . The results, in each test subject, are scaled to follow normal distributions, with mean around 500 score points and standard deviations around 100 score points for OECD countries. Due to the fact that only a sample of students from each country is tested, the estimates are accompanied with statistical uncertainty, meaning that it is impossible to assign an exact rank to each participating country, based on its mean score, but rather to place it within a range of positions (that is, between an upper and a lower rank).
To allow comparisons among countries and to conclude whether performance for each country is improved or not, PISA scores are reported at the same scale over time. Under these circumstances, both year-to-year comparisons are feasible and average trends over three or more assessment years can be calculated. Having this information, PISA identifies effective policies and practices that are implemented in economies that record high performance, or in economies that show significant improvement over time. It further, reinforces the participating members that are willing to engage in similar programs, acknowledging that there is no one education model that fits all members, since different economies share different characteristics.
PISA is established and enforced under the authority of education ministries through the PISA Governing Board, the PISA's decision-making body. All member countries plus partner countries with Associate status elect representatives to the Board, who are a mix of government officials and staff of research and academic institutions. The Board regulates the PISA's policy priorities and standards for data development, analysis and reporting, and supervises their implementation. PISA's financial support is exclusively derived from direct contributions from the participating economies' education ministries 11 .
The PISA database has been widely applied in many different studies since it provides the huge advantage to allow for cross-national comparisons of student performances. For instance, Jerrim, Oliver and Sims (2019) use the 2015 wave PISA data for England to investigate the relationship between students' achievement and the inquiry-based science teaching methodology. Zheng, Tucker-Drob and Briley (2018) examine how the family's and school's economic resources as well as the resources found at national level affect the association between science interest and science knowledge, using the 2015 PISA data. Applying the same dataset, S. Cumberworth and E. Cumberworth (2018) reveal if school socioeconomic composition is more strongly associated with standardized test scores among Lower-Socioeconomic Status students than it is among Higher-Socioeconomic Status students. Based on PISA database 2015, Tang (2018) discloses that immigrant adolescents face lower life satisfaction compared to their non-immigrant counterparts, however this gap declines when specific school and family factors are taken into consideration. Giambona and Porcu (2018) explain how Italian students' achievement is affected by school size, using PISA 2012 data. Yang and Ham (2017), adopting the same data, demonstrate how a well-established anti-discrimination legislation weakens the relationship between first-and second-generation immigrant students and school truancy. Sholderer (2017) utilize PISA 2015 database to explore the impact of social capital on the association between school autonomy and its performance. To inspect how intelligence (IQ) relates to happiness inequality and crime rates, Nikolaev and Salahodjaev (2016) and Burhan et. al. (2014), respectively, use two different databases whose common denominator is that the construction of IQ variable is based on distinct sources including PISA database. E. Erdogdu and F.Edrogdu (2015) show how home and school environment, access to ICT and student background influence students' academic achievements.
The present study is based on PISA 2015 data, when science was the subject of focus, and uses country level analysis. In this most recent cycle, the 2015 wave of PISA test, 72 participating countries were included for regional comparisons, but some countries are dropped in later analysis due to lacking data. Table 2 in Appendix lists all countries. The variable of interest is proxied by the students' performance in science. It is expressed in terms of mean score. Indicatively, students' performance in science ranges from 331,639 to 555,575 score points. The available number of variables in this dataset that can be used as regressors are 167 in total and are classified into 14 categories. However, due to the limited number of observations, Principal Component Analysis is applied to reduce the dimensionality of the dataset, while retaining as much as possible of the variation present in it. The result is a new transformed set of variables, 43 in total, named as principal components, which are uncorrelated, and which are ordered so the first few retain most of the variation present in all the original variables (Jolliffe, 2002). Descriptive statistics for all the OECD variables used in this study are presented in Table 3 in the Appendix.

Principal Component Analysis (PCA) -Technique
The derivation of Principal Components (PCs) is based on the eigenvectors and eigenvalues of the covariance and/or correlation matrix. An important drawback of PCA based on covariance matrix is that PCs are sensitive to the units of measurement used for the regressors. Thus, before applying the dimensional reduction procedure for multivariate data, all the regressors are standardized because they come with different units. Table 4 in Appendix provides details regarding the new transformed set of variables, namely the principal components. Finally, to regress the dependent variables on this new set of regressors, all the dependent variables have been centered.

Empirical Results
Gini-BMA approach is employed to investigate how pre-primary education and other determinants affect students' performance, by calculating the weighted average of model specific estimates using Gini estimates 12 . The weights attached to each estimate is identical to posterior model probabilities. As a baseline estimation, a universe of all potential models using 43 covariates is taken into consideration. The results are compared to the OLS ones, taken by default when applying BMA methodology. Being totally agnostic about whether any of these regressors is included in the true model, a prior probability of 0.5 is attached to each one, implying a uniform model prior. Regarding the parameter space, the unit information prior (UIP) is adopted, where the integrated likelihood is proxied by the Schwarz Information Criterion (SIC). Table 5 in Appendix displays the findings for the BMA analysis for the students' performance in science. The first column shows the posterior inclusion probability that each of the covariates is included in the truth model, while the second and the third columns present the BMA unconditional posterior mean (PSE) and posterior standard deviation (PSD) for each regressor. The remaining three columns provide, respectively, the same amount of information for the OLS case and for the other dependent variables. A covariate is identified as a "robust" determinant if the posterior inclusion probability exceeds 50% 13 .
Referring to the first panel, of the 43 potential/promising candidate regressors only 13 affect the students' performance in science under the Gini analysis. The results grounded in Gini coefficients, suggest that "attendance in pre-primary education" (i.e., PC11) is a robust determinant of students' performance in science, with a posterior inclusion probability 81.6%, but with an adverse effect on students' performance. However, this apparent negative result conceals the significant variation in years of pre-primary education. That is strongly reinforced by the finding that the "percentage of students who had not attended pre-primary education" (i.e., EducOut2) enters with high posterior inclusion probability, 73%, and its marginal effect is substantial as well: on average, an one percent increase in no-attendance in pre-primary education reduces the performance by 10.45 score points. In other words, the apparent negative result of PC11 found earlier reflects the significant trade -offs that may exist between entering pre-primary education at a very early age and missing out on parental care at these very early ages, while it is clear that some pre-primary education is crucial and important, but not at all costs. There is a broad opinion in the sense that early early childhood education interventions provide a cognitively simulating environment that enhance school readiness, academic performance, social integration and long-term skill development (Myers (1995), Entwisle and Alexander (1993), Waldfogel (2002), Brooks-Gunn (2003), Carniero and Heckman (2003)). However, along with the early childhood interventions, many studies have found that home conditions are another crucial determinant of child's educational achievement (Bjorklund and Salvanes,2011). Both Velez et.al (1993) and Wößmann (2005) agree that, apart from preschool attendance, parental involvement and family features are key components in students' performance. A child's development begins within the family and depends on the parents' educational and cultural levels (Wößmann, 2005). Waldfogel and Washbrook (2011, and press b) support that parents that are educated and receive high income, spend more time to prepare their children's reading skills. In contrast, parents with lower income and less education have more possibilities to engage in harsh and incompatible parenting teaching behaviours that may negatively affect child's progress. Becker (1981Becker ( , 1985 and Becker and Tomes (1986) embrace the theory of family to provide a reasonable justification for the failure of preschool education. Furthermore, the "percentage of students who had repeated a grade in primary, lower secondary or upper secondary school" (i.e., EducOut1), enters with a posterior inclusion probability, 53.8%, but plays a positive role on the students' performance: increasing EducOut1 by 1% raises the probability that the performance will increase by 4.39 score points. The optimistic results about the effectiveness of grade retention can be supported by other studies (Alexander et. al. (1994), Karweit (1999), Lorence, Dworkin, Toenjes and Hill (2002), Greene and Winters (2004), Jacobs and Lefgren (2004), Eide and Goldhaber (2005), Lorence and Dworkin (2006)).
Additional robust determinants are also identified for students' performance in science. "Difference in science performance between immigrant and non-immigrant students" (i.e., effect, if 99%<PIP<100% there is decisive evidence for the effect. These cut-offs form an approximation and are not based strictly in statistical theory. PC110b) and the "mean ratio between students and classroom characteristics" (i.e., PC25a) affect performance distinctly. Both appear with a posterior inclusion probability above 90%, (94% and 93% respectively) but with opposite impacts. While an increase in the PC110b coefficient adversely influences students' performance, an increase in the mean ratio is highly beneficial. "Expectations to work in science-related professional and technical occupations at age 30" (i.e., PC23a) exhibit important and negative effects on students' performance, with a posterior inclusion probability and a posterior mean equal to 78.8% and -11.63 respectively. In contrast, the "relative risk of boys expecting to work in science-related professional and technical occupations at age 30" (i.e., PC33b) enters with an inclusion probability 77% and with a considerable positive coefficient equal to 18.04.
The "number of students who are evaluated and assessed in alternative ways" (i.e., PC44), the "number of students with or without an immigrant background "(i.e., PC110a), the "number of science teachers who are qualified to teach science" (i.e., PC111a) and the "average time spent per week learning in regular science and no-science lessons" (i.e., PC25b), are four variables that are positively effective in performance, appearing with an posterior inclusion probability that ranges between 61% and 68% and posterior means that take values from 4.66 and 7.40. Although PC25b enters positively, the average time spent after school in studying science and no-science lessons (i.e., PC17) has a negative impact and a posterior mean equal to 5.87. Surprisingly, the third Principal Component that refers to the category of the "number of students who are evaluated and assessed in alternative ways" (i.e., PC34) plays a negative role: it enters with an inclusion probability of 56%, and a posterior mean equal to -4.73.
A comparison between the Gini-MA results and the OLS-MA ones, presented in the second panel, suggests that the determinants that are important under Gini analysis are not necessarily similar to the ones that are important under OLS analysis. Of the 43 potential/promising candidate regressors only 10 affect the students' performance in science under OLS. The "number of students who refer skipping/arriving late at classes two weeks prior to the test" (i.e., PC26c), and the "difference in science performance" (i.e., PC39c) appear, under this case, to affect performance. Entering with posterior inclusion probabilities 78% and 71%, respectively, with PC26c appear to have a negative effect, while PC39c appears to have a positive one. The "percentage of students who had not attend pre-primary education" (i.e., EducOut2) continues having a negative impact, although smaller (8.75 score points), with an inclusion probability of 84%. This is similar to the findings of the Gini-MA analysis, yet an important difference is that the effect of pre-primary education is not important with an inclusion probability less than 50% (0.31%).The case of "attendance in pre-primary education" (i.e., PC11) with the Gini-MA analysis was found to be strongly robust with a negative effect on students' performance and as was argued that apparent negative result may conceal the significant variation in years of pre-primary education and the potential benefits and costs between too many or too few years of pre-primary education. However, this is not the case for the "percentage of students who had repeated a grade in primary, lower secondary or upper secondary school" (i.e., EducOut1), which now appears with a negative posterior mean equal to -10.257 and much higher inclusion probability,75%. This is the most important difference so far since it implies that repeating a grade does not have a beneficial effect on performance.
The "number of students with or without an immigrant background" (i.e., PC110a) and the "mean ratio between students and classroom characteristics" (i.e., PC15a) enjoy strong posterior support for being important explanations for students' performance in science. Both receive an inclusion probability above 90%, and their posterior means are of the same magnitude but of opposite signs: the PC15a coefficient is positive and equal to 16.629, while the PC110a coefficient is negative and equal to 16.699. It certainly appears to be true that "expectations to work in sciencerelated professional and technical occupations at age 30" (i.e., PC13a), the "relative risk of boys expecting to work in science-related professional and technical occupations at age 30" (i.e., PC23b) and "average time spent per week learning in regular science and no-science lessons" (i.e., PC15b) contribute to science performance. The results found using Gini-MA approach are confirmed, since now, these variables enter with higher inclusion probabilities, 89%, 86% and 75% and with posterior means equal to -14.292, 22.123 and 6.324 respectively. In contrast, the "number of students who are evaluated and assessed in alternative ways" (i.e., PC34) seems to positively affect performance and enters with an inclusion probability equal to 57% and a posterior mean equal to 3.87. Table 6 summarizes all the robust determinants under the Gini and OLS analysis, respectively. The first four regressors (i.e., the "number of students with or without an immigrant background" (i.e., PC110a), the "number of students who are evaluated and assessed in alternative ways" (i.e., PC34), the "percentage of students who had repeated a grade in primary, lower secondary or upper secondary school" (i.e., EducOut1), "percentage of students who had not attended preprimary education" (i.e., EducOut2)) appear to be significant under both analyses. Only the last variable retains the negative coefficient, while for the rest, the sign changes between the two cases. Regarding the next four variables (i.e. "mean ratio between students and classroom characteristics" (i.e., PC25a, PC15a), "average time spent per week learning in regular science and no-science lessons" (i.e., PC25b, PC15b), "expectations to work in science-related professional and technical occupations at age 30" (i.e., PC23a, PC13a), "relative risk of boys expecting to work in science-related professional and technical occupations at age 30" (i.e., PC33b, PC23b)), all appear to retain their sign but each variable refers to a different principal component, when comparing between the two analyses. The rest five variables that appear in the second column, are robust determinants only under the Gini analysis (i.e., "difference in science performance between immigrant and non-immigrant students" (i.e., PC110b), "attendance in preprimary education" (i.e., PC11), "average time spent after school in studying science and noscience lessons" (i.e., PC17), "number of students who are evaluated and assessed in alternative ways" (i.e., PC44), "number of science teachers who are qualified to teach science" (i.e., PC111a)), while the rest two variables that appear in the fourth column are robust determinants only under the OLS analysis (i.e., "difference in science performance" (i.e., PC39c), "number of students who refer skipping/arriving late at classes two weeks prior to the test" (i.e., PC26c)).
Image plots in figures 1a. and 1b. in Appendix demonstrate the sign and the importance of the regressors in the universe of models. These graphical representations highlight how the estimated coefficients fluctuate for the top 100 models shown in the horizontal axis, scaled by their posterior model probability. Blue and red colors indicate the inclusion of the regressor with a positive and negative posterior mean, respectively. White color indicates non-inclusion (or a zero coefficient). Robust covariates retain the same sign pretty much throughout the model space.
The top panels of figures 2a. and 2b. in Appendix provide the distribution of model sizes for the baseline exercise for Students' performance in science. With 2 K possible variable combinations, a uniform model prior implies a common prior model probability of 21.5. However, the graphs show a posterior model size distribution equal to 19.3983 and 19.0188, for the Gini and OLS case, respectively. The low panels of figures 2a. and 2b. demonstrate the best 100 models encountered, ordered by their analytical posterior inclusion probability (red line) and plot their MCMC iteration counts (blue line) 14 . At 0.9308 and 0.83, these correlations are far from perfect but the differences from an exact likelihood approach are practically indiscernible and already indicate a satisfactory rate of convergence.

Conclusions
This paper identifies the robust determinants of students' performance in science. To ensure a comprehensive search, the analysis accounts for a rich set of possible regressors by employing Gini-BMA methodology. This approach constructs estimates that do not depend on a particular model specification, but rather they are conditional on the model space. A weighted average of Gini and OLS coefficients are calculated, respectively, where the weights are given by the posterior model probabilities.
Once model uncertainty is accounted for, the results can be summarized as follows. First, of the 43-promising candidate regressors only 13 affect the students' performance in science under the Gini analysis, while only 10 under OLS. Among the factors that are robust under both cases are: the "percentage of students who had not attend pre-primary education", the "percentage of students who had repeated a grade in primary, lower secondary or upper secondary school", the "mean ratio between students and classroom characteristics", the "expectations to work in science-related professional and technical occupations at age 30", the "relative risk of boys expecting to work in science-related professional and technical occupations at age 30", the "average time spent after school in studying science and no-science lessons" and the "number of students who are evaluated and assessed in alternative ways". However, in most cases, differences are observed in the signs of the coefficients.
Second, only under the OLS case, the "number of students who refer skipping/arriving late at classes two weeks prior to the test", the "difference in science performance" and the "number of students with or without an immigrant background" are considered important for the performance in science. In contrast, only under the Gini case, the "number of students who are evaluated and assessed in alternative ways", the "number of students with or without an immigrant background", the "number of science teachers who are qualified to teach science", the "difference in science performance between immigrant and non-immigrant students", the "average time spent per week learning in regular science and no-science lessons", and the "attendance in pre-primary education" affect performance.

Appendix A-Model Priors
Beginning with considerations for choosing model priors, p(M1),….,p(Mk), the most common approach is an non-informative prior which favors all candidate models equally. For a model with p independently included regressors and size Ξ, the model size follows a Binomial distribution with probability of success ξ: where p is the total number of candidate regressors and ξ is the prior inclusion probability for each variable. Based on the above, the prior of a model Mk with pk regressors is described by: Raftery (1988), Raftery et al. (1997), Fernandez et al. (2001 a,b) and George and McCulloch (1993), fix ξ to equal 0.5 so that every regressor has the same a priory probability 15 . This leads to the uniform model prior, which can be considered as a benchmark, that assigns equal prior probability to all models, implying that p(Mk)= 2 − for each p and that expected model size is p/2. Mitchell and Beauchamp (1988) introduced the more general model prior structure, when prior information about the relevance of a variable is applicable, namely: where ε [0 ,1] is the prior probability that variable Xj is included in the model and =1 if Xj is included in Mk and 0 otherwise. Usually, it is assumed that πj = π for j= 1,….,p. results. They pre-specify a prior mean model size, kbar, implying that each variable has a prior probability of inclusion equal to kbar/K, with K being the total number of potential regressors. As a special case, when kbar=K/2, equal probability is assigned to each possible model. corresponds to a uniform prior across model space, while π<0.5 imposes a penalty for large models. Assigning =1, guarantees the inclusion of variable j in all candidate models 16 . An extension on this approach was proposed by Brown et al (1998; and Ley and Steel (2009), who assign a hyperprior on the probability of inclusion, π, converting it into a random variable drawn from a Beta distribution 17 .
When little information is available about the relative validity of the candidate models, assuming independent inclusion of regressors a priori seems a "neutral" choice 18 . However, it might be quarrelsome in some circumstances (Chipman et al. (2001)). The uniform model prior does not take into consideration interrelations between different variables, replicating a problem comparable to the irrelevance of independent alternatives (IIA) in the discrete-choice literature 19 . When the goal is to evaluate the relative significance of distinct theories and to define noninformative model priors across theories, the uniform prior is inappropriate, since the researcher can change the prior weights across theories simply by introducing "redundant" proxy variables for each theory. George (1999b) proposed a dilution prior as a solution to the interrelations between variables. If the set of candidate regressors includes variables that represent the same concept, George's dilution prior increases the prior probabilities of models not containing these correlated predictors. However, this is not always the case, since variables are often measures of different ideas but are still correlated. Under this condition, the straightforward use of this prior penalizes larger models.
To deal with the interdependencies across theories due to the addition of "redundant" variables, Durlauf, Kourtellos and Tan (2007;2012) choose the prior probability that a particular theorydefined as the set of variables that are used as proxies for that theory -is included in the true model to equal 0.5. This assumption captures the non-informativeness (i.e. agnosticism) across theories but also ensures that the probability of inclusion of one theory in a model does not exclude other theories from being relevant. The question now is how to assign prior probabilities across the set of variables within each theory. To answer this, specification uncertainty should be taken into consideration. This problem is related to existence of correlations between potentially unrelated proxy regressors within theories. To handle this, they introduce a modified version of George's (1999b) dilution prior. Selecting a theory T, with pT regressors, as a priori proper, they construct a binary vector γT for each possible combination of these pT proxies. The conditional prior probability assigned to each γT is given by: 16 George and McCulloch (1993), Volinsky et al. (1997), and Madigan and Raftery (1994) apply this approach in the context of linear regressions, Cox models, and graphical models, respectively. 17 In that case, and according to Bernardo and Smith (1994, p.117), the prior on model size is a Binomial-Beta distribution. 18 Both the Binomial and the Binomial-Beta priors are based on this assumption.
where is the prior inclusion probability of each proxy variable in theory T, which is equal to =0.5 for j = 1,….,pT and | | is the correlation matrix for the set of variables included in the binary vector γT. When regressors are collinear, | | takes the value of zero, whereas when regressors are orthogonal, it is equal to 1. This structure penalizes models that include irrelevant variables and retains weights on informative models. A similar approach can be found in Brock and Durlauf (2001) and Brock et al. (2003) who focus on economic theories rather than individual regressors and use hierarchical tree structure to construct model priors. A related idea was expressed by Chipman et al. (2001), who assigns probability to neighborhoods of similar models. More recently, George (2010) develops dilution model priors classified in three distinct approaches: the tessellation defined dilution priors 20 , the collinearity adjusted ones and the model distance based. The key characteristic of these is to assign prior probabilities more uniformly across neighborhoods of models rather across models 21 .
Despite it seems a sufficient answer to the dilution property, this prior structure obliges a decision on which proxies are classified under a specific theory and which models belong to the same neighborhood. Such decisions are not within reach in most of the cases.

Parameter priors/Prior distributions of parameters
In the context of Bayesian framework, to complete the Normal linear regression model described in (2), a prior distribution for the parameters θk= (β0, β (k) , σ) is needed. This distribution will be given through a density function: and affects right away the posterior model probabilities p (Mk | D). Two challenging questions arise regarding the computation of p (Mk | D) and the influence of the assumptions made for prior distributions on the latter quantity. Analytical answer to the first one is provided in the next section. Apart from purely computational features, the choice of "rational" prior parameter distributions remains unresolved and depends mainly on the availability of prior information. When information about the parameters is given, informative priors can be constructed (e.g. Jackman and Western 20 Moser and Hofmarcher (2014) provide an extended analysis on the implementation of this approach. 21 Another promising approach to dilution prior construction is suggested by Garthwaite and Mubwandarikwa (2010), who construct prior model weights using the correlation matrix between models. This matrix reflects the similarities between models and assigns small weights to those who are highly correlated.
(1994)). However, under little or absence of prior information, choosing a distribution/density for (13) is a very complex task. Consequently, many efforts have been made to establish "default priors" or "reference priors" that can be applied in all such cases. Following Fernandez et al. (2001b), "non-informative" improper priors have been adopted for the common intercept and the scale σ, such

Informative priors for the regression coefficients
In general, direct use of improper noninformative priors for model-specific parameters is not allowable because their arbitrary norming constants remain in the integrated likelihoods and lead to uncertain model probabilities 22 (Jeffreys (1961); Berger and Pericchi (2001)). Conventional proper priors for regression coefficients have been relied on the natural-conjugate approach, which assigns a conditional normal prior on the k-th model's parameter (β (k) | σ 2 ) with zero mean and the variance proposed by Zellner (1986), leading to the following prior distribution (Fernandez et al. (2001b)): ( , ( ) , | ) ∝ −1 ( ( ) | 0, 2 ( ′ ) −1 ) (A.9) where (w | m, V) denotes the density function of a q-dimensional Normal-distribution of w with mean m and covariance matrix V, and g is a scalar that measures how important are the prior beliefs of the researcher 23 . The above prior distribution allows for exclusion of regressors from some of the models, represented by a prior point mass at zero (known as the "spike-and-slab" approach in Mitchell and Beauchamp (1988)).
Efficiently, the thorny problem of picking a prior distribution for β can be solved only by selecting a single parameter g. Under this "benchmark" prior structure, incorporating subjective prior knowledge into the analysis is not feasible or desirable, resulting in little influence on 22 To overcome this problem, Pericchi (1996) andO'Hagan (1995) apply intrinsic Bayes factors and fractional Bayes factors, respectively, but their approaches suffer from inconsistencies. 23 A large value of g implies a small true model, that is, many of the regressors equal zero, while a small g supports the existence of a large model. When g→0, the β estimator is the Least Squares estimator of the full/ "kitchen sink" model (George and Foster (2000)-Calibration and empirical Bayes variable selection). posterior inference. These "automatic" priors depend only on the number of independent variables and the sample size (Fernandez et al. (2001b)). In the same spirit, Wasserman (1995 andRaftery (1995) recommend "unit information" priors, which contain the same amount of information as a regular single observation. In contrast, Raftery et al. (1997) display "weakly informative" proper priors, which are data dependent through the response variable 24 .
To demonstrate the behavior of several notorious priors that belong to the above-mentioned categories (refer to the priors 1-11 and 14, presented in Table1 in Appendix), Eicher et all. (2007) compare their predictive performance by employing growth and simulated data and conclude that Unit information Prior in combination with uniform model prior surpass all the rest. In contrast, Ley and Steel (2009) recommend the avoidance of UIP in the context of growth regressions or under the presence of large number of potential regressors, proposing instead the use of prior g=1/k 2 (combined with the assumption that the inclusion of each regressor is independent and equal to 0.5).
According to Liang et al. (2008), when fixed g priors are used to construct Bayes factors, the result might suffer from the Bartlett's and the Information paradoxes. To overcome this complication, they investigate fully Bayes approaches and suggest three alternatives: Global and Local empirical Bayes procedures, the multivariate Zellner-Siow Cauchy priors (initially introduced by Zellner and Siow (1980)) and a family of prior-probabilities imposed on g 25 . Because this hyper-g prior family for g lacks model selection consistency, they provide a modification, known as hyper-g/n prior family 26 . In the same strand of literature, Ley and Steel (2012) attach a proper hyperprior to g, which corresponds to a shrinkage factor δ = g/(1+g) that follows a Beta prior distribution. This "benchmark" Beta prior with c=0.1 and a hyper-g/n prior with α=3 (and α=4), is compared with existing priors 27 in terms of model selection consistency, avoidance of information paradox and empirical behavior/performance. Feldkircher and Zeugner (2009) finalize the analysis presented by Liang et al. (2008), by adding the posterior distribution of β|y,X, its second moments, and the second moment of the shrinkage factor. Regarding the computation of these posterior expressions, they use algebraic transformations and implement accurate statistics to overcome possible errors that occur when Laplace approximations are applied. In a simulation study with noisy data, they show that hyper g-24 Tobias and Li (2004) apply this prior choosing the following values for the parameters: V=2.85 2 Ik, λ=0.28, v=2.58, μβ= 0. 25 Strawderman (1971)-Proper Bayes minimax estimators of the multivariate normal mean-and Cui and George (2007)-Empirical Bayes vs Fully Bayes variable selection-study priors that belong to this family of prior probabilities for g. 26 According to Liang et al. (2008), the three alternative solutions resolve the information paradox and are consistent under prediction. However, only Zellner-Siow priors are consistent for model selection. 27 These are Zellner and Siow (1980), Maruyama-George (2011), Bottolo-Richardson (2008) prior spreads the posterior mass more evenly among the candidate models compared to the "Benchmark prior" (FLS (2001b)). A completely different approach, known as the Bayesian Averaging of Classical Estimates (BACE) methodology, is given by Sala-i-Martin et al. (2004), whose analysis is not based on gparameter priors, but rather information is extracted from the data and the final estimates are the result of averaging OLS estimates across models. The weights assigned to each model is the logarithm of the likelihood function, agreeing to Schwarz model selection criterion 28 . 28 Although Sala-i-Martin et al. support that "BACE limits the effect of prior information", it is important to mention that even if prior assumptions are implicit now, this does not imply that the dependence on them becomes less important (Ley and Steel (2009)). Review of Economic Analysis forthcoming 13 (2021) www.RofEA.org  The log Bayes factor obtained under this prior behaves asymptotically like the Schwarz criterion. This prior leads to consistency.

Appendix C-Table
It assigns the same amount of information as a regular single observation. It is like "unit information prior" proposed by Kass and Wasserman (1995), but with zero mean instead of MLE.

2) =
The log Bayes factor obtained under this prior behaves asymptotically like the Schwarz criterion. This prior leads to consistency.
As more independent variables are added in the model, more information is attached to this prior (i.e. the discrete point mass at zero for β shrinks.)

3) = 1
The log Bayes factor obtained under this prior behaves asymptotically like the Schwarz criterion. This prior leads to consistency.
As more independent variables are added in the model, less information is attached to this prior.

4)
The penalty applied for selecting larger models is smaller compared to the Schwarz (BIC) criterion. This prior leads to consistency.
It is an in-between case of prior 1 and attributes smaller asymptotic penalty for selecting larger models.

5) = √
The penalty applied for selecting larger models is smaller compared to the Schwarz (BIC) criterion. This prior leads to consistency.
It is an in-between case of prior 2, more information is attached to this prior as the number of regressors increases (i.e. the discrete point mass at zero for β shrinks). This prior does not lead to consistency in general.
This natural conjugate prior structure is proposed by Laud and Ibrahim (1996)  This prior combines the consistent Bayes factors shaped under prior 1 with the remarkable small sample performance of prior 9.
It is a combination of prior 1 and prior 9 and is the most preferred prior for FLS (2001b) Unit Information prior (Kass and Wasserman (1995); Raftery (1995))
The posterior model probabilities (and thus the Bayes factors) are approximated by the Schwarz criterion (BIC). That is, log pr (D|Mk) ≈ c -1/2BICk where BICk = nlog(1-Rk 2 ) + pklog(n) c is a constant that does not change across models The g-UIP prior does not resolve the information paradox.
It assigns the same amount of information as a regular single observation.
The g-RIC prior does not resolve the information paradox.
It is related to Risk inflation Criterion (RIC) This prior is known as the "Benchmark prior" and it is called the "g-BRIC". Regarding predictive performance, it is the most favorite by FLS (2001b).
The g-BRIC prior does not resolve the information paradox.
It is a combination of prior 11 and prior 12.
Under the Zellner-Siow prior, there are not closed-form solutions for marginal likelihoods.
The Zellner-Siow prior resolves the information paradox and it is asymptotic consistent for prediction and model selection.
Usually in the literature, the shrinkage factor follows a Beta prior distribution Beta(b,c), leading the prior on g to follow a Gamma-Gamma distribution (Bernardo and Smith (1994), p. 120) or, alternatively, an inverted Beta distribution (Zellner (1971), p.375): This prior on g leads to the following prior on the regression coefficients: These are the multivariate Cauchy priors, introduced by Zellner and Siow (1980), where the prior on g follows an Inverse Gamma (1/2, n/2) distribution.
The posterior distribution of g under model Mk has a closed-form solution given by: The hyper-g prior derived from this family resolves the information paradox, under the constraint that 2< α ≤3 (when sample size is minimal). Also, it is asymptotic consistent for prediction but not for model selection.
This is a family of hyper-g priors proposed by Liang et al. (2008).
The shrinkage factor δ= g / (n +g) follows a Beta distribution Beta (b,c)= Beta (1 ,( a/2) -1 ) This family of hyper-g/n priors resolves the asymptotic inconsistency faced by the hyper-g priors in 16. This is a family of modified hyperg/n priors proposed by Liang et al. (2008).

18)
This g-prior is based on the Local Empirical Bayes approach.
This g-prior resolves the information paradox and is asymptotic consistent for prediction. However, it is not consistent for model selection.
A distinct g-prior is evaluated for each model. The estimate for g coincides with the maximum marginal likelihood estimate, under the restriction to be a positive number.

19)
This g-prior is based on the Global Empirical Bayes approach.
A common gprior is evaluated for all models. The estimate for g = >0 ∑ ( ) There is not closed-form solution for this g-prior since the marginal likelihood is not tractable. However, as suggested by George and Foster (2000), numerical optimization can be used as a solution. Liang et al. (2008) recommend an EM algorithm that provides a maximum likelihood estimation for g: Under this algorithm, Empirical Bayes posterior model probabilities can be attained at convergence.
This g-prior resolves the information paradox and is asymptotic consistent for prediction. However, it is not consistent for model selection. This g-prior is based on Fully Bayes approaches.
The variance is known.
To implement this approach, hyperpriors for two hyperparameters c and w should be defined.
c is a hyperparameter for the average size of the regression coefficients matrix for model γ (βγ).
w is a hyperparameter for the average number of nonzero coefficients in the model γ.
The term (1+c) for cϵ (0, ∞) follows an incomplete Gamma distribution (a, b), where a and b are hyperparameters, with default choices at b= 0 and a=1.
It is suggested by Cui and George (2007).

21)
where p≥6 and 0≤a<1 and if p=5 then 1/2≤a<1 These family of priors are used to provide proper Bayes minimax estimators when the mean of a multivariate normal distribution is estimated (under identity covariance matrix and a loss in squared errors).
The unconditional density for λ with respect to Lebesgue measure is given by − /(1 − ) for any a, 0≤a<1 It is introduced by Strawderman (1971).

24)
If the random shrinkage coefficient ki follows a Beta distribution Beta (b, c) It is introduced by Carvalho (2010) under the specification that b=c=1/2 then, the prior for the local shrinkage parameter λi is: and is known as the "Horseshoe" prior. He specifies the values for b, c = ½ such that the random shrinkage factor ki ~ Beta (1/2, 1/2) 25) The term 1/(1+g) follows a Beta distribution Be (α+1, z+1).

26)
The shrinkage factor follows a b=Beta distribution δ ~ Be (b, c). The term b=0 and by setting a= 2, the term c= a/2 -1= 0.
Following Ley and Steel (2012), Bottolo and Richardson (2008) adopt a hyper-g prior with a=2, but make it proper by truncating the right tail at max{n, k 2 }.

27)
They recommend the use of b=1 and c=1/2, while truncating g to be Bayarri, Berger, Garcia-Donato (2010).   Mean ratio defined between students and classroom characteristics.
Average time spent in learning (b) PC15b, PC25b PC15b is the first principal component (that explains most of the variation) and PC25b is the second in row principal component.
The average time spent after school in studying science and non-science lessons (expressed in hours).

6) Students' engagement, drive and beliefs
Benefits from science knowledge (a) PC16a, PC26a PC26a is the first principal component (that explains most of the variation) and PC26a is the second in row principal component.
Mean index of students who report that learning about science is beneficial in different ways.
Gender difference (boys-girls) in benefits from science knowledge (b)

PC16b
PC16b is the first principal component (that explains most of the variation).
Mean index of the gender difference (boys-girls) who report that learning about science is beneficial in different ways. Skip/arrive late in class (c) PC16c, PC26c PC16c is the first principal component (that explains most of the variation) and PC26c is the second in row principal component.
The number of students who refer skipping/arriving late at classes two weeks prior to the test.

Number of PC Description 7) After-School activities
Average time spent in studying after school PC17 PC17 is the first principal component (that explains most of the variation).
The average time per week spent after school in studying science and nonscience lessons (expressed in hours).

8) Access to ICT
Access to ICT ICT None Number of computers per student.

9) Performance and socio-economic status
Difference in science performance (c) PC19c, PC29c, PC39c PC13b is the first principal component (that explains most of the variation) and PC23b is the second in row principal component, PC33b is the third.
Difference in science performance associated with different reasons.
(expressed in score points).

10) Performance and Diversity
Students with an immigrant background (a)

PC110a
PC110a is the first principal component (that explains most of the variation) % of students with or without an immigrant background.
Difference in science performance between immigrant and nonimmigrant students (b)

PC110b
PC110b is the first principal component (that explains most of the variation).
Difference is science performance between immigrant and nonimmigrant students (expressed in scorepoints).

11) Resources for education
Resources for science course (a) PC111a, PC211a PC13b is the first principal component (that explains most of the variation) and PC23b is the second in row principal component.
% of science teachers who are qualified to teach science.

Shortage in resources (b) PC111b, PC211b
PC13b is the first principal component (that explains most of the variation) and PC23b is the second in row principal component.
Mean index of shortage in educational material.

12) Professional Development of teachers
Review of Economic Analysis forthcoming 13 (2021) www.RofEA.org

Number of PC Description Professional Development of Teachers
ProfDevTeach None % of teachers attended a programme of professional development in the previous three months.

13) School evaluation
School evaluation PC113, PC213 PC113 is the first principal component (that explains most of the variation) and PC213 is the second in row principal component.
% of students who use internal/external evaluation.

14) Governance
School autonomy (a) Governance1 None Mean index of school autonomy (% of tasks for which the schools have considerable responsibility) Area of residence as a criterion of admission

Governance2
None % of students in schools where residence in a particular area is always considered for admission to school Note: The values in yellow colour are above the 50% PIP and are determined as robust determinants

PC34 (-) PC34 (+)
Repetition of a grade EducOut1 (+) EducOut1 (-) Non-attendance in preprimary education Notes: There are two variables that refer to pre-primary education and are indicated with yellow color