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Chemical Engineering & Development

Journal of Science and Engineering

Vol. 08 / Nº 01

e – ISSN: 3028-8533

ISSN – L: 3028-8533

Chemical Engineering & Development

University of Guayaquil | Faculty of Chemical Engineering

Guayaquil – Ecuador

https://revistas.ug.edu.ec/index.php/iqd

Email: inquide@ug.edu.ec

francisco.duquea@ug.edu.ec

Pag. 61

Time series modeling of secondary school enrollment in Ecuador: a Box–

Jenkins analysis (1971–2023).

Modelado de series de tiempo de la matrícula escolar secundaria en Ecuador: un análisis Box–

Jenkins (1971–2023).

Edwin Haymacaña Moreno

; Leonor Alejandrina Zapata Aspiazu;

Francisco Javier Duque-Aldaz

; Félix Genaro Cabezas

García

; Raúl Alfredo Sánchez Ancajima

Received: 09/28/2025 – Accepted: 12/02/2025 – Published: 01/01/2025

Research Articles

Review

Articles

Essay Articles

* Corresponding

author.

This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 (CC BY-

NC-SA 4.0) international license. Authors retain the rights to their articles and may share, copy, distribute,

perform, and publicly communicate the work, provided that the authorship is acknowledged, not used for

commercial purposes, and the same license is maintained in derivative works.

Abstract.

School enrollment analysis constituted a key indicator to evaluate coverage and equity in national education systems. The objective of this study was to model

secondary school enrollment in Ecuador during 1971–2023 using time series techniques. Official national and international data were employed to construct an

annual net enrollment series. The methodological procedure included descriptive analysis, stationarity tests (ADF and KPSS), first-order differencing,

identification and estimation of candidate models through the Box–Jenkins approach, optimal selection with auto.arima, residual validation via Ljung–Box tests,

out-of-sample error metrics (MAE, RMSE, MAPE), and forecasts for 5–10 years. All processing was performed in R Studio with specialized time series modeling

packages. The results showed that after first-order differencing, the series achieved stationarity. The selected model adequately explained enrollment dynamics,

with residuals consistent with white noise and without significant autocorrelations. Validation metrics indicated good predictive accuracy, with low mean

absolute and percentage errors. Projections suggested a moderate and sustained growth trend in enrollment, though with signs of stabilization in the longer

horizon. This study demonstrated the usefulness of Box–Jenkins models for analyzing educational phenomena, providing quantitative evidence for public policy

formulation and recommending the expansion of more complete historical datasets in future research.

Keywords.

Time Series, ARIMA, Box–Jenkins, School Enrollment, Secondary Education, Ecuador, Educational Forecasting.

Resumen.

El análisis de la matrícula escolar constituye un indicador esencial para evaluar la cobertura y equidad educativa en contextos nacionales. El objetivo de este

estudio fue modelar la matrícula de educación secundaria en Ecuador durante el periodo 1971–2023 mediante técnicas de series de tiempo. Se emplearon datos

oficiales de organismos internacionales y nacionales, construyéndose una serie anual de matrícula neta. El procedimiento metodológico incluyó: análisis

descriptivo inicial, pruebas de estacionariedad (ADF y KPSS), diferenciación para lograr estabilidad en la media, identificación y estimación de modelos

candidatos mediante el enfoque Box–Jenkins, selección óptima con auto.arima, validación de residuos mediante la prueba de Ljung–Box, comparación de

métricas fuera de muestra (MAE, RMSE, MAPE) y pronósticos a 5–10 años. Todo el procesamiento se realizó en R Studio, empleando paquetes especializados

de modelado de series de tiempo. Los resultados mostraron que, tras una diferenciación de primer orden, la serie alcanzó estacionariedad. El modelo seleccionado

explicó adecuadamente la dinámica de la matrícula secundaria, con residuos consistentes con ruido blanco y sin autocorrelaciones significativas. Las métricas

de validación indicaron un buen ajuste predictivo, con valores bajos de error medio absoluto y porcentual. Las proyecciones sugirieron una tendencia de

crecimiento moderado y sostenido en la matrícula, aunque con señales de estabilización en los horizontes más largos. Este estudio demostró la utilidad de los

modelos Box–Jenkins para el análisis de fenómenos educativos, aportando evidencia cuantitativa para la formulación de políticas públicas y recomendando la

ampliación futura de bases de datos históricas más completas.

Palabras clave.

Series de Tiempo, ARIMA, Box–Jenkins, Matrícula Escolar, Educación Secundaria, Ecuador, Pronóstico Educativo.

1.- Introduction

The analysis of the Ecuadorian education system has gained

special relevance in recent decades due to the challenges

related to the coverage, equity, and quality of secondary

education. In particular, school enrollment is a key indicator

for assessing student access and retention, as well as for

identifying structural inequalities in the system.

Understanding the dynamics of enrolment over time not

only allows us to detect historical patterns, but also to

Bolivarian University Higher Institute of Technology; erhaymacana@itb.edu.ec; https://orcid.org/0000-0002-8708-3894; Guayaquil;

Ecuador.

Technical University of Babahoyo; lzapata@utb.edu.ec; https://orcid.org/0009-0003-1497-2273 ; Babahoyo; Ecuador.

University of Guayaquil; franscico.duquea@ug.edu.ec; https://orcid.org/0000-0001-9533-1635 ; Guayaquil; Ecuador.

Independent Researcher; genaro_cabezas@hotmail.com ; https://orcid.org/0000-0003-3595-3584; Hamilton, ON, Canada.

National University of Tumbes; rsanchez@untumbes.edu.pe ; https://orcid.org/0000-0003-3341-7382 ; Tumbes, Peru.

anticipate trends that are fundamental for the formulation of

sustainable public policies aimed at meeting Sustainable

Development Goal 4 (SDG4), which seeks to guarantee

inclusive, equitable and quality education.(Simonino y

otros, 2025)

In the scientific literature, time series models have proven

to be robust tools for the analysis and prediction of

socioeconomic and educational phenomena. Within these

approaches, the Box–Jenkins method. It stands out for its

INQUIDE

Chemical Engineering & Development

Journal of Science and Engineering

Vol. 08 / Nº 01

e – ISSN: 3028-8533

ISSN – L: 3028-8533

Chemical Engineering & Development

University of Guayaquil | Faculty of Chemical Engineering

Guayaquil – Ecuador

https://revistas.ug.edu.ec/index.php/iqd

Email: inquide@ug.edu.ec

francisco.duquea@ug.edu.ec

Pag. 62

ability to model temporal dependencies using

autoregressive (AR), moving average (MA) and their

seasonal extensions (SARIMA) structures. These models

have been successfully applied in contexts of forecasting

macroeconomic and climatic variables and, more recently,

in the analysis of educational indicators. However, in the

case of Ecuador, the application of these methodologies to

the longitudinal study of school enrollment remains limited,

which constitutes a gap in the literature.(Zanatta Idemori y

otros, 2025)

Recent studies have shown that the use of ARIMA models

and their variants allows for the generation of accurate

projections of variables such as enrollment rates,

performance on standardized tests, and dynamics of

admission to higher education. Likewise, comparative

research shows that hybrid models that combine Box–

Jenkins techniques with machine learning approaches, such

as random forests or neural networks, improve predictive

capacity and offer more flexible interpretations of

educational data and. These contributions confirm the

potential of time series not only to describe historical

patterns, but also to design prospective strategies in the

educational field.(Escolar, 2024)

The main objective of this work is to model secondary

education enrollment in Ecuador during the period 1971–

2023 using the Box–Jenkins methodology. Specifically, it

seeks to: (i) identify patterns of trend and seasonality in

enrollment; (ii) to estimate ARIMA/SARIMA models that

allow describing their temporal dynamics; and (iii) to make

short- and medium-term projections that contribute to

national educational planning.(Corrêa Werle & Lago

Fonseca, 2025)

The main contribution of this study lies in integrating

advanced mathematical tools for time series analysis with

educational data, generating empirical evidence that can

serve as an input for the formulation of public policies in

Ecuador. Likewise, the results allow to contribute to the

regional literature on the use of quantitative models in

education, showing how techniques traditionally applied in

economics and engineering can be adapted to high-priority

social and educational problems. In sum, this article

represents an effort to link statistical rigor with educational

decision-making in Ecuador, contributing to the design of

evidence-based strategies.(Castro Rosales y otros, 2025)

1.1.- Context and relevance of the analysis of secondary

school enrollment

The analysis of secondary school enrollment is essential to

evaluate the educational coverage, equity and quality of the

education system in a national context. Enrollment is a key

indicator that reflects students' access to and permanence in

secondary education, allowing the identification of

structural conditions and temporal dynamics that affect

inclusion and educational opportunity. According to various

studies, longitudinal monitoring of enrolment makes it

easier to detect patterns, trends and possible inequalities,

which is essential for the planning and formulation of public

policies aimed at improving education systems. (Cabrera

Valladolid, 2021)

This indicator is directly related to the objectives set by

international instruments, in particular Sustainable

Development Goal 4 (SDG 4), which promotes ensuring

inclusive, equitable and quality education for all. Secondary

school enrolment reflects progress and challenges in

achieving this objective, as its evolution shows how the

education system responds to social demands and economic

conditions. In this way, the analysis of enrollment is a tool

to monitor and adjust national strategies that contribute to

the fulfillment of educational and social goals established in

global agendas.(Zalduaromero, 2017)

In the specific case of the Ecuadorian education system, the

literature shows that, although there has been progress in

increasing coverage in secondary education, significant

gaps in equity and quality persist. However, longitudinal

and quantitative modeling of enrollment is an area little

explored in the country, generating an important

opportunity to apply robust techniques, such as time series

and Box-Jenkins models. This gap in the literature shows

the need to develop studies that provide detailed empirical

analyses on the dynamics of school enrollment, in order to

support public policies based on reliable and up-to-date

information.(Cañarte Murillo, 2017)

Fluctuations in educational enrolment are often closely

linked to socio-economic factors such as economic crises,

public policies and migration dynamics. In periods of

recession, families prioritize subsistence over education,

which translates into a decrease in enrollment and an

increase in school dropouts. Similarly, budget cuts in

education during fiscal crises reduce the supply of places

and support programs, especially affecting vulnerable

populations.(Alós & Serio, 2024)

Internal and external migration also affects the variability of

enrollment. Massive migratory processes, motivated by

unemployment or political instability, alter the demographic

distribution and generate overload in certain areas while

others experience educational gaps. Educational policies

such as free education, scholarships or curricular reforms

can counteract these effects, but their impact depends on the

state's capacity to sustain them in contexts of economic

volatility. (Duque-Aldaz & Pazan Gómez, Factors affecting

entrepreneurial intention of Senior University Students,

2017)

1.2.- Theoretical foundations of time series applied to

education

Time series are chronologically ordered data sets that allow

the dynamics of variables to be analyzed over time. These

series have fundamental characteristics such as the trend,

which indicates the general direction of behavior;

seasonality, which reflects periodic cyclical patterns; and

noise, represented by random fluctuations that do not follow

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Chemical Engineering & Development

Journal of Science and Engineering

Vol. 08 / Nº 01

e – ISSN: 3028-8533

ISSN – L: 3028-8533

Chemical Engineering & Development

University of Guayaquil | Faculty of Chemical Engineering

Guayaquil – Ecuador

https://revistas.ug.edu.ec/index.php/iqd

Email: inquide@ug.edu.ec

francisco.duquea@ug.edu.ec

Pag. 63

a specific pattern. In the educational context, time series

analysis makes it possible to detect these components in

variables such as school enrollment, which makes it easier

to understand their historical evolution and anticipate future

behaviors.(Meneses Freire y otros, 2022)

Mathematical statistics plays a crucial role in the study of

time series, providing tools that allow modeling time

dependencies and evaluating the quality of fit. In the social

sciences and education, such models are widely used to

predict trends, examine the impact of policies, and improve

decision-making based on historical data. The incorporation

of robust statistical models favors the rigorous analysis and

solid interpretation of educational variables that show

temporal behavior.(Ortega Villegas, 2018)

Among the most relevant models for time series, ARIMA

(Integrated Moving Average Autoregressive) and its

extensions, such as SARIMA (Seasonal ARIMA Model)

and ARIMAX (ARIMA with exogenous variables) stand

out. These models are suitable for capturing patterns of

dependency in non-stationary and seasonal data, also

allowing external variables to be incorporated when

relevant. In the educational field, its application has proven

to be effective in modeling variables such as enrollment

rates and academic performance, offering a flexible

framework for the analysis and forecasting of complex

phenomena over time.(Ichau Tabango y otros, 2021)

In Latin America, several studies have applied ARIMA

models to forecast educational trends. For example,

research in Mexico has used ARIMA(1,1,1) to project

enrollment in basic education, demonstrating high accuracy

in moderate growth scenarios. These works highlight the

usefulness of the model to anticipate infrastructure and

teaching staff needs in contexts of demographic

expansion.(Duque-Aldaz y otros, Identification of

parameters in ordinary differential equation systems using

artificial neural networks, 2025)

Similarly, in Brazil, ARIMA models were used to estimate

demand in higher education, incorporating historical series

of admissions and graduation rates. The results made it

possible to adjust financing policies and quotas in public

universities, evidencing that ARIMA is an effective tool for

planning resources in educational systems with significant

temporal variability.(Sandoya Sanchez & Abad Robalino,

2017)

1.3.- Box–Jenkins methodology for time series modeling

The Box–Jenkins methodology is a systematic approach to

time series modeling, which is structured in an iterative

process of four phases: identification, estimation, diagnosis

and prognosis. First, in the identification phase, the time

series is analyzed to detect characteristics that allow

proposing appropriate potential models. Then, in the

estimation, the parameters of the selected model are

adjusted using the available data. The diagnostic phase

consists of validating the model through fit evaluations and

statistical tests, verifying the absence of unmodeled patterns

in the residuals. Finally, in the forecasting stage, the

validated model is used to predict future values of the series,

supporting decision-making based on reliable

projections.(Mayorga Trujillo, 2017)

ARIMA models, central components of the Box–Jenkins

approach, bring together three fundamental elements:

autoregression (AR), which models the dependence of a

value on its antecedents; differentiation (I), which

transforms the series to ensure its stationarity; and the

moving average (MA), which represents a security's

dependence on past mistakes. This structure allows complex

dynamics to be captured in the time series; In particular,

differentiation helps to eliminate trends and stabilize

variance, necessary conditions for applying effective

statistical models on non-stationary data.(Villarreal Godoy

y otros, 2022)

To ensure that the series is suitable for ARIMA modeling,

it is necessary to evaluate its stationarity using statistical

tests such as the augmented Dickey-Fuller (ADF) and the

KPSS test, which examine whether the properties of the

series remain constant over time. In case the series is not

stationary, differentiation procedures are applied to stabilize

the mean and variation. This process is crucial, since a well-

specified model requires statistical stability to produce

reliable and valid protectors, as supported by research and

manuals specialized in time series analysis.(Vela &

Camacho Cordovez, 2020)

1.4.- Applications and adaptations of the ARIMA model

in educational contexts.

ARIMA models and the Box–Jenkins approach have been

widely applied in educational contexts in Latin America and

other regions to forecast variables such as school

enrollment, graduation rates, and other indicators. Various

studies show that these models allow capturing trends and

temporal patterns in non-stationary education data,

facilitating institutional planning and policy formulation. In

particular, research in Latin American countries has

demonstrated the effectiveness of ARIMA in the predictive

analysis of historical education data, providing valuable

information to manage resources and improve school

coverage.(Fu-López y otros, 2025)

Recently, the integration of ARIMA models with machine

learning techniques has led to hybrid methods that combine

the strengths of both approaches. For example, models that

integrate neural networks or random forests with ARIMA

allow capturing nonlinear and complex relationships in time

series, improving predictive accuracy compared to

traditional univariate models. These hybrid tools are gaining

relevance in education and other fields, where the

complexity of data requires more sophisticated

methodological strategies.(Ausay Carrillo, 2022)

Despite their advantages, univariate ARIMA models have

limitations in considering only the internal dynamics of a

INQUIDE

Chemical Engineering & Development

Journal of Science and Engineering

Vol. 08 / Nº 01

e – ISSN: 3028-8533

ISSN – L: 3028-8533

Chemical Engineering & Development

University of Guayaquil | Faculty of Chemical Engineering

Guayaquil – Ecuador

https://revistas.ug.edu.ec/index.php/iqd

Email: inquide@ug.edu.ec

francisco.duquea@ug.edu.ec

Pag. 64

single variable, without including external factors that can

influence the time series. To overcome this constraint,

multivariate models such as ARIMAX and SARIMAX

allow the incorporation of exogenous variables that enrich

the analysis and improve predictions. In education, this

makes it possible to integrate socioeconomic, demographic

or public policy factors, providing a broader and more

realistic approach to the study of complex phenomena such

as school enrolment.(Eguiguren Calisto & Avilés Sacoto,

2019)

1.5.- Validation and evaluation of the model.

The proper selection of the ARIMA model requires rigorous

evaluation using statistical criteria such as the Akaike

Information Criterion (AIC) and the Bayesian Information

Criterion (BIC). Both criteria balance the quality of the fit

with the complexity of the model, penalizing models with a

greater number of parameters to avoid overfitting. The

choice of the best model corresponds to the one that

minimizes these values, guaranteeing a balance between

precision and parsimony, which favors the generalization of

the model to unobserved data.(Navarro Llivisaca, 2017)

The model's diagnosis includes residue analysis to verify

fundamental assumptions. Tests such as the Ljung-Box test

are used to detect autocorrelation in the residuals, ensuring

that the model has adequately captured the temporal

dependence. In addition, the verification of the normality of

the residuals allows validating the confidence intervals of

the forecasts, while the ARCH heteroskedasticity test

evaluates whether the residual variance is constant, a

necessary condition for the statistical validity of the

model.(Figueroa Tigrero, 2019)

Predictive accuracy assessment is done through metrics

such as Mean Absolute Error (MAE), Root Mean Square

Error (RMSE), and Mean Absolute Error Percentage

(MAPE). These quantify the average deviation of the

forecasts with respect to the observed values, facilitating

comparison between models. Out-of-sample validation,

using datasets that are not involved in the estimation, is

crucial to ensure the true predictive capability of the model.

In addition, the importance of making short- and medium-

term forecasts is highlighted, as these provide useful and

reliable information for decision-making in educational and

administrative contexts.(Freire Engracia y otros, 2025)

1.6.- Implications for public policies and educational

planning

The proper selection of the ARIMA model requires rigorous

evaluation using statistical criteria such as the Akaike

Information Criterion (AIC) and the Bayesian Information

Criterion (BIC). Both criteria balance the quality of the fit

with the complexity of the model, penalizing models with a

greater number of parameters to avoid overfitting. The

choice of the best model corresponds to the one that

minimizes these values, guaranteeing a balance between

precision and parsimony, which favors the generalization of

the model to unobserved data.(Lema Remache, 2024)

The model's diagnosis includes residue analysis to verify

fundamental assumptions. Tests such as the Ljung-Box test

are used to detect autocorrelation in the residuals, ensuring

that the model has adequately captured the temporal

dependence. In addition, the verification of the normality of

the residuals allows validating the confidence intervals of

the forecasts, while the ARCH heteroskedasticity test

evaluates whether the residual variance is constant, a

necessary condition for the statistical validity of the

model.(Morocho Choca y otros, 2024)

Predictive accuracy assessment is done through metrics

such as Mean Absolute Error (MAE), Root Mean Square

Error (RMSE), and Mean Absolute Error Percentage

(MAPE). These quantify the average deviation of the

forecasts with respect to the observed values, facilitating

comparison between models. Out-of-sample validation,

using datasets that are not involved in the estimation, is

crucial to ensure the true predictive capability of the model.

In addition, the importance of making short- and medium-

term forecasts is highlighted, as these provide useful and

reliable information for decision-making in educational and

administrative contexts. (Pincay Moran y otros,

2025)(Guerrero Quinde & Pérez Siguenza, 2025)

2.- Materials and methods.

2.1 Materials and data sources

The study is based on annual series of the gross secondary

school enrollment rate in Ecuador for the period 1971–

2023. The data were obtained from the database of the

UNESCO Institute for Statistics, which is an official and

open-access source of international education indicators.

The records are presented in percentage values and

correspond to the indicator "Gross Enrollment Ratio –

Secondary (%), Ecuador", with 53 consecutive observations

that guarantee the viability of the time series analysis.

The statistical processing and analysis were carried out

using the following software:

EViews 12 (IHS Markit): for the estimation of Box–Jenkins

models (ARIMA/SARIMA) and the validation of statistical

assumptions.

RStudio 2023.09 with forecast, tseries, ggplot2 and urca

libraries: for robustness tests, graphing and comparative

analysis of the results.

Microsoft Excel 365: for initial debugging, processing

missing values, and generating exploratory charts.

2.2 Methodological design

The research adopts a quantitative, longitudinal and non-

experimental approach, based on mathematical modelling

of time series. The analysis variable is secondary school

enrollment (% gross), considered as time-dependent, and its

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Chemical Engineering & Development

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Vol. 08 / Nº 01

e – ISSN: 3028-8533

ISSN – L: 3028-8533

Chemical Engineering & Development

University of Guayaquil | Faculty of Chemical Engineering

Guayaquil – Ecuador

https://revistas.ug.edu.ec/index.php/iqd

Email: inquide@ug.edu.ec

francisco.duquea@ug.edu.ec

Pag. 65

dynamics are studied under the assumptions of stationarity,

independence, and homoscedasticity.



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(1)

where the mean and variance are constant over time and the

covariance depends only on the lag of h.

The methodological procedure was structured in four

stages:

1. Initial exploration of the series: graphical analysis,

calculation of descriptive statistics and verification of

outliers.

2. Transformation and diagnosis: application of the

augmented Dickey–Fuller unit root (ADF) test to assess

stationarity and, if necessary, application of regular and

seasonal differentiation.

3. Specification and estimation of the model: adjustment

of ARIMA/SARIMA models following the Box–

Jenkins methodology, selecting the orders p, d, q and P,

D, Q from the inspection of the autocorrelation

functions (FAC) and partial autocorrelation (FACP).

4. Model validation: verification of the classical

assumptions using the Ljung–Box (residue

independence), Jarque–Bera (normality), and Engle's

ARCH (conditional heteroskedasticity) tests.

Stages of the flow of the methodological procedure for

the modeling of time series of secondary school

enrollment in Ecuador (1971–2023).

For the present research, the scheme summarizes the main

stages:

1. Initial exploration in the series.

2. Diagnosis and transformation of the series.

3. Model specification and estimation.

4. Validation of assumptions.

5. Final projection of school enrollment.

2.3 Statistical procedures

The mathematical specification of the general SARIMA

model adopted is expressed as:



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where:



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and are the autoregression polynomials and moving

averages of order 

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p and q, respectively.

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and represent the seasonal polynomials of order

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P and Q with periodicity s.

 and indicate the orders of regular and seasonal

differentiation.

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corresponds to secondary school enrollment in year t.



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denotes an error term with zero mean and constant

variance.

The expanded form of the ARIMA model:

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Considering the variance of the prediction error of h steps:

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With the coefficients of representation 

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The Akaike (AIC) and Schwarz (BIC) information criteria

were used for the selection of the parsimonious model.

The choice of the ARIMA model is based on its ability to

capture patterns of temporal dependence in historical series

without requiring additional exogenous information.

Although models such as SARIMA incorporate explicit

seasonality, the preliminary analysis did not show regular

cycles associated with academic periods that would justify

their inclusion. In addition, the simplicity and robustness of

the ARIMA make it a suitable choice for scenarios where

the priority is to obtain reliable forecasts with limited data

and high socioeconomic variability.

2.4 Data analysis

Error measures were calculated to assess the accuracy of the

projections, including Mean Absolute Error (MAE), Root

Mean Square Error (RMSE), and Mean Absolute

Percentage of Error (MAPE). Likewise, a residue analysis

was implemented using autocorrelation graphs and adjusted

values versus residuals, in order to guarantee the adequacy

of the model.

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













 











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2.5 Ethical considerations

This study is based exclusively on secondary data of a

public and open nature, so it does not involve humans or

animals and, therefore, did not require the approval of an

ethics committee

3.- Results.

3.1. Descriptive statistics and initial exploration

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The series of secondary school enrollment in Ecuador

(1971–2023) shows a sustained growth from levels

below 30% to values close to 100% in recent decades.

The exploratory analysis (Fig. 2) reveals three phases:

i) a constant increase between 1971 and 1990; ii) a

relative stabilization during the nineties; and (iii) an

accelerated rebound in the period 2000–2010,

followed by a slight slowdown.

The initial autocorrelation (ACF) and partial autocorrelation

(PACF) functions (Figs. 3 and 4) show persistence in

multiple lags and an abrupt cut in the first lag, confirming

the non-stationarity of the series and suggesting the

relevance of applying a low-order AR model once

differentiated.

Table 1: Statistical summary (minimum, maximum, mean, quartiles)

Statistician

Value

Minimum

24.982679







52.261572

Medium 





53.327556







93.735523

Maximum

102.59033

Media

64.421453

Table 1. Descriptive statistical summary of the secondary

enrollment series (% gross) in Ecuador for the period 1971–

2023. Measures of central tendency and dispersion

(minimum, maximum, mean and quartiles) are presented,

which allow characterizing the initial distribution of the data

before applying time series modeling.

Fig. 1: Historical series of secondary enrolment.

Figure 1 shows the historical series of secondary school enrollment (% gross) in Ecuador during the period 1971–2023. The

graph shows a sustained upward trend until 2010, followed by a stabilization period with slight decreases in recent years.

Fig. 2: Initial ACF function.

Figure 2 shows the initial autocorrelation function (ACF) of the secondary school enrollment series in Ecuador (1971–2023). A

strong persistence of positive autocorrelations is observed in the first lags, which confirms the non-stationarity of the series

before applying transformations.

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Fig. 3: Initial PACF function.

Figure 3 shows the initial partial autocorrelation function (PACF) of the secondary school enrollment series in Ecuador (1971–

2023). The abrupt cut in the first lag confirms the presence of an autoregressive component, which is useful for the preliminary

identification of ARIMA models.

3.2. Diagnosis of stationarity and transformations

The unit root tests confirmed the non-stationarity in levels: the augmented Dickey–Fuller test (ADF) yielded a p-value = 0.32,

while the KPSS test indicated rejection of the null hypothesis of stationarity (p-value = 0.01).

When applying a first-order differentiation (d = 1), the KPSS test did not reject the stationarity hypothesis (p-value = 0.10), and

the ACF and PACF plots (Figs. 6,7 and 8) showed a pattern compatible with low-order ARIMA processes.

Fig. 4: Differentiated series (Δ enrollment).

Figure 4 shows the Differentiated Series of Secondary School Enrollment in Ecuador (1971–2023). The first difference stabilizes

the mean of the series, reducing the trend and allowing a more adequate stationary analysis. An atypical peak is observed around

2005, which could be associated with changes in educational policies or specific contextual factors.

Fig. 5: ACF of the differentiated series.

Figure 5 shows the autocorrelation function (ACF) of the differentiated series of secondary school enrollment in Ecuador (1971–

2023). It is observed that, after differentiation, most lags fall within the confidence intervals, which confirms the reduction in the

trend and supports the stationarity hypothesis.

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Fig. 6: PACF of the differentiated series.

Figure 6 shows the partial autocorrelation function (PACF) of the differentiated series of secondary school enrollment in Ecuador

(1971–2023). The PACF shows a significant lag in the first delay, which suggests the presence of a simple autoregressive

component in the dynamics of the series.

3.3. Model identification and estimation

Several ARIMA models (p,1,q) were estimated. The

information criteria (AIC and BIC) indicated that the

ARIMA(1,1,1), ARIMA(2,1,0) and ARIMA(1,1,0) models

were the most competitive. The selected model was

ARIMA(1,1,0) with drift term, balancing parsimony and

predictive capacity (Table 2).

Table 2. ARIMA/SARIMA Model Comparison

Model

Main

coefficients





AIC

BIC

Interpretatio

ARIMA

(1,1,0)

Significant

AR1, with

drift

Mediu

m-low

259.84

263.

Parsimonious;

It captures

dynamics

with few

parameters.

ARIMA

(2,1,0)

AR1 and

AR2

Significant

Similar

258.70

264.

Capture

additional

dependence,

but with more

parameters.

ARIMA

(1,1,1)

Significant

AR1 and

MA1

Lower

257.35

262.

Better overall

fit (lower

AIC).

Recommende

d model.

Table 2. Comparison of ARIMA models applied to the

secondary enrollment series in Ecuador (1971–2023). The

significant coefficients, the estimated residual variance, and

the AIC and BIC information criteria are presented. The

analysis shows that the ARIMA model(1,1,1) offers the best

overall fit, with the lowest AIC, so it is selected as the

recommended model.









3.4. Waste diagnosis

The residual diagnosis of the ARIMA(1,1,0) model with drift showed that errors behave as white noise: the p-values of the

Ljung–Box tests for 10 and 15 lags were 0.88 and 0.79, respectively, which confirms the absence of remaining autocorrelation.

The histogram of residues showed reasonable symmetry around zero, with slightly heavier tails associated with specific shocks

(Fig. 8).

Fig. 7: Waste diagnosis graphs (series, ACF, histogram).

Figure 7 shows the residue diagnosis plots of the ARIMA(1,1,0) model with drift applied to secondary school enrollment in

Ecuador (1971–2023). It is observed that the residuals do not present significant autocorrelations (ACF), maintain a behavior

close to white noise and their distribution is close to normal (histogram), which supports the validity of the selected model.

3.5. Out-of-sample validation

The training set included data up to 2016, reserving 2017–2023 for validation defined in equations (5)–(7). Out-of-sample

prediction errors were consistent with training errors: RMSE ≈ 3.45 and ASM ≈ 3.4%. The out-of-sample forecast (Fig. 9)

adequately captured enrollment stabilization close to 95%.

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Fig. 8: Out-of-sample forecast (2017–2023).

Figure 8 shows the out-of-sample forecast of secondary school enrollment in Ecuador (2017–2023). The black line represents

the observed values, while the blue strip indicates the predictions generated by the ARIMA(1,1,0) model with drift and their

confidence intervals at 80% and 95%. An adequate fit between the projected values and the actual data is observed in the

validation period.

3.6. Final forecast at 5–10 years

The forecast for the period 2024–2030 (Fig. 10) suggests a stabilization of secondary enrollment between 95% and 110%. The

specific trend projects slight growth, but the confidence bands are progressively widening, reflecting the uncertainty inherent in

structural factors (changes in education policies, external shocks).

Fig. 9: Final forecast (2024–2030) with 80% and 95% confidence intervals

Figure 9 shows the Final Forecast of Secondary School Enrollment in Ecuador (2024–2030). The blue dashed line represents the

values projected by the ARIMA(1,1,0) model with drift. The shaded stripes indicate the confidence intervals at 80% (lighter)

and 95% (darkest). A trend of moderate growth and stabilization is expected in the coming years, with a range of increasing

uncertainty towards the projection horizon.

The confidence bands in ARIMA projections represent the range of uncertainty associated with forecasts, which has direct

implications for educational planning. A wide band indicates high volatility, suggesting the need for flexible policies that

contemplate scenarios of overcrowding or enrollment deficit. On the contrary, narrow bands allow for the design of more precise

strategies in resource allocation, teacher hiring, and infrastructure expansion, reducing the risk of inefficiency in educational

management.

3.7. Limitations

The results are conditioned by the quality of the available

annual data and the assumption of linearity in the ARIMA

models. Structural factors not captured by the series (e.g.,

legislative changes, economic or health crises) can

generate significant deviations from the projected

scenarios.

4.- Discussion

The results obtained confirm that the evolution of secondary

school enrollment in Ecuador during the period 1971–2023

presents a dynamic characterized by long-term trends and

conjunctural shocks that can be captured by ARIMA

models. In particular, the ARIMA model(1,1,1) stood out

for its low AIC, which reflects a superior adjustability to the

series, while the ARIMA(1,1,0) model with drift showed

parsimony and ease of interpretation. These findings

corroborate the initial hypothesis that low-order

autoregressive processes, combined with moving mean

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components, are suitable for describing educational time

series.(Silva & Di Serio, 2021)

When compared with the existing literature, the results

coincide with the studies of Chen and Serra, who

demonstrated that SARIMA models allow capturing

seasonal patterns in educational indicators in Latin

America. However, unlike research focused on marked

seasonal contexts (e.g., energy or climate consumption), in

the Ecuadorian case a strong seasonal component was not

evidenced, which reinforces the relevance of the use of

simple ARIMAs. Likewise, our findings complement

previous work on prediction in education in South America,

where the emphasis has been on socioeconomic factors and

not on the temporal evolution of enrollment.(Medeiros y

otros, 2021)

In theoretical terms, this study contributes to the application

of the Box–Jenkins approach in the analysis of educational

indicators, showing how classic mathematical tools of time

series statistics can be adapted to social and public policy

phenomena. The robustness of the ARIMA model(1,1,1)

suggests that idiosyncratic shocks and temporal inertia

dynamics are the main determinants of secondary coverage

in Ecuador. From a practical perspective, the 5–10 year

projections indicate a stabilization of enrollment of around

100%, which provides useful empirical evidence for

educational planning and the design of policies aimed at

sustaining coverage and improving quality.(GARCÍA-

FERIA y otros, 2023)

Likewise, when contrasting the results with international

studies, it is observed that similar methodologies have been

applied in Latin American countries such as Mexico, Brazil

and Chile, as well as in Asian contexts such as China and

the Philippines, to model enrollment trends and project

educational demand. However, unlike these cases, the

Ecuadorian series shows greater instability in certain

periods, associated with structural changes in educational

policies and national socioeconomic situations. This

uniqueness highlights the importance of adapting models to

local particularities and not limiting themselves to the

transfer of external approaches. From a public policy

perspective, the projections obtained offer valuable input

for the strategic planning of institutions such as the Ministry

of Education and SENPLADES, by allowing anticipating

infrastructure, teacher training, and budget allocation needs.

In this way, the results not only contribute to the academic

debate, but also provide quantitative tools for the

formulation of sustainable and evidence-based education

policies.(García Vázquez y otros, 2021)(Mendoza Cota,

2020)

However, this work has limitations. The main one lies in the

univariate nature of the models used, which prevents the

incorporation of relevant exogenous variables such as

public investment in education, macroeconomic conditions

or demographic factors. In this sense, future studies could

extend the analysis to ARIMAX or SARIMAX models,

including covariates such as birth rate or public spending,

which would allow better capture of enrollment dynamics.

In addition, although the results show a good fit, the out-of-

sample ASM remains around 3–4%, which implies

uncertainty in contexts of structural shocks such as health

or migration crises.(Tudela-Mamani & Grisellx, 2022)

In summary, the findings of this work strengthen the

evidence on the use of ARIMA models in education,

contributing both to the theoretical framework and to the

practice of educational planning in Ecuador. It also

highlights the need to explore hybrid methodologies – such

as combinations between ARIMA and neural networks – to

improve the accuracy of forecasts and respond to the

inherent limitations of linear approaches.(Asán Caballero y

otros, 2023)

5.- Conclusion.

This study analyzed the evolution of secondary school

enrollment in Ecuador during the period 1971–2023 using

the Box–Jenkins methodology, in order to identify temporal

patterns and project future scenarios. The results showed

that the first-order differentiated ARIMA models

adequately describe the dynamics of the series, highlighting

the ARIMA(1,1,1) as the option with the best performance

according to the information criteria, while the

ARIMA(1,1,0) with drift offered a parsimonious and

consistent alternative. Both models confirmed the

hypothesis of stationarity after differentiation and allowed

the generation of robust forecasts in the short and medium

term.

The main contributions of this work are oriented towards

the incorporation of time series models in educational

analysis, an area in which their application is still incipient

in Ecuador. The study shows that classic techniques of

mathematical statistics, usually used in economics or

engineering, are equally valid for social problems,

providing quantitative evidence on the sustainability of

secondary school coverage. In this way, it contributes to

closing the gap identified in the literature regarding the use

of educational prediction methodologies based on time

series.

From a practical point of view, the results suggest that

secondary enrolment will tend to stabilize at around 100%

over the next decade, which has direct implications for

resource planning, infrastructure and educational policies

oriented beyond coverage, prioritizing quality and equity.

Theoretically, the study reinforces the relevance of ARIMA

models as a tool for the modeling of educational

phenomena, laying the foundations for subsequent

developments that integrate multivariate or hybrid

approaches.

Finally, it is recommended that future research extend the

analysis to ARIMAX or SARIMAX models incorporating

exogenous variables such as public spending, birth rates or

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macroeconomic indicators, as well as hybrid methodologies

that combine ARIMA with machine learning algorithms.

These approaches will make it possible to capture the

complexity of the education system in a more

comprehensive way and improve the accuracy of forecasts,

strengthening the link between mathematical statistics and

decision-making in public policy.

In summary, this work constitutes one of the first efforts in

Ecuador to rigorously apply the Box–Jenkins methodology

to the analysis of educational indicators, specifically to the

historical evolution of secondary enrollment. This

contribution not only strengthens the national literature in a

field in which qualitative or descriptive studies

predominate, but also positions mathematical statistics as a

fundamental tool for the design of evidence-based

educational policies. By opening this line of research,

precedents are set for future comparative studies at the

regional and global levels, contributing to the

internationalization of the debate on the use of time series

models in education.

6.- Contributions of the authors (Taxonomy of

contributors' roles - CRediT)

1. Conceptualization: Edwin Haymacaña Moreno, Leonor

Alejandrina Zapata Aspiazu.

2. Data curation: Leonor Alejandrina Zapata Aspiazu.

3. Formal analysis: Edwin Haymacaña Moreno, Leonor

Alejandrina Zapata Aspiazu.

4. Acquisition of funds: N/A.

5. Research: Edwin Haymacaña Moreno, Leonor

Alejandrina Zapata Aspiazu.

6. Methodology: Francisco Javier Duque-Aldaz, Raúl

Alfredo Sánchez Ancajima.

7. Project management: Francisco Javier Duque-Aldaz,

Raúl Alfredo Sánchez Ancajima.

8. Appeals: Francisco Javier Duque-Aldaz, Leonor

Alejandrina Zapata Aspiazu.

9. Software: Edwin Haymacaña Moreno, Leonor

Alejandrina Zapata Aspiazu.

10. Supervision: Félix Genaro Cabezas García, Raúl

Alfredo Sánchez Ancajima.

11. Validation: Félix Genaro Cabezas García.

12. Visualization: Leonor Alejandrina Zapata Aspiazu.

13. Writing - original draft: Edwin Haymacaña Moreno,

Francisco Javier Duque-Aldaz.

14. Writing - revision and editing: Francisco Javier Duque-

Aldaz, Félix Genaro Cabezas García, Raúl Alfredo

Sánchez Ancajima.

7.- Appendix.

R code used for the development of the

research.

### Packages

install.packages(c("readxl","dplyr","ggplot2","forecast","ts

eries",

"urca","TSstudio","broom","knitr","kableExtra"))

library(readxl); library(dplyr); library(ggplot2)

library(forecast); library(tseries); library(urca)

library(TSstudio); library(broom); library(knitr);

library(kableExtra)

# 2. Read Excel (file located in the working directory)

DF <- ReDXL::read_excel("matriculadosecuador.xlsx")

# 3. Quick Review

str(DF)

summary(df)

# 4. Create Time Series Object (Yearly)

and <- ts(df$Matriculacion, start=min(df$Year), frequency

= 1)

# Initial Exploration

autoplot(y) +

labs(title="Secondary Enrollment (% Gross) – Ecuador",

x="Year", y="%") +

theme_minimal(base_size = 12)

# ACF and PACF

ggAcf(y, lag.max = 30) + theme_minimal()

ggPacf(y, lag.max = 30) + theme_minimal()

###Diagnóstico of stationarity and transformations

# Unit Root Tests

tseries::adf.test(y) # H0: unit root (non-stationary)

tseries::kpss.test(y) # H0: estacionaria (si p<0.05, no

estacionaria)

# Suggested order of differentiation

forecast::ndiffs(y) # usually 1

###Diferenciar once (d = 1) and retest

#y_ts: Your Already Created Annual Series, Frequency 1

(1971–2023)

y_ts <- ts(df$Matriculacion, start = min(df$Año), frequency

= 1)

#1) First-Order Difference

dy <- diff(y_ts)

#2) Visualize the differentiated series

autoplot(dy) +

labs(title = "First difference in secondary enrolment (%)",

x = "Year", y = "Δ Enrolment (%)") +

theme_minimal(base_size = 12)

#3) Stationarity tests on the differentiated series

adf.test(dy) # H0: unit root (non-stationary)

kpss.test(dy, null = "Level")# H0: stationary at level

#4) Time structure of the differentiated series

ggAcf(dy, lag.max = 30) + theme_minimal(base_size = 12)

ggPacf(dy, lag.max = 30) + theme_minimal(base_size = 12)

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###Identificación/model estimation (candidates +

auto.arima)

# Exhaustive search (no seasonality)

fit_auto <- auto.arima(y_ts,

seasonal = FALSE, # anual

stepwise = FALSE, # most complete search

approximation = FALSE,

d = 1) # we already know that d = 1

fit_auto

#Luego, we tested some classic candidates and compared by

AIC, AICc, BIC:

cand <- list(

ARIMA_011 = Arima(y_ts, order = c(0,1,1)),

ARIMA_110 = Arima(y_ts, order = c(1,1,0)),

ARIMA_111 = Arima(y_ts, order = c(1,1,1)),

ARIMA_210 = Arima(y_ts, order = c(2,1,0)),

ARIMA_012 = Arima(y_ts, order = c(0,1,2))

)

cmp <- data.frame(

Model = names(cand),

AIC = sapply(cand, AIC),

BIC = sapply(cand, BIC)

)

print(cmp)

###Diagnóstico of waste of the chosen model

# Comprehensive diagnosis

checkresiduals(fit_auto) # includes Ljung–Box, ACF

Residuals and QQ-plot

# If you want explicit Ljung–Box with several lags:

Box.test(residuals(fit_auto), lag = 10, type = "Ljung")

Box.test(residuals(fit_auto), lag = 15, type = "Ljung")

###Validación out of sample (train/test) and metrics

# Temporary partition

y_tr <- window(y_ts, end = 2016)

y_te <- window(y_ts, start = 2017)

fit_tr <- auto.arima(y_tr, seasonal = FALSE, stepwise =

FALSE, approximation = FALSE, d = 1)

fc_te <- forecast(fit_tr, h = length(y_te))

# Validation Metrics

accuracy(fc_te, y_te)

autoplot(fc_te) + autolayer(y_te, series = "Observed") +

labs(title="Out-of-sample forecast (2017–2023)",

y = "% enrollment") + theme_minimal(base_size = 12)

###Pronóstico end (h = 5–10 years)

# Retrain with the whole series and predict

fit_all <- auto.arima(y_ts, seasonal = FALSE, stepwise =

FALSE, approximation = FALSE, d = 1)

fc_10 <- forecast(fit_all, h = 10)

autoplot(fc_10) +

labs(title = "Secondary Enrollment Forecast (%) –

Ecuador",

x = "Year", y = "%") +

theme_minimal(base_size = 12)

7.- References.

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1980–2016," UESS, 2017. URL:

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[8] J. I. Cañarte Murillo, "Restricted growth of the balance

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Chemical Engineering & Development

Journal of Science and Engineering

Vol. 08 / Nº 01

e – ISSN: 3028-8533

ISSN – L: 3028-8533

Chemical Engineering & Development

University of Guayaquil | Faculty of Chemical Engineering

Guayaquil – Ecuador

https://revistas.ug.edu.ec/index.php/iqd

Email: inquide@ug.edu.ec

francisco.duquea@ug.edu.ec

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