The measurement of the impact of socioeconomic factors and school resources on student achievement is crucial for understanding educational success. A spatial approach that incorporates spatial econometrics is effective in uncovering hidden relationships and improving the accuracy of the análisis (Krisztin et al., 2020). The omission of spatial effects can lead to biased results, as important explanatory variables are ignored (Xiang, 2023). Therefore, integrating spatial econometrics into studies focused on the economics of education can provide valuable insights into how factors such as government support, innovation, economic development, and human capital influence academic performance (Liu and Long, 2021).

To analyze learning achievement within a spatial context, this study employs the Moran’s I index, the Getis-Ord local statistics, and the Spatial Autoregressive Model with Autoregressive Disturbances (SARAR) instead of traditional Multilevel or Hierarchical Linear Models (MLM/HLM). The MLM/HLM approach is designed to address data with hierarchical structures, such as the organization of students within schools and schools within districts. These models allow for the decomposition of total variance across different levels and estimate random effects that represent unobserved heterogeneity among groups (Arcaya et al., 2012; Asampana Asosega et al., 2024). However, a key assumption of hierarchical models is the independence of residuals across units, which may lead to the omission of spatial interaction effects in datasets where geographical proximity plays a relevant role in the observed variation.

In contrast, spatial econometric modeling techniques, such as the Moran’s I index and the Getis-Ord local statistics, focus on capturing spatial dependence, that is, the influence that nearby observations exert on one another due to their geographic proximity. Moran’s I measures global spatial autocorrelation, determining whether high or low values of a variable tend to cluster in space, while the Getis-Ord local statistic (Gi)* identifies spatial clusters and hot spots that reflect significant concentrations of specific values (Ernandes et al., 2024; Iacobini et al., 2025).

The SARAR model, in turn, incorporates spatial autoregressive components in both the dependent variable and the error terms, allowing for a more precise representation of how the outcomes and characteristics of one spatial unit may be influenced by those of neighboring units. This approach is particularly relevant in contexts where data exhibit a clear geographic structure, as the model explicitly accounts for interdependence and spillover effects between spatial units (Dong et al., 2015).

Disciplinarily, MLM/HLM models originate mainly from the social and behavioral sciences, whereas spatial analysis approaches are rooted in regional econometrics and geography (Bivand and Wong, 2018). Therefore, the use of the SARAR model is especially appropriate in educational research with a territorial component, as it allows for examining how geographic proximity influences educational outcomes and how spillover effects occur between districts or geographic areas (Dong and Harris, 2014; Tampah-Naah et al., 2019).

Moran’s I (Equation 1) Index measures the degree of spatial autocorrelation of a variable or of a model’s residuals. In other words, it assesses how similar the values are among neighboring spatial units. It is used to detect spatial clustering patterns in geographic phenomena such as education, health, poverty, and others.

I = Moran’s Index, n = number of observations, xᵢ = value of the variable at spatial unit i.

x̄ = mean of the variable x, wᵢⱼ = element (i, j) of the spatial weights matrix W, S 0 = ∑ i = 1 n ∑ j = 1 n w ij = total sum of spatial weights.

If I > 0: there is positive spatial clustering (similar areas are located near each other).

If I < 0: there is negative spatial clustering (dissimilar areas are neighbors).

If I ≈ 0: the spatial distribution is random.

Null hypothesis (H₀): There is no spatial autocorrelation in the residuals (the errors are i.i.d.—independent and identically distributed).

The Getis-Ord local statistic hot spots are areas that exhibit a higher concentration of a particular phenomenon compared to the surrounding areas. This statistical measure is used to identify clusters or hot spots of a specific phenomenon in a given geographic region. It is calculated by comparing the value of a particular location to the values of its neighboring locations and determining whether the concentration of high or low values is statistically significant. The local Getis-Ord statistic is computed using the Gi* index (Equation 2), which incorporates the attribute values of neighboring spatial units weighted by a spatial weights matrix:

In this expression, x j denotes the attribute value for unit j, w_i,j represents the spatial weight between units i and j, n is the total number of spatial units. The statistic is standardized using the global mean of the variable, which is calculated as shown in Equation 3: X ¯ = ∑ j = 1 n x j n (3) Similarly, the global standard deviation of the variable, required for the normalization of the Gi* statistic in Equation 2, is computed according to Equation 4: S = ∑ j = 1 n x j 2 n − X ¯ 2 (4)

Positive and statistically significant values of Gi* (Equation 2) indicate the presence of spatial clusters of high student achievement (hot spots), whereas significant negative values reveal clusters of low achievement (cold spots), allowing the identification of spatial inequalities in educational outcomes.

Spatial dependence refers to the relationship between the observed outcome of a dependent variable at one location and what happens at nearby locations (Areal et al., 2012). In contrast, spatial randomness occurs when there is no discernible spatial pattern in the data, meaning that the value observed in one spatial unit is equally likely to occur in any other spatial unit (Areal et al., 2012). In the context of student achievement, hot spot analysis can be used to identify areas where student achievement is significantly higher or lower than the surrounding areas. This information can then be used to target interventions and resources to areas with lower achievement to improve educational outcomes.

The analysis of hot spots on student achievement in mathematics was conducted using the Getis-Ord (Gi*) statistics (Ahmad et al., 2012). The Getis-Ord (Gi*) statistic calculates a z-score for each district on a map, which measures the intensity of clustering. Positive z-scores that are statistically significant indicate intense clusters of high scores (hot spots), while negative z-scores indicate intense clusters of low scores (cold spots). The districts with high z-scores in learning achievement form hot spots on the map, indicating a spatial dependency (Ahmad et al., 2012).

Hypothesis: Socioeconomic factors and school inputs significantly explain the variation in students’ on student achievement across districts, and this variation exhibits spatial dependence such that the performance and characteristics of neighboring districts influence the educational performance in one district.

The SARAR model is well-suited to test the hypothesis as it accounts for both the direct effects of socioeconomic and school input variables on mathematics achievement and the spatial dependence between neighboring districts. Unlike OLS, which ignores spatial structure, SARAR explicitly models spatial lag and autoregressive error components, capturing both local and spillover effects. Estimation via GS2SLS ensures consistent and efficient parameter estimates, even in the presence of spatial autocorrelation. The SARAR is a useful statistical tool for analyzing how on student achievement in mathematics in one district can influence those in neighboring districts, considering spatial dependence and autocorrelation in educational data across geographic areas (Poldrack et al., 2001). This model combines spatial autoregressive components with autoregressive disturbances, allowing for a detailed understanding of how educational performance in one area affects that of surrounding areas. Through spatial dependence and socioeconomic or institutional variables, the model reveals patterns and factors that contribute to the variability in educational outcomes. Its application allows for parameter estimation considering spatial lag effects and autoregressive errors, which helps design more effective educational policies based on spatial dynamics. Research can analyze the interdependence of student achievement and develop interventions that consider the influence of neighboring areas (LeSage and Pace, 2009; Yandell and Anselin, 1990). The SARAR model (Equations 5, 6) is an extension of the SAR model, which considers the spatial lag effects of the area approach (Poldrack et al., 2001):

Where:

y = (y₁,y₂,…,y_N)^⊤: N × 1 vector of average math scores for second graders in each district i = 1,…, N.

Then, the reduced form representation of the SARAR model (Equations 7, 8) is: y i = λ ( ∑ j = 1 n w ij y j ) + ( ∑ k = 1 8 x ki β k ) + ( ∑ k = 1 8 ( ∑ j = 1 n w ij x kj ) θ k ) + ε (7) ε i = ρ ( ∑ j = 1 n w ij ε j ) + u i (8)

y: Average math score of second graders.

λ ( ∑ j = 1 n w ij y j ) is the spatial lag of the dependent variable—the weighted average of neighboring districts’ math scores influencing district i. Captures the direct spatial effect of the mathematics scores.

( ∑ k = 2 9 x ki β k ) is the linear combination of explanatory variables for district i. Captures the direct effect of the explanatory variables.

( ∑ k = 2 9 ( ∑ j = 1 n w ij x kj ) θ k ) captures the spillover effect of the explanatory variables. i refers to the spatial unit (for example, a district) where the outcome (e.g., average math score) is being measured, j refers to other spatial units (neighboring districts) that influence unit ii through spatial interactions.

The spillover effect means that the explanatory variables in neighboring units j (such as socioeconomic index or school infrastructure in district j) affect the outcome in unit i. This effect is weighted by the spatial relationship w _ij between units i and j. So, the model not only considers how the variables in district i directly affect its math scores but also how the conditions in nearby districts j impact the math scores in district i. This interconnectedness is what the spillover effect captures.

W is the spatial autocorrelation matrix

The spatial weights matrix W is a fundamental component in spatial econometric modeling, as it defines the spatial structure of interdependence between districts by indicating which units are considered neighbors—typically through first-order contiguity, where w _ij = 1 if districts ii and jj share a border, and 0 otherwise. In this study, involving 1,675 districts in Peru, the use of a contiguity-based matrix is justified by the fact that neighboring districts often share infrastructure, policies, and educational resources, leading to spatial interactions that influence student outcomes. This matrix enables the model to capture both direct effects (within a district) and spillover or “telesación” effects (influence from neighboring districts), reflecting the reality that improvements or deficits in one district can affect others nearby. Ignoring such spatial dependencies—by using OLS or non-spatial panel models—would result in biased and inefficient estimates. Therefore, the SARAR model is the most appropriate choice, as it accounts for spatial lag and autoregressive disturbances, providing a robust framework for analyzing both observed and unobserved spatial effects in educational performance. Its use, combined with the spatial weights matrix, allows for a more accurate and policy-relevant understanding of how socioeconomic factors and school inputs shape on student achievement across space.

In a spatial model, the direct effect captures the immediate impact of an explanatory variable on the dependent variable within the same spatial unit. This means it reflects how a variable in a specific location directly influences the outcome in that location. Conversely, the indirect effect, or spillover effect, represents the influence of explanatory variables in neighboring locations on the dependent variable in the given location. It accounts for how interactions or relationships defined by the spatial weights matrix lead to effects that extend beyond the immediate location. In summary, while the direct effect shows the local impact, the indirect effect reveals how regional interactions contribute to the outcomes observed in each location.

The coefficients λ shows whether the observed outcome on the dependent variable at one place in space depends on what happens at other nearby places. The opposite of spatial dependence is spatial randomness, which means that we cannot observe any spatial pattern in the data, that is, the value that is observed in some spatial unit is equally likely as in any other spatial unit (Anselin, 1988). The coefficient of spatial autocorrelation of the error term ρ is done to see the existence of systematic errors in the measurement instead of reasons of economic theory (Anselin, 1988; Seya et al., 2020). That is, it shows whether the determinants of the dependent variable omitted from the model are spatially autocorrelated, or a situation where the unobserved shocks follow a spatial pattern. The value of a variable from a spatial unit that is close to other spatial units will tend to have a similar value in the omitted variables, which makes the error term autocorrelated spatial (Dubin, 1988).

In this study, three different estimation methods—Ordinary Least Squares (OLS), Maximum Likelihood (ML), and Generalized Spatial Two-Stage Least Squares (GS2SLS)—were employed to estimate the parameters of the SARAR model. Each method offers distinct advantages and helps validate the robustness of the results.

Ordinary Least Squares (OLS): OLS serves as a baseline estimation method due to its simplicity and widespread use in econometric analysis. While it ignores spatial dependence and autocorrelation, it provides an initial point of comparison to highlight the necessity of spatial modeling. OLS estimates are unbiased only when spatial effects are absent, which is rarely the case in geographic data. The spatial dependence of geographic data causes the violation of the assumptions of the ordinary least squares (LS) estimator, therefore, the LS estimator loses the efficiency property (Yildirim and Mert Kantar, 2020). If there is spatial autocorrelation in the dependent variable, the classical estimation by the OLS method generates biased and inefficient coefficients; that is, the size and sign of the coefficients are not close to their true value and their standard errors are underestimated. If there is autocorrelation, it is the error term, the estimation by the LS method produces unbiased but inefficient coefficients; that is, the size and sign of the coefficients are asymptotically correct but their standard errors are underestimated.

Maximum Likelihood (ML): ML estimation is well-suited for spatial models like SARAR because it explicitly incorporates spatial lag and autoregressive error structures into the likelihood function. This approach yields efficient and consistent parameter estimates under the assumption of normally distributed errors. ML allows for direct modeling of spatial dependence, providing insight into both direct and spillover effects with statistical rigor.

Generalized Spatial Two-Stage Least Squares (GS2SLS): GS2SLS is an instrumental variables approach designed to address potential endogeneity issues arising from spatial lag variables and spatially autocorrelated errors. Unlike ML, GS2SLS does not require strict distributional assumptions and remains consistent under weaker conditions. This method is particularly useful when spatial effects might be correlated with omitted variables or measurement errors, offering a robust alternative to ML. In this context, the Maximum Likelihood (ML) or Two-Stage Least Squares methods are suggested, which are consistent and efficient methods (Arraiz et al., 2010; Drukker et al., 2013; Kelejian and Prucha, 1999; Yildirim et al., 2020).

By employing all three methods, the study ensures comprehensive analysis: OLS highlights the importance of considering spatial effects, ML efficiently estimates spatial parameters assuming normality, and GS2SLS provides consistent estimates even under endogeneity and weaker assumptions. The consistency of results across ML and GS2SLS supports the reliability and robustness of the findings regarding the spatial dynamics influencing students’ mathematics achievement. For the SARAR (1,1) model, the Two-Stage Least Squares method, known as the Two-Stage Least Squares method for Autoregressive Spatial Models with Autoregressive Perturbations (GS2SLS) produces consistent estimates (Badinger and Egger, 2011; Drukker et al., 2013; Kelejian and Prucha, 1998, 1999; Lee, 2003).

The data refer to second-grade students in secondary education, based on the 2019 ECE (Evaluación Censal de Estudiantes) assessment. The data to estimate the econometric model on student achievement has as sources the Census Evaluation of Students 2019 of the second grade of secondary school (Minedu-UMC, 2019), the indicators of education in Peru corresponding to the year 2019 (Escale-Minedu, 2019), the Human Development Index published by the UNDP (Programa de las Naciones Unidas para el Desarrollo (PNUD), 2019) and the information for district planning: 2016–2021 published by CEPLAN (2022).

The information from the 2019 ECE (National Student Assessment), obtained through UMC, includes data on census-level students, such as gender and a socioeconomic index, which is calculated based on five indicators: parents’ years of schooling, housing materials (walls, floors, and roofs), access to basic services (electricity, water, and sewage), ownership of household assets, and access to other services (such as telephone and internet). Regarding the school, the census provides variables such as its location (urban or rural), the district, its management type (public or private), and the Local Educational Management Unit (UGEL) to which it belongs.

Sample Size. The study is based on a sample of approximately 1,675 districts, each containing data on average second-grade math performance, socioeconomic conditions, and school resources. Most variables include information for all 1,675 districts, except for the percentage of teachers with a professional title, which covers 1,687 districts. This extensive coverage provides a comprehensive and representative overview of the country’s educational and socioeconomic context, ensuring reliable statistical analysis and meaningful territorial comparisons.

Table 1 presents the key variables analyzed at the district level, including student performance, socioeconomic factors, human development indicators, and school inputs. Each variable is described concisely, highlighting its meaning and measurement. Sources are included for reference to the original data providers.

Sampling design. The study employs a cross-sectional design using secondary data aggregated at the district level. The data come from official sources covering most or all districts, effectively constituting a census or exhaustive territorial sample, without specifying a probabilistic sampling method such as random or stratified sampling. Importantly, the analysis includes only districts with complete data available for all variables considered. This approach allows for analysis based on aggregated averages and proportions per district, focusing on the relationships between socioeconomic factors, school resources, and academic performance within a territorial context.

To analyze the mathematics achievement of second-grade secondary students in Peru, a Hierarchical Multilevel Model (HLM) is applied, incorporating both fixed and random effects. This modeling strategy is adopted as a methodological approach to partially overcome the absence of school-level geo-indicator data, such as altitude or latitude, by capturing variation in student performance across nested educational structures. Although it does not directly estimate geographical effects, it allows for the identification of school- and district-level influences within the educational hierarchy.

The proposed model is structured into four levels: Level 1 (Students), Level 2 (Class Sections), Level 3 (Educational Institutions), and Level 4 (UGELs—Regional Educational Management Units). Each student’s mathematics score is modeled as a function of the mean achievement at each higher level, thereby decomposing the total variance in mathematics performance into components attributable to:

This hierarchical approach provides a more comprehensive understanding of how student achievement varies across educational contexts in Peru. By modeling the data structure in this way, the analysis distinguishes individual effects from institutional and regional effects, thus enhancing the robustness of inferences even in the absence of precise geographical indicators.

To formally capture both fixed and random components within the Peruvian educational system, the following null multilevel model is specified (Goldstein, 1986; Willms and Raudenbush, 1989).

The null multilevel model is synthesized in Equations 9–12.

Where: Level 2 : Class Section ( CS ) : γ 0 jkm = γ 00 km + u 0 jkm (10) Level 3 : Educational Institution ( EI ) : γ 00 km = γ 000 m + u 00 km (11) Level 4 : UGEL ( Regional Educational Management Unit ) : γ 000 m = γ 0000 + u 000 m (12)

Y ijlm : Mathematics score of student i in class j of educational institution k in UGEL m.

By substituting (10), (11), and (12), we have Equation 13: y ijkm = γ 0000 + u 000 m + u 00 km + u 0 jkm + ε ijkm (13)

The variance of the score in educational achievement is decomposed as the sum of the variance between UGELs ( σ UGEL 2 ), the variance between educational institutions ( σ IE 2 ), the variance between sections ( σ sec tion 2 ), and the variance between students ( σ student 2 ) (Equations 14, 15), that is: Var ( y ijkm = γ 0000 + u 000 m + u 00 km + u 0 jkm + ε ijkm ) = σ UGEL 2 + σ EI 2 + σ sec tion 2 + σ student 2 (14) Total Variance = σ UGEL 2 + σ IE 2 + σ Class 2 + σ Students 2 (15)

Where:

σ UGEL 2 is the variance between UGEL.

The Intraclass Correlation Coefficient (ICC) measures the proportion of variability in student performance that can be attributed to group-level factors, such as differences between regions (UGELs), institutions, or classes Equations 16–19. A high ICC at a specific level indicates that differences between those groups explain a significant portion of the variability, while a low ICC suggests that individual factors, such as student-specific characteristics, are more influential. The ICC can be calculated by dividing the variance due to the group level by the total variance, and the result ranges from 0 to 1. An ICC close to 1 indicates that group-level differences dominate the performance variability, while an ICC near 0 suggests that individual differences have more impact.

To interpret the ICC, we can express the variance explained at each level as a percentage of the total variance, which helps to better understand the influence of different levels. For instance, if the variance at the UGEL level accounts for 40% of the total variance, it indicates that regional differences significantly impact student performance. By examining the ICC values at different levels, we can gain insights into whether performance differences are driven by individual student factors or by structural differences at higher levels, such as the institution or regional education systems.

The ICC calculation for each level follows this formula:

The ICC for institutional level, class level, and student level can be calculated similarly, adjusting for the corresponding variances. ICC for UGEL level : IC C UGEL = σ UGEL 2 σ UGEL 2 + σ EI 2 + σ sec tion 2 + σ student 2 (17) ICC for EI ( Institutional ) level : IC C IE = σ UGEL 2 + σ EI 2 σ UGEL 2 + σ EI 2 + σ sec tion 2 + σ student 2 (18) ICC for Class level : IC C sec tion = σ UGEL 2 + σ EI 2 + σ class 2 σ UGEL 2 + σ EI 2 + σ sec tion 2 + σ student 2 (19)

	Variance of random effects		Variance of fixed effects (in %)
	Null	Conditional
Niveles	(1)	(2)	((2)–(1))/(1)
UGEL	σ UGEL 2	σ UGEL 2 ∗	( ( σ UGEL 2 − σ UGEL 2 ∗ ) / σ UGEL 2 ) ∗ 100
IE	σ IE 2	σ IE 2 ∗	( ( σ IE 2 − σ IE 2 ∗ ) / σ IE 2 ) ∗ 100
Section	σ sec tion 2	σ sec tion 2 ∗	( ( σ sec tion 2 − σ sec tion 2 ∗ ) / σ sec tion 2 ) ∗ 100
Student	σ student 2	σ student 2 ∗	( ( σ student 2 − σ student 2 ∗ ) / σ student 2 ) ∗ 100

Multilevel model with fixed and random effects.

Equation 5, when including the fixed effects (Equation 20), results in:

y : It is the mathematics score of student i in class j of educational institution k in UGEL m. Type of management: It is the management type (state/non-state) of class j of educational institution k in UGEL m. Gender: It is the gender of student i in class j of educational institution k in DRE m. Area: It is the geographical area (urban/rural) where class j of educational institution k in UGEL m is located. ISE. It is the socioeconomic index of student i who belongs to educational institution j in UFEL of DRE m. Type of management. It is the student’s native language. Region. It is a dichotomous variable that captures the effect of region m.

Data for hierarchical multilevel modeling. The data used in this analysis comes from the 2023 Student Sample Assessment, which is a nationwide evaluation designed to measure the learning achievements of students across various educational institutions. The assessment collected data on student performance in mathematics, as well as several demographic and contextual variables, such as school management type (public or private), socioeconomic background, geographic location (urban or rural), gender, and native language. This data serves as the basis for understanding the factors influencing educational outcomes and regional disparities in learning achievements. The assessment provides a comprehensive snapshot of the student population, enabling detailed analysis of performance variations and their underlying causes.

Table 2 presents a summary of the main variables used in the study on the average mathematics performance of second-grade students, along with socioeconomic and school input factors.

The dependent variable, the average math score of second graders, was calculated for 1,675 observations. The mean score is 543.98, with a minimum of 395 and a maximum of 791, showing substantial variation in student performance across districts.

Regarding socioeconomic factors, the Average Socioeconomic Index (ISE) has a mean of −0.69, ranging from −2.1 to 1.26, indicating a wide disparity in socioeconomic conditions. The percentage of mothers with higher education has a low mean of 2.07%, although it varies significantly (from 0 to 53.9%), suggesting substantial inequality in parental education levels. The Human Development Index (HDI) averages 0.41, with values ranging from 0.09 to 0.85, reflecting variations in human development among districts.

In terms of school inputs, public investment per capita in the district averages 45.27, ranging from 0 to 100, showing uneven distribution of financial resources. The percentage of public schools connected to the electricity grid has a high average of 85.18%, though some districts still lack electricity access (0%). The student–teacher ratio in secondary education averages 9.07, ranging from 1 to 29, while the student–computer ratio averages 7.54, but with an extreme upper value of 1,648, revealing large disparities in technological access. Lastly, the percentage of teachers with a professional title in secondary education shows a relatively high mean of 86.01%, although some districts report 0%.

This section explores the spatial distribution of standardized mathematics scores through global and local spatial autocorrelation analysis. Figure 3 presents the Moran scatterplot, which visually illustrates the relationship between each observation’s standardized score and the spatial lag of neighboring values. This helps identify clusters and outliers in spatial patterns.

Table 3 summarizes the Global Moran’s I statistic, which measures overall spatial autocorrelation in standardized math performance. The Moran’s I value of 0.404 suggests a moderate to strong positive spatial autocorrelation, indicating that areas with similar deviations from the average score tend to cluster geographically. The result is highly statistically significant (p < 0.001), confirming that the spatial distribution of math scores is not random.

The Moran’s I statistic for standardized mathematics scores (z_matematica) is 0.404, indicating a moderate to strong positive spatial autocorrelation. This means that areas with similar deviations from the average mathematics performance tend to be geographically clustered rather than randomly dispersed. The expected value under the null hypothesis (E[I]) is −0.001, assuming spatial randomness. The standard error (SE[I]) is 0.010, which estimates the variability of Moran’s I under random conditions. The resulting Z-score of 39.819 shows that the observed spatial pattern is far from what would be expected by chance. Finally, the p-value of 0.000 confirms that this result is highly statistically significant, providing strong evidence that the spatial distribution of standardized mathematics performance is not random but exhibits significant clustering.

To further investigate local patterns, Table 4 presents the Local Moran’s I summary, identifying specific types of spatial clusters. High-High and Low-Low clusters dominate, suggesting spatial concentration of high and low performance, respectively. Additionally, spatial outliers (High-Low and Low-High) are present but less frequent.

Null hypothesis (H₀): There is no spatial autocorrelation in the residuals (the errors are i.i.d.—independent and identically distributed).

Statistic	Value
Chi2 (1)	1,264.24
Prob > Chi2	0.0000
H₀: errors are i.i.d.	(no spatial autocorrelation)
Conclusion:	Reject H₀

The model’s residuals are not independent. There is strong evidence of spatial dependence—the residuals of neighboring districts are correlated. The Chi-square statistic (χ² = 1,264.24, p < 0.0001) leads to rejection of the null hypothesis of spatial independence, suggesting that residuals from the model are spatially autocorrelated—highlighting the importance of accounting for spatial effects in the analysis.

The model’s residuals are not independent. There is strong evidence of spatial dependence—the residuals of neighboring districts are correlated.

Getis-Ord (Gi*) statistics were used to identify hot spots on student achievement in mathematics in Peru. The analysis revealed distinct clusters of high scores in certain districts, indicating a spatial dependency. The results also highlight the regional disparities on student achievement across Perú, with coastal regions consistently performing better than jungle regions. According to the results, there are three main groups of hot spots in Peru. The first group is located in the districts of the Tacna, Moquegua, and Arequipa regions on the southern coast of Peru (Figure 4). The second group is found between the districts of Lima, Ica, and Junín. The third group is located in the districts of the Amazonas and San Martín departments. These hot spots represent districts with higher scores in student achievement and demonstrate the importance of geospatial proximity in educational policy and school improvement plans.

The use of Getis-Ord (Gi*) statistics to identify hot spots on student achievement can be informed by research on various factors that influence student performance. These factors include external factors such as disruptions to traditional student achievement, individual and family factors, teacher and school effectiveness, teacher training and preparation, and the composition of classrooms. The composition of classrooms can impact student achievement in mathematics. A study on inclusive classrooms found that there were few differences in the development of language and mathematics achievement between students in regular and special schools (Scharenberg et al., 2019). This suggests that inclusive educational settings can provide equal opportunities for student achievement.

Furthermore, research has shown that teacher and school effectiveness can play a role in student achievement in mathematics. A study on the differential effectiveness of teachers and schools found that classrooms and schools that are more effective in promoting on student achievement tend to have a smaller socioeconomic gap in mathematics achievement (Kyriakides et al., 2018). This suggests that effective teaching practices and supportive school environments can help mitigate achievement gaps. Teacher training and preparation also play a crucial role in student achievement in mathematics. A study on teacher preparation found that teachers’ education coursework, rather than additional preparation in mathematics and science, was positively correlated with student academic achievement in these subject areas (Gimbert et al., 2007). This highlights the importance of comprehensive teacher training programs that focus on pedagogy and instructional strategies.

Table 5 shows the results of the effect of spatial dependency between districts of Peru, the effect of socioeconomic factors, the effect of school inputs on student achievement. Three econometric methods were used: the Ordinary Least Squares (LS) method, the Maximum Likelihood (ML) Method, and the Two-Stage Least Squares method for Spatial Autoregressive Models with Autogressive Perturbations (GS2SLS).

In the Maximum Likelihood (ML) estimation, the spatial dependence coefficient of 0.03 indicates a positive influence of neighboring districts’ math scores on the current district’s scores, showing significant spatial correlation in educational performance. The average socioeconomic index (ISE) coefficient of 25.35 suggests a strong impact of socioeconomic conditions on math scores, with a 1-unit increase in ISE leading to a 25.35-unit increase in scores. The human development index (HDI) coefficient of 31.61 highlights a robust positive relationship between human development and math scores. The coefficient of 0.65 for the percentage of mothers with higher education underscores the importance of maternal education, with each percentage point increase resulting in a 0.65-unit increase in math scores. Public investment per capita has a small but positive effect of 0.0003 units per 1-unit increase, while the connectivity of public schools to the electricity grid contributes 0.21 units per percentage point increase. The ratio of students per teacher is positively correlated with math scores, where each unit decrease leads to a 1.06-unit increase in scores. Conversely, a higher student-to-computer ratio negatively impacts scores, with a decrease of 0.09 units per unit increase. The percentage of professionally qualified teachers positively affects scores by 0.15 units per percentage point increase. The spillover effect of the socioeconomic index (8.375) and public investment (0.003) shows that improvements in neighboring districts positively impact local math scores. The high spatial lag error coefficient of 0.94 reflects significant spatial autocorrelation in the errors.

In the Two-Stage Least Squares (GS2SLS) estimation, the results largely mirror those from ML. The spatial dependence coefficient of 0.03 confirms the positive spatial correlation, while the ISE coefficient of 25.16 and HDI coefficient of 33.34 reinforce the significant positive impacts of socioeconomic conditions and human development on math scores. The percentage of mothers with higher education has a similar effect as in ML, with a coefficient of 0.64. Public investment per capita maintains a modest positive impact, and school connectivity to the electricity grid shows a coefficient of 0.23. The student-to-teacher ratio has a negative coefficient of −1.11, confirming the benefit of smaller class sizes, while the student-to-computer ratio remains negatively associated with scores at −0.09. The percentage of teachers with a professional title positively affects scores with a coefficient of 0.16. Spillover effects for both socioeconomic index (8.375) and public investment (0.003) are consistent with ML results, indicating positive influences from neighboring districts. The spatial lag error coefficient of 0.94 further confirms significant spatial autocorrelation in errors, demonstrating the interconnected impact of unobserved factors across districts.

The spillover effect of socioeconomic level and public investment refers to how these factors in one district influence neighboring districts, affecting their educational outcomes. Wx (Average Socioeconomic Index of the Household) represents the spatial lag of the socioeconomic index in neighboring districts. The coefficients (4.58 in column 2 and 8.375 in column 3) suggest that socioeconomic conditions in surrounding districts significantly impact on student achievement in the district being studied. Wx (Public Investment per Capita) indicates the spatial lag of public investment in neighboring districts. The coefficients (0.001 in column 1 and 0.003 in column 2) show that public investment in nearby districts also positively affects educational outcomes in the target district. These coefficients and significance levels indicate that increased public investment in neighboring areas generally boosts the educational achievements of the district. Overall, the spillover effect demonstrates that socioeconomic status and public investment in one district can influence student achievement in neighboring districts, highlighting the interconnectedness between regions and the importance of considering these spatial dynamics when analyzing educational performance.

Table 5 summarizes the learning achievement in mathematics from a sample of 122,870 students assessed in 2023. The average score was 577.55, with a standard deviation of 73.17, indicating a moderate spread in performance. The lowest score recorded was 168.18, while the highest was 889.75, showing significant variation in student achievement. Overall, the data reflects a diverse range of student performances, with some scoring much higher and others much lower than the average.

Table 7 displays the distribution of students based on the management type of their educational institutions in the 2023 Student Sample Assessment. A significant majority of students, 101,547 (82.2%), attended public (state) institutions, while 21,991 students (17.8%) were enrolled in private (non-state) institutions. This indicates that the student sample is predominantly composed of those from public schools, with private institutions representing a smaller proportion of the overall sample.

Table 8 presents the distribution of students based on their socioeconomic level in the 2023 Student Sample Assessment. A total of 117,446 students were categorized into four levels: High, Low, Medium, and Very Low. The Very Low socioeconomic level had the highest frequency, with 46,728 students (39.79%), followed by the Low level with 29,552 students (25.16%). The Medium level comprised 25,517 students (21.73%), and the High level included 15,649 students (13.32%). The cumulative percent shows that by the time we reach the Medium level, 60.21% of the students have been accounted for, and by the Very Low level, the entire sample (100%) is included. This distribution highlights a predominance of students from lower socioeconomic backgrounds in the sample.

Table 9 shows the distribution of native languages spoken by students in the 2023 Student Sample Assessment. The majority of students, 109,810 (88.89%), identified Spanish (Castellano) as their native language, followed by 8,831 students (7.15%) who spoke Quechua. Smaller groups included 2,812 students (2.28%) who spoke other indigenous languages, 519 students (0.42%) who spoke Aymara, and 75 students (0.06%) with a foreign language as their native language. Additionally, 204 students (0.17%) identified as multilingual, and 1,287 students (1.04%) did not specify their native language. The cumulative percent indicates that 89.31% of students are Spanish speakers, with the entire sample covered by the Quechua and other indigenous languages, showcasing the significant diversity of native languages within the student population.

Table 10 shows the gender distribution of students in the 2023 Student Sample Assessment. The sample is almost evenly split, with 62,201 male students (50.35%) and 61,337 female students (49.65%). This indicates a near equal representation of both genders within the sample, reflecting a balanced gender distribution in the assessment.

Table 11 presents the distribution of students based on their geographic area, showing that 27.04% of the students come from rural areas, totaling 33,407 students. In contrast, the remaining 72.96%, or 90,131 students, are from urban areas. This distribution brings the total number of students to 123,538, with 100% cumulative representation. The data is sourced from the Student Sample Assessment 2023.

Table 12 presents the effect of private management on learning achievement in mathematics, with additional control variables considered in both the Null model and the Conditional model. Overall, private management positively influences learning outcomes in mathematics, even after accounting for other factors such as socioeconomic status, gender, geographic location, and regional differences. This finding highlights the importance of school management type in shaping educational achievements.

In the Conditional model, which accounts for control variables, Private Management (D = 1) has a significant positive effect on learning achievement, with a coefficient of 20.65 and a t-value of 16.13 (p-value = 0.00). This indicates that students in private-managed institutions score 20.65 points higher on average in mathematics compared to students in public institutions, with the result being highly statistically significant.

The Socioeconomic Index (ISE) has a coefficient of 10.81 with a p-value of 0.00, indicating that students from higher socioeconomic backgrounds tend to perform better in mathematics. Specifically, for each unit increase in the socioeconomic index, students score an additional 10.81 points on average.

The variable of gender (Female = 1, Male = 0) shows a significant negative effect with a coefficient of −5.70 and a p-value of 0.00. This suggests that female students score 5.70 points lower in mathematics than male students, all other factors being equal.

Geographic area (Rural = 1, Urban = 0) also has a significant effect, with a coefficient of −25.11 and a p-value of 0.00, indicating that students from rural areas score 25.11 points lower in mathematics compared to students from urban areas.

For the Spanish language variable (D = 1), the coefficient is 22.19 with a p-value of 0.00, showing that students who speak Spanish perform better, with an average increase of 22.19 points in mathematics compared to those who do not.

Private management (coded as 1 for private institutions) shows a positive coefficient of 20.65 with a p-value of 0.00, meaning students in private institutions score 20.65 points higher in mathematics on average compared to those in other types of educational management.

It seems like there was a mix-up with the interpretation. If private management has a positive coefficient, that would typically mean higher scores, not lower. Let me know if this is what you intended!

Regarding specific regions, Apurimac (D = 1) has a positive and statistically significant coefficient of 31.18 with a p-value of 0.00, indicating that students in Apurimac score 31.18 points higher in mathematics compared to the reference region. Other regions like Arequipa, Ayacucho, Callao, and Madre de Dios also show positive effects, with students scoring significantly higher than the baseline region, though the size of the effect varies. For example, Arequipa students score 21.43 points higher, while Ayacucho students score 18.25 points higher, though Ayacucho’s result is only marginally significant (p-value of 0.08).

On the other hand, Lima shows a negative coefficient of −44.68 with a p-value of 0.00, meaning students in Lima perform significantly worse than those in the reference region, with a decrease of 44.68 points in mathematics achievement.

Other regions, such as Loreto, Moquegua, and Puno, exhibit smaller effects, with Loreto having a positive but not significant impact (p-value of 0.56) and Moquegua and Puno showing minimal effects (with p-values of 0.14 and 0.80, respectively).

Finally, San Martin and Tacna exhibit strong positive coefficients of 50.28 and 575.66, respectively, with p-values of 0.00, indicating that students in these regions perform significantly better in mathematics compared to those in the reference region.

Overall, the fixed effects model shows significant regional disparities in math achievement, with certain regions such as Apurimac, Arequipa, and San Martin showing higher performance, while Lima exhibits a significant negative effect.

Table 13 provides insights into the effect of private management on mathematics learning achievements across various regions. The effect is measured as the difference in average scores between students in private institutions and those in other types of educational management, with a positive coefficient indicating better performance in private management.

Regions with significant effects: Most regions show a significant positive effect (with p-values < 0.05), meaning students in private institutions generally perform better in mathematics compared to students in other educational management systems. For example, Loreto shows the highest effect at 65 points, followed by Ucayali (47 points) and San Martín (43.09 points). In these regions, students in private schools score significantly higher in math.

Average scores: The average math scores for students in private institutions vary across regions. For instance, Lima, which has a relatively lower effect (21.61 points), has the highest average score at 609 points, whereas Loreto, despite having a large effect (65 points), has a lower average score of 520 points.

Regions with no significant effects: Some regions like Apurímac, Amazonas, Pasco, and Madre de Dios show no significant differences in math achievement between private and non-private institutions (p-values > 0.05). Despite this, students in these regions still achieve a wide range of scores, with Pasco having an average of 579 points and Amazonas having the lowest at 537 points.

Private management has a generally positive effect on students’ math scores in most regions, although the degree of the effect varies. Regions with a stronger positive effect tend to have lower average scores, while regions with weaker effects show higher average scores. However, in a few regions, there is no significant difference between the educational management types.

Table 14 illustrates the variance of random and fixed effects at various levels: UGEL, IE (Institution), Section, and Student. The results show how fixed effects influence the overall variance at each level.

UGEL level: The variance of random effects is initially 1,238.19 in the null model, and after applying the conditional model, it decreases significantly to 190.24. This represents a reduction of 1,047.95, with fixed effects accounting for 84.64% of the total variance, indicating a strong influence of fixed effects at this level.

IE level: At the IE level, the random effects variance drops from 896.76 to 315.92 in the conditional model, with a difference of 580.84. Fixed effects explain 64.77% of the variance, showing a notable impact, although slightly less than at the UGEL level.

Section level: At the section level, the reduction in variance is smaller, from 136.51 to 111.89, with a difference of 24.62. Fixed effects explain only 18.04% of the variance, suggesting a lower level of influence at this level.

Student level: The variance at the student level shows the smallest change, from 3,374.22 to 3,225.41, with a difference of 148.81. Fixed effects explain only 4.41% of the variance at this level, indicating a minimal impact of fixed effects on student-level variance.

Table 15 presents the Intraclass Correlation Coefficients (ICC) for different levels of variance (UGEL, Educational Institution (EI), and Section) in both the Null model and the Conditional model, along with their associated standard errors and z-values.

The SARAR spatial econometric model reveals that mathematics achievement among Peruvian students is influenced by interrelated socioeconomic, educational, and geographic factors. The analysis, based on Ordinary Least Squares (LS), Maximum Likelihood (ML), and Generalized Spatial Two-Stage Least Squares (GS2SLS) estimations, identifies significant spatial dependence between districts (ρ ≈ 0.03) and a strong spatial error correlation (λ ≈ 0.90–0.94), indicating that student performance in one district is correlated with that of neighboring districts.

Socioeconomic variables exert a strong influence: the household socioeconomic index (ISE) and Human Development Index (HDI) show substantial positive effects (≈25 and ≈32 points, respectively), while the percentage of mothers with higher education also contributes positively (≈0.64 points per percentage increase). Public investment per capita, school connectivity to electricity, and the proportion of professionally qualified teachers enhance achievement, whereas higher student–teacher and student–computer ratios reduce it. Spillover effects of socioeconomic status (Wx ISE ≈ 8.375) and public investment (Wx Inv. ≈ 0.003) confirm that improvements in surrounding districts foster local academic performance.

The spatial pattern reveals that the highest mathematics achievement occurs in coastal districts, particularly in southern Peru, where socioeconomic conditions and maternal education levels are higher (Marsh and Parker, 1984). This spatial dependence highlights that family socioeconomic background outweighs school resources in explaining achievement differences (Ombay, 2018). The HDI’s influence operates through hidden contextual factors and mediates inequalities (Alkire and Foster, 2010; Giménez Beut et al., 2021; Savasci and Tomul, 2013).

Regarding school inputs, previous studies confirm that greater educational investment and better infrastructure enhance performance (Frndak, 2014; Patrinos, 2013; Wenglinsky, 1997). Access to electricity and sanitation in schools improves attendance and achievement (Adeyemi, 2014; Alimi et al., 2012). Smaller class sizes and more qualified teachers positively affect mathematics outcomes, while excessive reliance on computers may not (Chang and Kim, 2009; Tian and Yao, 2020; Warschauer et al., 2004).

The findings align with prior evidence that family socioeconomic conditions are major predictors of academic success (Gómez and Suárez, 2020; Vázquez, 2024; Yang, 2023), by cultural capital and motivation (Hendrawijaya, 2019; Edgerton and McKechnie, 2023) students is shaped by a complex interplay of socioeconomic, school-related, and geographic factors, in close alignment with a broad body of empirical and theoretical literature.

First, the study reaffirms that household socioeconomic status is a strong predictor of academic performance, as demonstrated by Gómez and Suárez (2020), Vázquez (2024) and Yang (2023). The positive effects of socioeconomic index and maternal education level found in this research reinforce those conclusions. Family cultural capital, household wealth, and parental expectations emerge as key mechanisms explaining these associations. Additionally, psychological and motivational factors such as growth mindset and learning motivation also mediate this relationship, as discussed by Hendrawijaya (2019) and Edgerton and McKechnie (2023).

From a school-based perspective, this study empirically validates the crucial role of educational inputs. Significant positive effects were observed for per capita public investment, access to infrastructure (such as electricity), and the share of certified teachers—findings consistent with those of Glewwe et al. (2011), Dewi (2022), and Bornaa et al. (2022). Conversely, high student–teacher and student–computer ratios negatively impact performance, supporting the conclusions of Ramadan et al. (2021) and highlighting the need for better allocation and quality of school resources. As emphasized by Hansen and Strietholt (2018) and Suyati et al. (2022), a positive school climate and strong pedagogical relationships are critical for mitigating the effects of poverty.

Geographic location emerges as a key structural factor in academic performance. Evidence shows that students in economically developed regions with better educational infrastructure—such as southern coastal areas of Peru (Tacna, Moquegua, Arequipa, Lima, and Callao)—consistently outperform their peers in the Andean and Amazon regions, including Apurímac, Loreto, and Tumbes. This pattern aligns with the findings of Annette and Muliro (2019), Li (2023), and Owoseni et al. (2020), who emphasize how school location conditions access to qualified teachers, technologies, and educational materials.

These regional patterns reflect deeply embedded structural inequalities, as also identified by Faggian and Franklin (2014), Kalaycıoğlu (2015), and Olsen and Huang (2021). More geographically isolated regions, with higher poverty rates and lower connectivity, face significant barriers to student achievement—from school transport to access to basic services. This underscores the urgent need for policies that address territorial diversity and educational equity (Yu et al., 2021).

Methodologically, the study confirms the value of spatial modeling in capturing regional patterns and dependencies in academic achievement, following approaches by Arauzo-Carod (2007), Ahmed (2022), and Oliveira et al. (2021). The statistically significant spatial autocorrelation coefficient (ρ ≈ 0.03) and spatial error (λ ≈ 0.90) suggest that student performance in one district is correlated with that in neighboring districts. Furthermore, positive spatial spillover effects in Wx variables (socioeconomic status and per capita public investment) reinforce the argument by Novitasari and Iskandar (2022) on the importance of spatially targeted investments to reduce poverty and improve educational outcomes.

At a broader level, Wang et al. (2020) argue that the unequal distribution of educational facilities not only impacts academic outcomes but also shapes long-term urban and economic development. Similarly, Feigenson et al. (2013) and Murphy (2019) demonstrate that students with greater access to technology and adequate learning environments are more likely to achieve high academic performance. This study provides concrete evidence from the Peruvian context that reinforces those international findings.

Finally, the interaction between geographic location, school conditions, and socioeconomic factors—as argued by Alamdarloo et al. (2013)—reveals that addressing any one of these dimensions in isolation is insufficient. Instead, comprehensive strategies are required that simultaneously consider family context, educational resources, and territorial conditions to reduce disparities and foster a more equitable and sustainable educational system.

The analysis of public investment per capita reveals its significant role in enhancing student achievement, particularly in mathematics. Specifically, the findings indicate that there is a positive relationship between increases in public investment in education and higher levels of student performance in mathematics. This aligns with existing literature that emphasizes the critical impact of educational funding and resources on academic successgarcia (Díaz-García et al., 2023). However, I was unable to find robust evidence to support claims about spatial spillover effects in education. Therefore, the statement claiming that educational outcomes in a district can be positively influenced by public investments made in neighboring areas has been removed. As the findings indicate a correlation between public investment and student achievement, policymakers should prioritize increasing funding for educational institutions. This funding should be equitably distributed, particularly targeting disadvantaged regions that historically receive less investment. Enhanced funding can be allocated to improve school infrastructure, hire qualified teachers, and provide necessary educational materials. In light of the observed correlations, policies should be developed to create educational programs that account for the socioeconomic contexts of various districts. Initiatives such as collaborative funding models can be established, where prosperous districts contribute to the improvement of education in less affluent areas, thus enhancing overall regional educational standards. It is critical for the government to implement robust mechanisms for evaluating the effectiveness of public investment in education. Establishing performance indicators that track student achievement relative to public spending will ensure that investments are used efficiently and improve learning outcomes. Regular assessments can help in reallocating resources where they are most needed (Norrahman, 2024). With the growing emphasis on technology in education, part of the public investment should be directed toward digital infrastructure, ensuring that all schools are equipped with the necessary technology to enhance learning. This includes not only hardware but also training for teachers on effectively integrating technology into their teaching practices.

The findings from the analysis on the effect of private management on learning achievement in mathematics reveal several noteworthy insights that underscore the relevance of school management type and its influence on educational outcomes. The data presented indicates that students in private-managed institutions significantly outperform their peers in public schools, reinforcing existing literature on the advantages of private education management.

The conditional model reveals a statistically significant positive relationship between private management and learning achievement in mathematics, evidenced by a coefficient of 20.65 and a t-value of 16.13 (p-value < 0.001). This suggests that, on average, students in private institutions score 20.65 points higher in mathematics than those in public schools, controlling for other relevant factors such as socioeconomic status, gender, and geographic location. This finding aligns with previous studies that suggest a correlation between private schooling and enhanced academic performance due to improved resources and a more effective educational environment (Shehzadi et al., 2022).

Importantly, the results indicate that socioeconomic factors also play a significant role in shaping educational outcomes. The Socioeconomic Index (ISE) coefficient suggests that students from higher socioeconomic backgrounds tend to achieve better results. This confirms existing research linking economic stability with educational success (Neuman, 2022). Conversely, the negative impact of gender, with female students scoring 5.70 points lower than male students, reiterates findings of gender disparities in academic achievement (Ngware et al., 2012). Rural students also show a marked disadvantage, scoring 25.11 points lower than their urban counterparts.

The analysis of the fixed effects model reveals critical insights into the determinants of mathematics achievement among students, with several observable variables having significant influences. The findings highlight a pronounced impact of socio-economic status (SES) on math performance, wherein a unit increase in the Socioeconomic Index corresponds to an average improvement of 10.81 points in mathematics scores (p-value = 0.00). This robust correlation underscores the role of SES as a significant predictor of academic success, particularly in mathematics (Bialystok, 2018; Madrid Fernández and Julius, 2020). Education systems aiming to bridge the academic achievement gap must prioritize interventions that address SES disparities, focusing on resources and support for students from lower socioeconomic backgrounds (Fryer and Levitt, 2004; Karademir and Yilmaz, 2023).

In terms of gender differences, the model indicates that female students, on average, score 5.70 points lower than their male counterparts (p-value = 0.00). This finding aligns with prior research suggesting that socialization and systemic biases may contribute to a persisting gender gap in mathematics achievement (Cornwell et al., 2013; Fryer and Levitt, 2004). Discussions surrounding gender and math performance often involve exploring the influences of societal expectations and internalized stereotypes that may discourage female engagement and confidence in mathematics-related tasks (Gunderson et al., 2011; Tomasetto et al., 2011). It is essential that educational strategies include initiatives to counteract these biases and promote equitable learning environments for all genders.

Geographic disparities also emerge as a compelling aspect of the results, as students from rural locations score an average of 25.11 points lower compared to urban peers (p-value = 0.00). This disparity may be attributed to differences in educational resources and support structures available in rural areas compared to urban settings (Bialystok, 2018; García et al., 2016). Urgent policy measures are required to ensure that rural schools receive adequate funding and resources to provide equitable educational opportunities, thus enhancing students’ mathematics achievements and overall educational outcomes.

Interestingly, the analysis reveals a positive correlation with Spanish language proficiency, where students who speak Spanish perform 22.19 points higher in mathematics than those who do not (p-value = 0.00). Such results suggest that bilingual education programs can have significant benefits on mathematics achievement, supporting findings that indicate a link between bilingualism and enhanced cognitive skills in problem-solving and reasoning tasks (Pinzón and Centenó, 2021; Shook and Marian, 2013). Implementation of bilingual education frameworks could facilitate better academic achievements for bilingual students, thereby supporting their integration and success within the mainstream educational framework.

The impact of educational management is also notable, with public management institutions associated with a decrease of 20.65 points in mathematics scores relative to other management types (p-value = 0.00). This finding raises critical questions about the quality of instruction and available resources in public educational institutions, particularly in light of previous studies depicting systemic challenges faced by public schools (Karademir and Yilmaz, 2023; Madrid Fernández and Julius, 2020). Effective management strategies and resources are essential for public institutions to foster improved mathematics outcomes.

Analysis of regional differences emphasizes the nuanced effects of private management on learning achievement. Notably, regions such as Apurimac and Arequipa exhibit significantly positive coefficients, suggesting that students in these areas benefit substantially from private management. However, the negative coefficient observed for students in Lima presents a contrast, highlighting complexities within urban educational dynamics.

This regional analysis supports findings indicating variations in educational effectiveness across different regions and reinforces the need for targeted educational policies that can address these disparities effectively. The data suggest that while private management generally leads to improved outcomes, the extent of this improvement can be significantly impacted by local socioeconomic conditions and regional educational policies (Pandey et al., 2009).

The variances in the random and fixed effects observed in the models provide additional depth to the understanding of how differing educational management types influence student performance. The notable reduction in variances linked to UGEL and institutional effects in the conditional model underscores the strong influence of fixed effects, meaning that once control variables were accounted for, the disparities between educational institutions diminished considerably. This suggests that socioeconomic factors, gender, and geographic location significantly mediate the effects of school type on learning outcomes, echoing findings relating to inequities in educational achievement in varied contexts (Callaman and Itaas, 2020; Long and Freese, 2006).

Furthermore, the Intraclass Correlation Coefficients (ICC) data indicate that while UGEL differences initially accounted for a notable percentage of variance in achievement, this figure drops significantly when controls are applied, illustrating that institutional disparities become less pronounced when accounting for individual student characteristics. Such insights highlight the importance of contextualized approaches in education policy development that recognize the multifaceted nature of student achievement (Sumantri and Whardani, 2017; Watanabe et al., 2020).

Further analysis reveals marked regional variations, with students from Apurimac achieving 31.18 points higher than the reference region (p-value = 0.00), in contrast to students from Lima, who exhibit a significant decline in scores by 44.68 points (p-value = 0.00). These regional differences highlight the need for targeted educational interventions that are tailored to the unique socio-economic and cultural contexts of each area, particularly in metropolitan regions such as Lima where scholastic challenges may be compounded by a host of other socio-economic factors (Bialystok, 2018; García et al., 2016). Conversely, regions like San Martin and Tacna show exceptional performance, indicating potential best practices that might be replicated elsewhere.

To enhance future analyses, it is recommended to collect and use georeferenced data from educational centers, including latitude and altitude. This would allow for more precise and detailed spatial analyses, enabling the evaluation of how specific geographic factors affect student performance and the identification of local performance patterns. Moreover, with this information, it would be possible to design more targeted and context-adapted educational interventions and policies, taking into account each school’s exact location and geographic environment.