Implementation of the ARIMA model for prediction of economic variables: evidence from the health sector in Brazil

The study addresses the gap in empirical research on quantitative models to forecast economic crises and their impacts on health systems, particularly in emerging economies such as Brazil. Economic downturns—characterized by GDP contractions, inflation volatility, and rising unemployment—exert pressure on both public (SUS) and private (SSS) health sectors. Brazil has experienced multiple crises over the last five decades, with notable effects on healthcare utilization patterns, including beneficiary shifts between private and public systems during the 2019 unemployment crisis. The research question centers on whether ARIMA models can effectively forecast key economic time series relevant to healthcare management during crises. The paper proposes applying ARIMA to five series (GDP, IPCA, unemployment, total health plan beneficiaries, and individual health plan beneficiaries) from 2000–2020 to provide accurate, actionable forecasts for strategic planning in the health sector.

Design and data: Ex-post-facto analysis using secondary data from the Institute of Applied Economic Research (IPEA) and the National Supplementary Health Agency (ANS). The dataset spans 2000–2020 for five series: GDP, IPCA (inflation), unemployment rate, total health plan beneficiaries (SUS + SSS), and individual health plan beneficiaries (SSS). Modeling approach: The Box-Jenkins ARIMA framework was followed: (1) series analysis and stationarity assessment; (2) differencing to achieve stationarity (order d); (3) identification of AR (p) and MA (q) orders via ACF/PACF; (4) parameter estimation by maximum likelihood (MLE); (5) model selection using information criteria (AIC) with penalty for added parameters; (6) diagnostic checking via residual analysis and Portmanteau tests to ensure white-noise residuals. Specification details: ARIMA(p,d,q) models were considered with low orders (0–2) typical for practice. Formal expressions for AR, MA, ARMA, and ARIMA were provided, including the ARIMA(1,1,1) difference-equation form. Seasonal forms and forecasting equations were illustrated conceptually. Model selection was guided by minimizing AIC (or its approximation where needed), and out-of-sample performance metrics. Training/validation/forecast protocol: Models were trained on 2000–2010, validated against 2011–2015 (out-of-sample), then re-estimated using 2000–2015 to generate forecasts for 2016–2020. Software: GRETL. Error metrics: Mean Absolute Percentage Error (MAPE) and Theil’s U (one-step-ahead inequality coefficient). Lower MAPE and Theil’s U indicate better performance. Selected model orders by series (p,d,q): GDP (1,0,2); IPCA (2,2,1); Unemployment (0,1,2); Total beneficiaries (1,1,2); Individual beneficiaries (2,2,1).

- Across the five series, ARIMA models achieved high predictive accuracy during 2011–2015 validation and were then used to forecast 2016–2020. - Selected ARIMA orders and performance (out-of-sample 2011–2015; Table 2): • GDP: ARIMA(1,0,2); MAPE 9.16; Theil’s U 0.617. • IPCA: ARIMA(2,2,1); MAPE 15.04; Theil’s U 0.167. • Unemployment rate: ARIMA(0,1,2); MAPE 4.14; Theil’s U 0.031; validation accuracy reported at 95.86%. • Total health plan beneficiaries (SUS + SSS): ARIMA(1,1,2); MAPE 1.27; Theil’s U 0.069; validation accuracy reported at 98.93%. • Individual health plan beneficiaries (SSS): ARIMA(2,2,1); MAPE 2.89; Theil’s U 0.577; validation accuracy reported at 97.11%. - For all series, residual diagnostics based on ACF/PACF with a 5% significance threshold supported adequate model fit, with statements that prediction accuracy reached at least 95% in figures. - The approach provides actionable forecasts of key health-relevant economic indicators, particularly unemployment and beneficiary counts, informing strategic planning in Brazil’s health sector during downturns. - Descriptive context: GDP variability (mean ~1.67%, SD 2.76%), IPCA variability (mean ~6.83%, SD 2.55%), unemployment mean ~8.43% (SD 1.33%); rising trend in total and collective beneficiaries, with variability noted for individual plans.

The findings demonstrate that parsimonious ARIMA models can capture the dynamics of key economic indicators that influence Brazil’s public (SUS) and private (SSS) health systems. By accurately forecasting unemployment and beneficiary trends, decision-makers can anticipate shifts from private to public coverage during economic stress, plan capacity, and allocate resources. The models’ validation performance (MAPE and Theil’s U) indicates reliable short-term predictive capability across heterogeneous series with differing degrees of differencing and AR/MA components. The study’s results address the core research question by evidencing that ARIMA, selected via ACF/PACF and AIC and validated out-of-sample, yields sufficiently accurate forecasts to inform strategic planning in health services during crises. These insights are particularly salient for episodes like the 2015–2016 downturn and the 2019 unemployment crisis, where forecasting can help mitigate pressures on SUS and guide SSS operators’ strategies.

The study implemented ARIMA models to forecast five critical economic time series—GDP, IPCA, unemployment, total health plan beneficiaries, and individual beneficiaries—relevant to Brazil’s health sector during crises (2000–2020). Using a structured training (2000–2010), validation (2011–2015), and forecasting (2016–2020) protocol, the selected ARIMA orders—(1,0,2), (2,2,1), (0,1,2), (1,1,2), and (2,2,1)—produced accurate out-of-sample predictions, with MAPE and Theil’s U indicating robust performance and reported accuracies often exceeding 95%. The results underscore ARIMA’s practicality, transparency, and ease of implementation for health-sector planning, especially to anticipate unemployment dynamics and shifts between SSS and SUS. Future research should compare ARIMA with non-linear and AI-driven approaches (e.g., ARIMAX, machine learning, fuzzy logic), and assess performance under major shocks (e.g., COVID-19), to enhance robustness and capture external drivers. Refining ARIMA parameters for specific sub-sectors and integrating exogenous variables could further improve forecasting utility for policy and operations.

- External validation constraints: For several series, observed data for 2016–2020 were not available at the time to validate ex-post forecasts, limiting confirmation of long-horizon performance. - Model assumptions: ARIMA requires stationarity after differencing and assumes that past patterns persist; abrupt structural breaks (e.g., severe crises, pandemics) may degrade accuracy. - Parameter uncertainty: MLE parameter estimates carry uncertainty; model selection via information criteria may still yield misspecification under regime changes. - Linear structure: ARIMA does not incorporate exogenous drivers by default and may underperform when external shocks dominate; ARIMAX or non-linear/ML methods could address this. - Forecast horizon: Accuracy may decline with longer horizons; ARIMA is typically strongest for short-term forecasts.