6.1.1 Model identification.

As shown in Fig 1, the original time series exhibited a non-stationary mean. To stabilise the mean of COVID-19 cases, we first applied a logarithmic transformation to the data, followed by both first and second-order differencing. The stationarity of the transformed series was then assessed using the ADF and KPSS tests.

The ADF test produced a test statistic of -13.457 with a lag order of 9 and a p-value of 0.01, leading to the rejection of the null hypothesis of non-stationarity, confirming that the series is stationary. Similarly, the KPSS test returned a test statistic of 0.0062082 with a truncation lag parameter of 6 and a p-value of 0.1. Since the p-value exceeds conventional significance levels (0.05), we fail to reject the null hypothesis, indicating that the series is level stationary. Together, these results confirm the stationarity of the transformed series.

All subsequent statistical analyses were conducted on the transformed COVID-19 case data. Based on the distributional characteristics and autocorrelation structure, five ARIMA models were evaluated: ARIMA(5,2,7), ARIMA(4,2,7), ARIMA(6,2,7), ARIMA(6,2,5), and ARIMA(7,2,6). Model performance was compared using both information criteria (AIC, AICc, and BIC) and out-of-sample forecast accuracy metrics (RMSE, MAE, and MAPE), as presented in Table 4.

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Table 4. Comparison of precision measures for models.

https://doi.org/10.1371/journal.pone.0328202.t004

While several models exhibited similar RMSE and MAE values, ARIMA(5,2,7) achieved the lowest AIC and AICc, indicating the best overall trade-off between model fit and parsimony. Although ARIMA(4,2,7) and ARIMA(6,2,7) produced comparable forecast errors, their higher BIC values suggested less efficient parameterization. The final selection of ARIMA(5,2,7) therefore balanced goodness-of-fit with predictive stability, supported by the ACF and PACF structures shown in Figs 5 and 6. The ACF displayed significant correlations up to lag 7, reinforcing the inclusion of a moving average component of order 7, while the PACF confirmed the suitability of an autoregressive component of order 5.

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Fig 5. Second-order differences of log-transformed COVID-19 cases (per 1,000,000 pop.).

https://doi.org/10.1371/journal.pone.0328202.g005

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Fig 6. Autocorrelation (ACF, top) and Partial Autocorrelation (PACF, bottom) plots for the second-order differenced COVID-19 cases, with 99% confidence intervals.

https://doi.org/10.1371/journal.pone.0328202.g006

6.1.2 Model diagnosis.

The normal probability plot in Fig 7 showed that the residuals followed the theoretical normal line closely, with only minor deviations in the tails, supporting approximate normality. The histogram, together with the kernel density estimate, confirmed this impression by displaying a symmetric, bell-shaped distribution centered around zero and without extreme outliers. The plot of residuals versus fitted values exhibited no systematic trend or funnel shape, indicating that the assumption of constant variance was reasonable. Finally, the plot of residuals against observation order revealed no obvious cycles or drifts, suggesting that the residuals were largely independent.

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Fig 7. Residual diagnostics for the fitted model.

The top panel shows the standardised residuals over time. The bottom left panel displays the ACF of the residuals, and the bottom right panel presents the histogram of residuals to assess the normality of residuals.

https://doi.org/10.1371/journal.pone.0328202.g007

This is supported by the Ljung-Box statistics (Table 5), which yield non-significant p-values. Specifically, Table 5 displays the modified Box-Pierce (Ljung-Box) statistic. It is evident that for all lags, the p-values exceed the 0.05 significance level. This non-significance suggests that the model is appropriate. Therefore, the assumption of independence is not violated. Given that the assumptions are satisfied, the model can be considered valid for prediction.

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Table 5. Modified Box-Pierce (Ljung-Box) statistic.

https://doi.org/10.1371/journal.pone.0328202.t005

Furthermore, Table 1 in the appendix compares observed COVID-19 case values with the ARIMA forecasts, showing absolute percentage errors between 4–12%, while Table 2 reports the corresponding 99% prediction intervals, which widen across the forecast horizon and illustrate the growing uncertainty of projections.

To evaluate short-term predictive performance, a rolling-origin cross-validation procedure was applied using a 3-day-ahead forecast horizon. The ARIMA(5,2,7) model was sequentially re-estimated over an expanding training window of 365 days, producing 421 out-of-sample forecasts. Forecast accuracy was assessed using the RMSE, MAE, and MAPE, see Table 6.

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Table 6. Rolling-origin 3-day-ahead forecast accuracy for Morocco COVID-19 cases.

https://doi.org/10.1371/journal.pone.0328202.t006

Fig 8 shows that the forecasts align closely with the observed infection dynamics, capturing both the timing and magnitude of Morocco’s main COVID-19 waves between 2021 and 2022. The relatively low RMSE and MAE values indicate that the ARIMA(5,2,7) model achieves strong short-term predictive accuracy, while the MAPE of 9.78% confirms reliable performance in tracking daily case variations. Forecasts tend to slightly overestimate during sharp outbreak peaks, reflecting the inherent difficulty of modelling sudden, high-velocity surges in infection rates. Nonetheless, the model provides robust 3-day-ahead forecasts, demonstrating that short-term COVID-19 case dynamics in Morocco can be effectively captured using autoregressive time-series techniques.

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Fig 8. Rolling-origin 3-day-ahead forecasts of Morocco COVID-19 cases using the ARIMA (5,2,7) model.

The solid blue line represents observed daily cases (per 1,000,000 population), and the dashed red line shows 3-day-ahead predictions.

https://doi.org/10.1371/journal.pone.0328202.g008

6.1.3 Discussion.

The ARIMA(5,2,7) model achieved a MAPE of 9.19%, indicating strong predictive accuracy. This aligns with comparable Moroccan studies: for instance, [29] reported a much higher MAPE of 41.18%, while [30] obtained 51.25% using ARIMA(3,2,0). Other approaches have produced mixed results: [31] reported MAPEs of 29.31% (LSTM), 37.01% (Prophet), 215.87% (Auto-ARIMA), and 83.41% (Random Forest), whereas [32] achieved values between 6.12% and 9.82% using Prophet with a longer forecast horizon. At a regional scale, [19] found an RMSE of 11.78 in Maghreb countries using Random Forest, and [33] reported a MAPE of 184.81% in Morocco with exponential smoothing. Deep learning methods have also been tested: [34] implemented LSTM-based models with 98.6% accuracy, though they highlighted issues such as vanishing and exploding gradients. Compared with these diverse approaches, our ARIMA model is parsimonious, transparent, and delivers competitive accuracy.

Several factors explain this performance. Our analysis used more than 1,500 daily observations—far exceeding the minimum 30 recommended by [35]—enhancing robustness. In addition, AIC values below –2000 indicate a superior fit relative to models in [29], where tentative models exceeded 3000. The three-day forecast horizon was deliberately chosen, as ARIMA is known to lose reliability under structural breaks such as lockdowns or vaccine rollouts [36].

From a policy perspective, these findings demonstrate that ARIMA-based short-term forecasts could support Morocco’s National Institute for Hygiene in resource allocation, hospital preparedness, and daily monitoring. By serving as an early-warning tool, such forecasts may guide timely interventions and reduce uncertainty for decision-makers. Similar short-term forecasting strategies have proven effective in Guangdong and Zhejiang [37] and elsewhere [38].

Nevertheless, limitations remain. Our validation was limited to in-sample diagnostics, with no independent out-of-sample testing. Rolling-window cross-validation, as recommended in epidemic modeling [39], should be explored in future work. The model also cannot capture abrupt policy shocks, and data derived from passive surveillance systems are subject to underreporting biases.

Environmental factors further complicate forecasts. The correlation between infectious diseases and climate is well established—e.g., dengue [40,41], COVID-19 [42], and tuberculosis [43]. In Morocco, [20] emphasised the importance of including meteorological covariates in epidemic modeling. Future research should therefore test ARIMAX models with temperature, humidity, and immunity-related variables to capture seasonality and improve robustness [44].

In summary, our ARIMA model provides strong short-term predictive accuracy, compares favourably to both traditional and machine-learning methods, and offers practical value for Moroccan public health policy. However, its utility is limited to short horizons, and future extensions should incorporate exogenous covariates and rolling validation for improved generalisability.

6.2.1 Model identification.

Fig 9 illustrates that the original time series displayed a non-stationary mean. To stabilise this, a logarithmic transformation was applied to the COVID-19 case data, followed by first-order differencing. The stationarity of the resulting series was evaluated using both the ADF and KPSS tests.

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Fig 9. Residual diagnostics for the fitted model.

The top panel shows the standardized residuals over time. The bottom left panel displays the ACF of the residuals, and the bottom right panel presents the histogram of residuals to assess the normality of residuals.

https://doi.org/10.1371/journal.pone.0328202.g009

The ADF test yielded a test statistic of -6.7809 with a lag order of 8 and a p-value of 0.01, allowing us to reject the null hypothesis of non-stationarity and conclude that the series is stationary. Similarly, the KPSS test produced a test statistic of 0.27656 with a truncation lag parameter of 6 and a p-value of 0.1. Since this p-value exceeds conventional significance levels, we fail to reject the null hypothesis, indicating level stationarity. These results collectively confirm the stationarity of the transformed series.

All further statistical analyses were performed on the transformed COVID-19 case data. Based on the observed distributional properties, five ARIMA models were considered: ARIMA(3,1,7), ARIMA(2,1,7), ARIMA(4,1,7), ARIMA(5,1,7), and ARIMA(7,1,3). Model performance was assessed using a combination of information criteria (AIC, AICc, and BIC) and forecast accuracy measures (RMSE, MAE, and MAPE), as shown in Table 7.

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Table 7. Comparison of precision measures for models.

https://doi.org/10.1371/journal.pone.0328202.t007

Although ARIMA(3,1,7) achieved a slightly lower AIC than ARIMA(2,1,7), the difference was marginal, while ARIMA(2,1,7) produced a distinctly lower BIC value. Consequently, ARIMA(2,1,7) was selected as the final specification, offering the best balance between fit, interpretability, and generalization. The chosen configuration indicates that short-term fluctuations in the MASI index depend primarily on two autoregressive lags and seven moving-average components, effectively capturing rapid market adjustments and persistent sentiment effects characteristic of emerging markets.

6.2.2 Model diagnosis.

For the MASI opening price series in Fig 9, the normal probability plot indicated that residuals aligned closely with the theoretical normal line, with only moderate tail deviations, supporting the assumption of approximate normality. The histogram, complemented by the kernel density estimate, displayed a centered, symmetric distribution without severe skewness or outliers. The residuals versus fitted values plot showed no systematic patterns or funnel effects, suggesting homoscedasticity. Finally, the residuals versus order plot showed no visible autocorrelation, confirming that the residuals were largely independent over time.

Compared with the COVID-19 case series in Fig 7, the MASI residuals appeared slightly tighter around zero, consistent with financial market data being more stationary after differencing. By contrast, the COVID-19 residuals displayed heavier tails and more variability, reflecting the influence of epidemic waves and policy shocks. Together, the diagnostics suggest that while both models are statistically adequate, forecasts for COVID-19 carry higher uncertainty due to underlying structural volatility, whereas MASI forecasts remain more stable.

The Ljung–Box test results in Table 8 further confirm model adequacy, as all p-values exceed 0.05, supporting the absence of serial correlation. These results imply that the ARIMA(2,1,7) model successfully captures the main temporal dependencies in daily MASI price dynamics.

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Table 8. Modified Box-Pierce (Ljung-Box) statistic.

https://doi.org/10.1371/journal.pone.0328202.t008

To evaluate short-term predictive performance, a rolling-origin cross-validation procedure was applied using a 3-day-ahead forecast horizon. The ARIMA(2,1,7) model was sequentially re-estimated over an expanding training window of 250 trading days, producing 288 out-of-sample forecasts. Forecast accuracy was assessed using the RMSE, MAE, and MAPE, see Table 9.

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Table 9. Rolling-origin 3-day-ahead forecast accuracy for the MASI opening price.

https://doi.org/10.1371/journal.pone.0328202.t009

Fig 10 shows that the forecasts closely follow the observed MASI trajectory, successfully capturing both the upward trend and the sharp decline in early 2022. The model achieved an RMSE of 154.64, an MAE of 114.25, and a MAPE of 0.89%, indicating excellent short-term predictive accuracy for financial time series. When compared with the COVID-19 forecast evaluation (RMSE = 12.56, MAPE = 9.78%), the MASI model exhibits substantially lower relative error, which is expected given the smoother and more stable behaviour of market price movements compared to the highly volatile infection data. Nevertheless, both models demonstrate strong short-horizon forecasting ability within their respective domains, confirming the suitability of ARIMA-based approaches for capturing short-term autoregressive patterns. Minor divergences during sudden corrections highlight the MASI model’s limited responsiveness under abrupt volatility, suggesting potential benefits from integrating volatility-sensitive extensions such as ARIMAX–GARCH in future research.

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Fig 10. Rolling-origin 3-day-ahead forecasts of the MASI opening price using the ARIMA(2,1,7) model.

The solid blue line represents observed prices, and the dashed red line shows 3-day-ahead predictions.

https://doi.org/10.1371/journal.pone.0328202.g010

6.2.3 Discussion.

The MASI index experienced a sharp decline in March 2020, followed by a relatively rapid recovery between late 2020 and early 2021. This pattern reflected heightened uncertainty, sector-specific shocks (notably a 50% fall in property development and a one-third loss in banking) [24], and later improvements in sentiment driven by low bond yields, vaccine rollouts, and resumed dividends.

Our ARIMA(2,1,7) model achieved strong short-term predictive performance with a MAPE of 6.56%. This compares favourably with [18], who reported 26.7% using ARDL, and with deep learning approaches such as LSTM, where [45] found RMSE = 105.5 and MAE = 72.8, versus our values near 1. Other comparative studies also show ARIMA’s relative efficiency: [46] found ARDL outperformed LSTM and XGBoost, while [47] reported higher errors for SVR and hybrid models.

Economically, these results suggest that short-run MASI dynamics during COVID-19 were primarily driven by short-term expectations and sentiment responses, rather than long-term fundamentals. The dominance of moving-average terms indicates that shocks to returns (e.g., government announcements, infection surges) had transient but measurable impacts that decayed within roughly a week.

From a policy and investor perspective, accurate daily forecasts of MASI volatility can serve as a short-term risk management tool. For regulators, this offers the potential to anticipate market stress, allocate liquidity support, and monitor contagion effects across sectors. For investors, such models can aid in position sizing, stop-loss calibration, and timing strategies during crisis episodes.

Despite these strengths, limitations remain. Validation was limited to in-sample diagnostics; rolling-window or out-of-sample tests should be used in future work to assess temporal stability. Moreover, as financial markets exhibit nonlinearity and regime shifts, ARIMA’s linear framework cannot fully capture sudden structural breaks or policy shocks. As [48,49] emphasise, financial markets are inherently nonlinear and noisy, making consistent prediction challenging.

Future research should therefore integrate exogenous variables—such as oil prices, exchange rates, and COVID-19 case numbers—into an ARIMAX framework to capture cross-market linkages. Alternatively, hybrid ARIMA–machine learning models could be explored to combine statistical transparency with adaptive learning, improving predictive resilience under high volatility.

6.3.1 Unit root testing: Augmented Dickey-Fuller (ADF) test.

The ADF test regression equation is

(1)

Here, t is the time trend, p is the number of lagged differences, and denotes the first difference of y_t. The null hypothesis is

• H₀: (the series has a unit root; it is non-stationary),
• H₁: (the series is stationary).

If the null hypothesis is rejected, the series is considered stationary.

6.3.2 Vector Autoregression (VAR) model.

Once the stationarity properties of the series are understood, a Vector Autoregression (VAR) model can be constructed to explore the dynamic interdependencies among the variables. The VAR(k) model is specified as

(2)

where Y_t is a vector of endogenous variables, are coefficient matrices, and is the vector of white noise errors. The optimal lag length k is selected using AIC or Schwarz Information Criterion (SIC).

6.3.3 Causality analysis: Toda-Yamamoto and Granger Causality tests.

To test for causal relationships between variables, we use both the Toda-Yamamoto (1995) and standard Granger causality approaches. The Toda-Yamamoto method is robust to the integration and cointegration properties of the series. First, the maximum order of integration () is determined using unit root tests (e.g., ADF). Then, the VAR is estimated with lags. The system of equations becomes

(3)

(4)

The null hypothesis is tested using the Modified Wald (MWALD) test

• H₀: (no causality from X to Y),
• H₁: (causality from X to Y).

If the Wald statistic exceeds the critical value from the Chi-Square distribution with k degrees of freedom, we reject the null hypothesis. Alternatively, standard Granger causality testing estimates

(5)

(6)

We test whether past values of w_t significantly help to predict z_t, and vice versa.

To determine the optimal lag length for the VAR model, we primarily use the AIC as it is considered the best indicator for balancing model fit and complexity. In Table 5 (see the appendix in S1 File), we see that the AIC suggests a lag length of 17 , but this is relatively large for our analysis. To address this, we select the smallest lag with a similar AIC value, which is 10, still yielding a value of -15.12. Based on this lag length, we proceed to test for potential causal relationships between the two variables using the Toda-Yamamoto approach to Granger causality.

The results of the Toda-Yamamoto Granger causality tests in Table 10 provide key insights into the relationship between COVID-19 case counts and the MASI index. Our findings indicate a causal relationship from COVID-19 cases to the MASI index, as the null hypothesis is rejected with a p-value of 0.035, suggesting that fluctuations in COVID-19 cases in Morocco Granger-cause movements in the MASI index. However, no such causality is observed in the reverse direction, as the p-value for the test from the MASI index to COVID cases is 0.348, failing to reject the null hypothesis. This suggests that changes in the MASI index do not Granger-cause changes in reported COVID-19 cases. Therefore, while there is evidence of causality from COVID-19 cases to the MASI index, no causal relationship is found between the MASI index and COVID-19 case counts at the 5% significance level.

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Table 10. Granger causality Wald tests (Toda-Yamamoto).

https://doi.org/10.1371/journal.pone.0328202.t010

[18] explored the impact of COVID-19 on the Moroccan financial market using an ARDL model and found a one-way causal relationship from COVID-19 cases to the MASI index, supported by a significant Toda-Yamamoto causality test result (p = 0.0005). They also reported no causality in the reverse direction, from the MASI index to COVID-19 cases (p = 0.0857). Our findings align with these results, as our Wald test statistic similarly failed to show evidence of causality in either direction. The consistency between these studies enhances the robustness of our conclusions, suggesting that fluctuations in COVID-19 cases may influence the MASI index, but not vice versa.

Moreover, their analysis identified a significant long-term relationship between the two variables, expressed as

(7)

This negative coefficient suggests that an increase in COVID-19 cases is associated with a long-term decline in the Moroccan stock market index. Furthermore, [51] highlighted the broader economic repercussions of the pandemic, particularly in commodity markets. Crude oil—often regarded as the most critical global commodity—experienced sharp price declines due to reduced global demand, stemming from lockdowns and travel restrictions. These demand shocks not only depressed oil prices but also contributed to wider inflationary and deflationary pressures across economies.

The link between COVID-19 incidence and economic indicators like Crude Oil and stock prices is both direct and profound. As confirmed cases rose, public health measures tightened, leading to reductions in industrial activity and transportation. This, in turn, decreased fuel consumption and commodity demand, driving prices down [52]. Similarly, increased uncertainty prompted mass sell-offs in equity markets, leading to stock price declines. These dynamics highlight the interconnectedness of health, energy, and financial markets and the importance of including multi-source data for improved forecasting.

This study contributes to the emerging literature on the financial implications of the COVID-19 pandemic by investigating the relationship between daily Moroccan stock prices, the MASI index, and COVID-19 cases using ARIMA models. While previous research has modeled the effects of pandemics on macroeconomic indicators or global markets, there is a notable lack of work focusing on North African economies or on a daily timescale. Our model adds to the discourse by providing a localised, high-frequency time series analysis of how the pandemic and oil price fluctuations interact with financial market performance in Morocco.

Future work could benefit from incorporating exogenous variables into the forecasting model, particularly using ARIMAX or machine learning-based architectures such as LSTM. As shown by [53], oil markets are intricately linked to both economic health and infection trends, and including such indicators may improve forecast accuracy.

This research also underscores the importance of localised studies. While global modeling efforts are valuable, national economies like Morocco require specific, data-driven strategies to understand their vulnerabilities and resilience during crises. We encourage future research to explore regional differences in epidemic impact, especially in underrepresented financial markets.