1. Introduction

2. Literature Review

3. Overview of the Moroccan and Bahraini Stock Markets

4. Materials and Methods

4.1. Econometric Approach

4.1.1. The Autoregressive Conditional Heteroscedastic (ARCH) model

ARCH models involve time series characterized by changing volatility over time (heteroscedasticity), which is conditional on the autocorrelation of previous lags (autoregressive conditional). Referring to the seminal paper by Engle (1982), a variable $Y_t$ follows an ARCH (p) process if:

$$\left.y_t\right|{\psi }_{t-1}~N\left(0,h_t\right)$$

(1)

$$h_t={\alpha }_0+{\sum }_{i=1}^p{\alpha }_i{y}_{t-1}^2$$

(2)

where, ${\psi }_{t-1}$ is the information set available at time t-1, $h_t$ is the conditional variance function, p is the order of ARCH process and $\alpha$ is a vector of unknown parameters.

To guarantee positive variance, the parameters must satisfy some conditions, such that, $\forall t\in \mathbb{Z},{\alpha }_0>0and{\alpha }_i>0$. Furthermore, recent past have to had more influence than older lags, so: ${\alpha }_1>{\alpha }_2>\dots >{\alpha }_P$.

To estimate our ARCH model, we will follow the steps below:

Figure 1. Illustration of ARCH model estimation steps.

Figure 1. Illustration of ARCH model estimation steps.

4.1.2. The Generalized Autoregressive Conditional Heteroscedastic (GARCH) Model

GARCH models were introduced in 1986 by Bollerslev as an extension of ARCH models. They incorporate both autoregressive and moving average terms in the volatility equation. In addition to past squared residuals, GARCH models also include lagged conditional variances in the model specification. This allows them to capture both short-term volatility clustering and long-term persistence in volatility, making them more flexible and capable of modeling various volatility patterns observed in financial time series (Francq and Zakoian 2019).

Moreover, GARCH models provide a parsimonious alternative to high order ARCH models, which can become problematic when estimating many ARCH effects ((Bhowmik and Wang 2020).

The formalities of GARCH models involve both terms p and q, where in ARCH you only have the p term. So, a variable $Y_t$ follows a GARCH (p,q) process if:

$$\left.y_t\right|{\psi }_{t-1}~N\left(0,h_t\right)$$

(3)

$$h_t={\alpha }_0+{\sum }_{i=1}^p{\alpha }_i{y}_{t-1}^2+{\sum }_{i=1}^q{\beta }_i{h}_{t-1}$$

(4)

where ${\alpha }_0>0,{\alpha }_i>0and{\beta }_i>0$.

To estimate our GARCH model, we will follow the steps below:

Figure 2. Illustration of GARCH model estimation steps.

Figure 2. Illustration of GARCH model estimation steps.

In summary, the conditional variance equation in ARCH and GARCH models serves to capture the time-varying nature of volatility in financial market data. This specification is essential in financial contexts, where analysts seek to predict future volatility based on past observations and information. In these models, the conditional variance at each time point is estimated as a weighted average of several components (Abdalla and Winker 2012). Firstly, a long-term average (represented by the constant term) provides a baseline estimate of volatility. Secondly, the GARCH term incorporates information from the previous period’s forecast variance, contributing to the persistence of volatility over time. Finally, the ARCH term captures the impact of unexpected asset returns on volatility, adjusting the estimate of variance based on the magnitude and direction of these returns. This dynamic interplay between past volatility, current information, and unexpected returns allows ARCH and GARCH models to effectively capture the complex dynamics of financial market volatility.

4.2. Deep Learning Approach

4.2.1. LSTM Network

Modeling and predicting future values of time series using artificial neural networks presents a challenge in retaining past information for accurate forecasting. Recurrent Neural Networks (RNNs) address this challenge by incorporating memory mechanisms in their hidden layers, allowing them to store and recall past states. However, simple RNNs are prone to gradient explosion, where gradients grow uncontrollably during training, affecting model stability (Kanai et al. 2017). To overcome this limitation, Long Short-Term Memory (LSTM) networks were developed. LSTM networks feature gated mechanisms that regulate the flow of information, preventing gradient explosion and enabling effective long-term memory retention. These networks learn to prioritize relevant information for prediction, enhancing forecasting accuracy. LSTM networks have proven effective for financial time series forecasting due to their ability to handle sequential data and long-term dependencies (Yu et al. 2019). However, training an LSTM network requires careful parameter selection to optimize performance (Sako et al. 2022).

In our study, we’ve developed and trained a neural network with a range of parameters, as depicted in the Table 1 below:

4.2.2.1. D-CNN Network

A Convolutional Neural Network (CNN) is a versatile machine learning algorithm applied extensively in diverse fields, such as image processing, speech recognition, and time series analysis. Originally tailored for image data, CNNs necessitate two-dimensional input. However, given that time series data typically unfolds in one dimension, a specialized adaptation known as a one-dimensional CNN has emerged (Markova 2022). This model operates on a single sequence, potentially incorporating multiple convolutional layers and a pooling layer to distill key features. Subsequently, a fully connected dense layer interprets these features, facilitated by a flattening layer to reduce dimensionality. To enhance performance, we meticulously curated and refined parameters using our comprehensive training dataset, with the objective of minimizing prediction errors, as illustrated in the following table:

Table 2. CNN Network parameters.

Table 2. CNN Network parameters.

Network Parameters	Values
Data standardization formula	$x^{\mathrm{\text{'}}}=\raisebox{1ex}{$(x-x_{min})$}\!\left/ \!\raisebox{-1ex}{$(x_{max}-x_{min})$}\right.$
Optimization Algorithm	Adam
Activation function	ReLU
Number of Iterations (Epochs)	1000
Gradient Threshold	ReLU
Number of filter units	64
Number of kernels	2
Batch size	32
Dense units	1
Training Rate	0,9
Testing Rate	0,1

5. Data and Descriptive Statistics

Figure 3. (a) Volatility clustering for BAX index; (b) Volatility clustering for MASI index.

Figure 3. (a) Volatility clustering for BAX index; (b) Volatility clustering for MASI index.

Table 3. Descriptive statistics of the BAX and MASI return series.

6. Empirical Results and Discussion

6.1. Empirical Results

6.1.1. Testing for Stationarity

To test the stationarity of our two-return series, we employed the Augmented Dickey-Fuller (ADF) unit root test. This test is commonly used to determine whether a time series is stationary or non-stationary by assessing the null hypothesis regarding the presence of a unit root in the data. The results of the test are summarized in the table below, providing insights into the stationarity properties of the BAX and MASI return series:

Table 4. Stationarity test results for the BAX and MASI return series.

Table 4. Stationarity test results for the BAX and MASI return series.

Variables	ADF Test	T-Statistics	P-Values	Hypothesis
BAX	Intercept	-15,88660*	0,0000	Null hypothesis rejected
	Trend and intercept	-15,88105*	0,0000	Null hypothesis rejected
	None	-15,81436*	0,0000	Null hypothesis rejected
MASI	Intercept	-21,16787*	0,0000	Null hypothesis rejected
	Trend and intercept	-21,15932*	0,0000	Null hypothesis rejected
	None	-21,17456*	0,0000	Null hypothesis rejected

Notes: * indicates significance at the 5% level.

The Augmented Dickey-Fuller (ADF) test results indicate that the null hypothesis for both the BAX and MASI return series is rejected across all specifications. This suggests that the return series for both indices are stationary in level, indicating that they exhibit stable mean and variance over time.

After analyzing the correlogram and evaluating various specifications, we have concluded that the most suitable model for representing the mean equation of both series is ARMA (1,1).

6.1.2. Testing for Heteroscedasticity

We conducted the heteroscedasticity test using the ARCH effect specification, under the null hypothesis of no existing ARCH effect in the residual of ARMA (1,1) equation up to 1 lag. The results for both series are summarized in the table below:

Table 5. Heteroscedasticity test results for the BAX and MASI residual return series.

Table 5. Heteroscedasticity test results for the BAX and MASI residual return series.

	F-Statistics	Prob.	Chi Square-Statistics	Prob.	Hypothesis
BAX	182,6009*	0,0000	159,1391*	0,0000	Null hypothesis rejected
MASI	37,99301*	0,0000	36,92633*	0,0000	Null hypothesis rejected

The results of the ARCH test, as indicated by the F-Statistics and Chi Square-Statistics, reveal that the null hypothesis of no ARCH effect is rejected for both the BAX and MASI residual series. This suggests a pattern of volatility clustering, where significant changes are often succeeded by further significant changes, regardless of their direction, while minor changes tend to be followed by additional minor changes (Joshi 2010). Such findings have significant implications for modeling and forecasting, as ignoring heteroscedasticity can lead to biased parameter estimates and inaccurate predictions.

6.1.3. Estimation Results

The following Table 6 summarizes the predictive performance of the various models employed in our analysis. The selection of the ARCH (1) and GARCH (1,1) models was based on their ability to capture time-varying volatility in our financial return series. In our analysis, we extended the lag length beyond the first lag, observing that additional lags did not contribute significantly to the ARCH effect. This observation was validated by conducting the ARCH-LM test statistics at various lag lengths. The statistically insignificant p-values obtained for lag lengths beyond the first lag suggest that there is no ARCH effect remaining in the models. This implies that the chosen ARCH (1) and GARCH (1,1) specifications adequately capture the dynamics of volatility in the BAX and MASI return series.

Table 6. Historical volatility prediction: comparative analysis for different models.

Table 6. Historical volatility prediction: comparative analysis for different models.

	ARCH (1)		GARCH (1,1)		LSTM		1D CNN
	BAX	MASI	BAX	MASI	BAX	MASI	BAX	MASI
RMSE	0,0055	0,0081	0,0055	0,0081	0,0447	0,0810	0,0447	0,0812
MAE	0,0033	0,0049	0,0034	0,0049	0,0346	0,0453	0,0346	0,0455
MAPE (%)	154,913	166,035	161,52	157,854	86,638	88,013	86,145	89,399

Observing the results, we can see that the ARCH (1) and GARCH (1,1) models consistently outperform the LSTM and 1D CNN models across all metrics and for both indices. This is evident from the lower RMSE, MAE, and MAPE values achieved by the ARCH and GARCH models compared to the LSTM and CNN models.

These findings suggest that the parsimonious ARCH and GARCH models are more effective in capturing the volatility dynamics of the BAX and MASI indices compared to the more complex LSTM and CNN models. Despite their computational complexity, the deep learning models do not exhibit superior predictive performance in this context.

7. Conclusion and Perspective

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