Estimating Volatility of Saudi Stock Market Using Hybrid Dynamic Evolving Neural Fuzzy Inference System Models

Abstract

This paper examines the volatility risk in the KSA stock market (Tadawul), with a specific focus on predicting volatility using the logarithm of the standard deviation of stock market prices (LSCP) as the output variable. To enhance volatility prediction, it proposes the combined use of the dynamic evolving neural fuzzy inference system (DENFIS) and the nonlinear spectral model, maximum overlapping discrete wavelet transform (MODWT). This study utilizes a dataset comprising 4609 observations and investigates the inputs of lag 1 of the close stock price (LCP), the natural logarithm of oil price (Loil), the natural logarithm of cost of living (LCL), and the interbank rate (IB), determined through autocorrelation (AC), partial autocorrelation (PAC), correlation, and Granger causality tests. Regression analysis reveals significant effects of variables on LSCP: LCP has a negative effect, and Loil has a positive effect in the ordinary least square (OLS) model, while LCL and IB have positive effects in the fixed effect model and negative effects in the random effect model. The MODWT-Haar-DENFIS model was developed as we found that the model has the potential to be an effective model for stock market forecasting. The results provide valuable insights for investors and policymakers, aiding in risk management, investment decisions, and the development of measures to mitigate stock market volatility.

1. Introduction

Volatility risk refers to the potential for large and rapid fluctuations in the prices of financial instruments or assets. It represents the uncertainty and instability in the market, reflecting the degree of variation or dispersion in the returns of an investment over a specific period. This risk is a crucial aspect of financial markets and affects various market participants, including investors, traders, and financial institutions. It is influenced by a variety of factors, including market sentiment, economic indicators, geopolitical events, and market liquidity. Volatility risk is a key consideration in risk management. Investors and financial institutions employ risk management techniques such as diversification, hedging, and position sizing to mitigate the adverse impacts of volatility. By identifying and analyzing volatility risk, investors can make informed decisions to manage and control their exposure to market fluctuations (Hull 2023).

Tadawul is recognized as the largest financial market among developed countries. Its prominence stems from the KSA’s heavy reliance on oil as a significant source of income. Consequently, the stock market is prone to notable volatility, as it closely follows the fluctuations in oil prices. In addition to its economic significance, the KSA holds a prominent position as a member of the Group of Twenty (G20), a forum for international economic cooperation that includes leaders from 19 individual countries and the European Union. This membership highlights Saudi Arabia’s active engagement in global economic affairs and its role in shaping international economic policies. Furthermore, the KSA is currently undergoing a comprehensive set of economic reforms and actively pursuing global integration. These efforts demonstrate the country’s commitment to diversifying its economy and reducing its dependence on oil. By embracing economic reforms and opening up to the world, the KSA aims to attract foreign investment and foster sustainable economic growth (Alshammari et al. 2020; Belanès et al. 2024).

However, Tadawul has faced significant fluctuations throughout various periods, primarily due to its susceptibility to external crises. The market’s exposure to global events and uncertainties has led to considerable volatility in stock prices and investor sentiment. Factors such as the global financial crisis of 2008, the Arab Spring unrest in 2011, oil price volatility from 2014 to 2016, the COVID-19 pandemic, and geopolitical tensions have all contributed to the fluctuations experienced by Tadawul. The authors of (Alsabban and Alarfaj 2020) investigate overconfidence behavior among investors of Tadawul. The study uses monthly data from 2007 to 2018. Previous research suggests that positive past market returns influence overconfidence and higher trading turnover. The research applies a market-wide Vector Autoregression (VAR) model to examine the lead–lag relationship between market returns and market turnover as a test for overconfidence behavior. The findings indicate that investors in the Saudi stock market exhibit overconfidence. The authors of (Belkhir and Abbes 2024) investigate the risk spillover among different sectors of Tadawul during the COVID-19 pandemic. The findings reveal that the Tadawul All-Share Index played a dominant role as a net transmitter of risk for various sectors prior to the COVID-19 outbreak. The COVID-19 Panic Index significantly influences the net pairwise connectedness, particularly impacting sectors such as energy, banking, development, materials, real estate, transport, and utilities. The insurance industry serves as a primary receiver of spillovers from sectors such as pharmaceuticals, media, utilities, and retail throughout the specified period, signaling the increased influence of the pandemic on these sectors. The conclusions drawn from the research offer substantial implications for portfolio managers, investors, and policymakers, delivering crucial insights into the potential effects of the pandemic on the stock market.

Various studies have spotlighted that stock market volatility exhibits time-varying characteristics, indicating that volatility changes are not random. This recognition has led practitioners and financial econometricians to develop diverse models for time-varying volatility (Poon and Granger 2003). These models aim to capture the dynamic nature of market volatility and enhance our understanding of its patterns and behavior. In Tadawul, (Abdalla and Suliman 2012) focuses on modeling stock return volatility in Tadawul using daily closing prices of the “Tadawul All Share Index” from 1 January 2007 to 26 November 2011. Various univariate specifications of the generalized autoregressive conditional heteroscedastic (GARCH) model, including symmetric and asymmetric models, are employed. The application of the GARCH (1,1) model indicates persistent time-varying volatility. The results support the positive correlation hypothesis between volatility and expected stock returns, suggesting the existence of a positive risk premium. Additionally, the asymmetric GARCH models provide significant evidence of asymmetry in stock returns, confirming the presence of a leverage effect. The authors of (Alshammari et al. 2020, 2023) aim to enhance the accuracy of forecasting stock market volatility patterns in Tadawul. They utilize daily closed price index data spanning from October 2011 to December 2019 (2048 observations) and propose a combination of the MODWT-Bl14 function with the fitted GARCH model in one study, and the EGARCH model in another study. Both studies demonstrate the effectiveness of the proposed models in analyzing stock market data, identifying highly volatile events, and improving forecast accuracy. Comparative analysis with other mathematical models and evaluation using error functions such as MAPE, MASE, and RMSE further support the findings of enhanced forecasting accuracy in predicting stock market volatility patterns. The authors of (Belanès et al. 2024) examine the dynamic relationship between oil prices, the USA dollar exchange rate, and the Saudi stock market index. They used a dynamic simulated autoregressive distributed lag (ARDL) model on weekly data spanning from 2010 to 2021. Their findings reveal that a cointegration relationship exists between oil prices and the Saudi stock market index. Fluctuations in oil prices strongly impact the Saudi stock market in both the short and long run. The influence of the USA dollar exchange rate on the Saudi stock market is relatively minor. Simulations suggest that the Saudi stock market index experiences a long-run upward trend after an oil price shock, while the dollar index shows a moderate increase. The COVID-19 pandemic led to a significant decline in the Tadawul index, primarily due to a substantial drop in oil prices.

In recent years, several studies have focused on exploring the role of artificial neural networks (ANNs) in the field of economics. Notable examples include the works of (Al-Araj et al. 2022; Al-Gasaymeh et al. 2023; Hatamlah et al. 2023), which delve into the application and impact of ANNs in various economic contexts (Al-Araj et al. 2022; Al-Gasaymeh et al. 2023; Hatamlah et al. 2023). ANN models have emerged as powerful tools for analyzing stock markets, offering unique capabilities in capturing complex patterns and relationships within financial data. As the global financial markets become increasingly dynamic and interconnected, traditional analytical approaches often struggle to capture the intricate dynamics of stock market movements. ANN models, with their ability to learn from data and adapt to changing market conditions, have gained substantial importance in unraveling the complexities of stock market analysis. By leveraging their computational power and advanced algorithms, these models provide valuable insights into market trends, volatility, and risk, empowering investors and analysts to make the right decisions in the ever-evolving landscape of stock markets. According to (Alenezy et al. 2021, 2022, 2023), a series of studies focused on forecasting stock market behavior using Tadawul data. The researchers employed MODWT and various modeling techniques, including the MODWT-LA8-ANFIS, MODWT-LA8-FIR.DM, MODWT-LA8-FS.HGD, and MODWT-LA8-HyFIS models. These models combined MODWT with fuzzy inference systems. The dataset used in the studies consisted of daily closing prices from the Tadawul stock market spanning from August 2011 to December 2019. Input variables, such as oil price and repo rate, were selected based on multiple regression, tolerance, and variance inflation factor tests. The logarithm of the Tadawul served as the output variable. The results of the studies demonstrated the superior forecasting accuracy of the models, outperforming traditional approaches and effectively decomposing stock market patterns.

In contrast to previous studies, there is a research gap regarding the application of a novel approach that combines the use of MODWT with DENFIS to model volatility and enhance forecasting accuracy in the Tadawul. Notably, there has been a lack of focus on this specific combination in the literature over the past decade. However, the present study aims to address the following research problem: How can the combination of MODWT and DENFIS be applied to model volatility and improve forecasting accuracy in the Tadawul? This study intends to bridge the research gap by exploring the effectiveness of this novel hybrid model in capturing the volatility patterns and dynamics of the market. Additionally, this study aims to investigate the impact of new variables, such as the cost of living in Saudi Arabia, on volatility, in order to gain a more comprehensive understanding of the market dynamics. This study distinguishes itself from previous research in several aspects. Firstly, the dataset used in this study covers an extended period from 1 January 2006 to 30 April 2024, encompassing significant events such as the global financial crisis in 2007, the COVID-19 pandemic in 2020–2021, and geopolitical conflicts like the Ukraine–Russia war and the Gaza war in October 2023. This broader timeframe consisting of 4609 observations provides a comprehensive analysis of volatility patterns in the Tadawul market. Secondly, this study introduces new variables, including the cost of living in KSA, to explore their impact on volatility. By incorporating these additional factors, this research aims to capture a more comprehensive understanding of the market dynamics. Thirdly, a novel hybrid model is employed in this study, combining five functions of MODWT with DENFIS. This hybridization allows for a more robust and accurate modeling of volatility, leveraging the strengths of both MODWT and DENFIS. Finally, the measure of volatility used in this study is the LSCP for the Tadawul All-Share (TAS) Index. This measure provides insights into the volatility dynamics of the market, considering the fluctuations in stock prices over the specified period.

The structure of this article is organized as follows: Section 2 presents a comprehensive literature review, highlighting the existing research in the field. Section 3 outlines the methodology and mathematical models utilized in this study. Section 4 focuses on the dataset used and presents the results of the analysis. Finally, Section 5 concludes the article, summarizing the key findings and implications of the study.

2. Literature Review

According to the Efficient Market Hypothesis (EMH) proposed by (Fama 1970), stock prices follow a random walk and are considered unpredictable. However, empirical evidence from global events has shown that stock markets do react to significant shocks, challenging the notion of complete market efficiency (Al-Awadhi et al. 2020). In the past two decades, an increasing number of studies have examined the impact of major international events on financial markets worldwide, including financial and economic crises, terrorist attacks, and medical catastrophes.

One notable recent event that has garnered significant attention from researchers is the war and terrorism. The authors of (Kollias et al. 2013) investigate the impact of war and terrorism on the covariance between oil prices and the stock market indices of four major markets: S&P500, DAX, CAC40, and FTSE100. Non-linear BEKK-GARCH models are employed to analyze the relationship. The results indicate that war has a significant effect on the covariance between stock and oil returns. The study suggests that the two wars examined in this research have a more profound and longer-lasting impact on global markets, influencing investors and market agents. On the other hand, terrorist incidents, which are unanticipated security shocks, only affect the co-movement between CAC40, DAX, and oil returns. No significant impact is observed in the relationship between S&P500, FTSE100, and oil returns. This difference in reaction implies that the latter markets may be more efficient in absorbing the impact of terrorist attacks. The authors of (Hudson and Urquhart 2015) focus on studying the impact of World War II on the British stock market. The findings indicate limited evidence of strong links between war events and market returns, although there is support for the “negativity effect”.

The author of (Verdickt 2020) examines the impact of war on the Brussels Stock Exchange during the pre-1914 era. The findings reveal that managers responded to war risk by cutting dividends and companies postponed IPOs. Foreign companies were more likely to delist after wars, and investors reacted negatively to increased war news coverage. Mean reversion was observed after war threats, and a negative trend followed the start of war. Proximity to military conflicts played a crucial role. Overall, managers and investors became more risk-averse due to war-related news. The authors of (Derindere Köseoğlu et al. 2024) analyze the impact of the Russian–Ukraine war on the Moscow Exchange index. It finds that the war has a detrimental effect on the index, particularly at the beginning. The actual stock index consistently underperforms compared to counterfactual predictions. However, a reconvening pattern is observed when the stock market resumes activities. The estimated decline in the Moscow Exchange index during the conflict is statistically significant, reaching a cumulative decrease of −51%. This highlights the importance for economic sectors to monitor the conflict’s effects on the stock market and make informed investment decisions and policy adjustments.

The second recent event that has garnered significant attention from researchers is the coronavirus health crisis, commonly referred to as COVID-19, which emerged in 2019. Researchers such as (Chen et al. 2018) have found that previous epidemics, like the SARS outbreak, had a significant impact on stocks in the Taiwanese hospitality sector and regional stock markets in Asia. Moreover, (He et al. 2020) confirm that while the financial markets of China and other Asian countries experienced adverse effects in the early stages of the COVID-19 pandemic, the impact was relatively limited. However, the stock markets of the United States and Europe were negatively influenced by the pandemic. The authors of (Belkhir and Abbes 2024) explore risk spillover among the COVID-19 Panic Index, the insurance industry, the Tadawul All-Share Index, and sector indices in the Saudi stock market. The Tadawul All-Share Index acted as a significant transmitter of risk to various sectors before the pandemic. The COVID-19 Panic Index had a notable impact on connectedness, particularly affecting the Tadawul All-Share Index, energy, banking, development, materials, real estate, transport, and utilities sectors. The insurance sector received spillovers from the utilities, media, pharma, and retail sectors. The pandemic significantly affected health-related sectors, leading to increased volatility. These findings hold implications for investors, portfolio managers, and policymakers, providing insights into the pandemic’s impact on the stock market.

The third event that has garnered significant attention from researchers is the credit crisis in 2007. The crisis had a profound effect on the stock market, leading to significant disruptions and turmoil. The crisis originated from the subprime mortgage market in the United States, with the collapse of several major financial institutions and a widespread loss of confidence in the financial system. One of the primary impacts on the stock market was a sharp decline in stock prices. As the crisis unfolded, investors became increasingly concerned about the health of financial institutions and the overall stability of the market. This led to a widespread sell-off of stocks, causing major stock indices to plummet. Many investors experienced substantial losses as the value of their portfolios declined rapidly. The credit crisis also resulted in increased market volatility. Uncertainty and fear gripped the market as investors grappled with the potential implications of the crisis. Volatility measures, such as the VIX index, spiked significantly during this period, reflecting the heightened levels of market unpredictability. Furthermore, the credit crisis had a cascading effect on various sectors and industries. Financial institutions, in particular, were severely impacted, with many facing insolvency or requiring government bailouts. This, in turn, had a ripple effect on other sectors of the economy, as businesses faced difficulties accessing credit and consumers curtailed their spending. The credit crisis also led to a decline in investor confidence and a loss of trust in the financial system. As a result, market participants became more risk-averse and cautious in their investment decisions. This sentiment persisted for an extended period, contributing to a prolonged period of market uncertainty and subdued economic growth (Hull 2023).

According to behavioral theories, as a consequence of sentiment, investors may acquire optimistic or pessimistic attitudes, which leads to uninformed (noise) trading and, as a result, stock prices drift away from their intrinsic values (De Long et al. 1990). The authors of (Alnafea and Chebbi 2022) investigate the link between investor mood and the risk of a stock price drop. From 2011 to 2019, our datasets contain 131 companies incorporated on Tadawul. The findings suggest that high levels of investor emotion enhance managers’ tendency to keep negative news from investors, increasing the chance of a stock price fall. Furthermore, they investigate the liquidity effect by separating samples into subsamples with better and worse liquidity, and they discover that companies with poor liquidity have a significantly bigger positive influence on investor mood. The authors of (Salisu and Vo 2020) found that news and information patterns can anticipate stock returns and volatility. Some papers believe that only poor news influences investment decisions (Akıncı and Chahrour 2018; Cohen et al. 2018), whilst others demonstrate that both good and bad news influence investment decisions (Phan and Narayan 2020).

The volatility has been affected statistically significantly and positively by the pandemic which is considered as a source of systematic risk (Bai et al. 2021; Sharif et al. 2020). The stock market volatility increased throughout the pandemic time, although the rise was higher in the US, UK, and European indexes. They also discovered that gold had the lowest volatility according to (Salisu and Vo 2020). The cryptocurrency market has high volatility (Corbet et al. 2021). The increase in lockdown restrictions during the COVID-19 pandemic, as well as the number of confirmed cases, causes additional volatility in the Saudi stock market return (Atassi and Yusuf 2021). In Tadawul, only 9 of 21 sectors suffered significant volatility impacts. Five of the nine sectors had a significant increase in volatility, while four saw significant decreases. An examination of the influence of trade volume on volatility reveals that increased investor sentiment influences volatility only in sectors that have higher volatility (Wasiuzzaman 2022).

Many theories, including the Quantitative Theory of Money (QTM), the Cash Wallet Model (CWM), and the Efficient Market Theory (EMT), have examined the relationship between macroeconomic factors and stock prices. However, the cost of living refers to the amount of money required to cover basic expenses and maintain a certain standard of living. It encompasses essential needs such as housing, food, healthcare, transportation, and other everyday expenses. The cost of living can vary across different regions and countries due to factors like inflation, wages, housing costs, and the overall economic environment. The cost of living and stock prices have a complex relationship. Increased cost of living can impact consumer spending, inflation, sector performance, investor sentiment, and macroeconomic factors, all of which can influence stock prices. The authors of (Alhakimi and Sharaf-Addin 2022) used the autoregressive distributed lag (ARDL) technique to demonstrate long-run causation between the exchange rate, return on investment, and oil price and financial market efficiency. In the near run, however, only inflation and return on investment have a causal influence on financial market performance. Furthermore, there is no connection between the exchange rate and oil price and economic market efficiency. In (Alenezy et al. 2021, 2022, 2023), the researchers focused on forecasting stock market behavior using Tadawul data. The dataset encompassed daily closing prices from the Tadawul stock market between August 2011 and December 2019. They utilized the MODWT model combined with different ANN models such as MODWT-LA8-ANFIS, MODWT-LA8-FIR.DM, MODWT-LA8-FS.HGD, and MODWT-LA8-HyFIS. Input variables, including oil price and repo rate, were selected through regression, tolerance, and variance inflation factor tests. The models exhibited superior forecasting accuracy, surpassing traditional approaches and effectively decomposing stock market patterns.

The factors that influence the dynamics of financial markets are as follows: Investor sentiment, characterized by the emotions and attitudes of market participants, stands out as a significant driver of market volatility. Market psychology, which delves into collective behavioral patterns such as herd mentality and the fear of missing out (FOMO), underscores the importance of understanding the human element in market movements. External factors like economic indicators, geopolitical events, and regulatory changes also play a crucial role in shaping stock market volatility. Regulatory environments, market liquidity, and information dissemination further contribute to the intricate web of influences on market behavior and volatility, highlighting the multifaceted nature of financial markets (Aouadi et al. 2013; Gavrilakis and Floros 2024; Vo and Phan 2019). Stock price volatility mirrors investor behavior by reflecting their collective emotions, decisions, and sentiments during turbulent market periods. Heightened volatility often amplifies investor actions driven by fear, greed, and uncertainty, leading to rapid buying or selling trends that intensify market fluctuations. The MODWT models, acting as filters to distinguish detailed (non-smooth or high frequency) and approximate (smooth) data, will help identify outliers in investor behavior and focus on smoother data for market analysis. Subsequently, the DENFIS model incorporates the output variable (smooth data from MODWT) in conjunction with input variables to forecast stock market volatility.

According to the mentioned literature review for the past decade, no research has focused on the application of MODWT functions which are Haar, Daubechies (d4), coiflet (c4), least symmetric (LA8), and best localized (Bl14) in combination with suitable DENFIS to model and improve the forecasting accuracy in the Tadawul. Because of its adaptability, the DENFIS has been successfully used in a wide range of hydrological modeling applications, including rainfall–runoff modeling, runoff forecasting, reference evapotranspiration modeling, and river water level forecasting (Lee et al. 2022). The structure of DENFIS is similar to some popular fuzzy ANNs, such as ANFIS. To produce the fuzzy membership functions, the DENFIS uses an evolving clustering technique (ECM) for online clustering or a constrained ECM (ECMc) for offline clustering (MF). The DENFIS comes in three categories: (i) the online DENFIS, which employs ECM as the clustering method and first-order TSK as the inference engine; (ii) the first-order offline DENFIS, which employs ECMc as the clustering method and first-order TSK as the inference engine; and (iii) the higher-order offline DENFIS, which employs higher-order TSK as the inference engine instead (Kasabov and Song 2002; Ye et al. 2013). The main objective of this project is to develop predictive models using DENFISs in the Tadawul in combination with MODWT.

3. Research Design and Mathematical Models

The objective of this research is to construct a hybrid model that effectively predicts volatility risk in the Tadawul. The study covers the period from 1 Jan. 2006 to 30 Apr. 2024. The proposed model combines two distinct approaches, namely the DENFIS model and the MODWT-Haar model. The performance of the hybrid model is evaluated using an accuracy measure, which allows for an assessment of its effectiveness in volatility risk forecasting. By integrating these approaches, this study aims to provide an enhanced and reliable methodology for predicting volatility risk in the Tadawul stock market. Furthermore, the MODWT model is utilized to convert the original data into a time-scale domain.

Figure 1 depicts the various steps encompassed in the MODWT prediction model. The key aim is to reduce statistical error criteria in the dataset, both pre-transformation and post-transformation. Moreover, the MODWT model segregates the data into two specific categories: the detail series and the approximation series. In cases where the original financial data demonstrate significant fluctuations, these two categories are employed as they effectively capture and explain the underlying behavior of the dataset. The suggested hybrid model relies on a robust methodology, which involves several key steps. First, data collection: gather closing price data from the Tadawul. Second, calculation of the LSCP: calculate the LSCP using the stock closed price data. Third, decomposition with the MODWT model: utilize the MODWT function to decompose the LSCS data. This decomposition splits the data into two groups: the low-fluctuated data (approximation coefficient) and the high-fluctuated data (details coefficient).

Fourth, in the process of selecting input variables (LCP, Loil, LCL, and IB), various statistical tests are employed. The AC test assesses whether past values influence current values, while the PAC test identifies direct relationships between variables while controlling for intervening variables. Correlation analysis evaluates the relationships between input variables to avoid multicollinearity, whereas Granger causality tests determine predictive relationships between variables. OLS regression optimizes the model fit, and fixed effects models control for time-invariant factors, with random effects models capturing general unobserved heterogeneity across panel units. In fixed and random effects models, time series data are transformed into panel data using years to observe how input variables impact the output variable over time. To enhance the predictive capabilities of our models, we intend to amalgamate the selected input variables with the approximate coefficient component extracted from the MODWT model into the DENFIS model, which encompasses both the input variables and the output variable. This integration strategy aims to harness the synergies between these methodologies to improve the precision and robustness of our output variable predictions.

Fifth, the approximation coefficient (LSCP) for each MODWT function is utilized with input variables (LCP, Loil, LCL, and IB) within the DENFIS model in the new hybrid model called MODWT-DENFIS. Finally, a comparative study is conducted between the best-performing MODWT-DENFIS model and alternative MODWT-DENFIS functions, as well as traditional models such as ARIMA and DENFIS. This comparative analysis allows for an assessment of the effectiveness and superiority of the MODWT-DENFIS hybrid model in volatility risk forecasting in comparison to other models. To assess the performance of the proposed model, an evaluation is conducted using a 90% training and 10% test data split. The training phase utilizes 90% of the original data to train multiple models and identify the most effective one. Subsequently, the chosen model is applied to the remaining 10% of the data as test data to evaluate its performance and accuracy.

3.1. Wavelet Transform Formula

The wavelet transform (WT) is a mathematical technique used to analyze time series data by representing it in a different domain called the time-scale domain. Unlike the traditional Fourier Transform, which decomposes signals into frequency components, the WT decomposes signals into both time and scale components. This means that it can capture localized variations in the data at different scales. It is particularly suitable for analyzing non-stationary data such as stock market data (Jaber et al. 2023; Yaacob et al. 2021). The WT encompasses various types, including DWT, CWT, and MODWT. These types of WTs share the common goal of decomposing signals into different scales or resolutions, enabling the analysis of localized features in time series data. DWT provides a discrete representation by dividing the signal into approximate and detailed components, while CWT offers a continuous representation by analyzing the signal at all possible scales. The MODWT model, an extension of DWT, allows for a flexible analysis of signals with arbitrary lengths, offering better time localization of wavelet coefficients. However, one key distinction between DWT and MODWT is that DWT requires a specific number of data points (2 raised to the power J), while MODWT can accommodate datasets of any size, as seen in the following formula (Alenezy et al. 2023):

$$\mathrm{X}\left(t\right)=\sum S_{j,k}{\varphi }_{j,k}\left(t\right)+\sum d_{j,k}{\phi }_{j,k}\left(t\right)+\sum d_{j-1,k}{\phi }_{j-1,k}\left(t\right)+\cdots +\sum d_{1,k}{\phi }_{1,k}\left(t\right)$$

(1)

$$D_j\left(t\right)=\sum d_{j,k}{\varphi }_{j,k}\left(t\right) S_j\left(t\right)=\sum S_{j,k}{\varphi }_{j,k}\left(t\right)$$

(2)

In the context of a time series dataset $\mathrm{X}\left(t\right)$, the WT computes the approximation coefficient using Equation (2), where $S_j\left(t\right)$ represents the smooth coefficient and $D_j\left(t\right)$ represents the detailed coefficient. The smooth coefficients capture the crucial characteristics and essential features of the original data, while the detailed coefficients are utilized to identify and analyze the principal fluctuations or variations present in the original data.

The WT encompasses a range of popular transform functions, including Haar, d4, c6, LA8, and bl14 functions (Gençay et al. 2001). These functions exhibit various key characteristics. (1) With the exception of the Haar model, the WT functions are generally arbitrary and regular, providing flexibility in their application. (2) Interestingly, WT functions, except for the Haar model, lack explicit mathematical expressions, making their implementation more intricate. (3) They utilize real numbers and possess properties such as orthogonality, compactness, support for arbitrary zero moments, and the inclusion of a scale function, enabling a wide range of analyses such as orthogonal analysis, bio-orthogonal analysis, continuous/discrete transformation, exact reconstruction, and fast algorithms. (4) The Haar model demonstrates symmetry, while LA8 and d4 exhibit asymmetry, and near symmetry is associated with C6 and bl14 functions. These different WT functions offer trade-offs in terms of properties and capabilities, allowing researchers and practitioners to choose the most suitable function based on their specific requirements and desired outcomes.

3.2. Evolving Clustering Method (ECM)

In this approach, a dynamic and online clustering method known as the ECM is introduced for scatter partitioning of the input space, specifically for generating fuzzy inference rules. The ECM offers two modes: an online mode, typically used in online learning systems, and an offline mode, more suitable for offline learning systems. The online ECM is employed in the DENFIS online model, while the offline ECM with constrained minimization (ECMc) extends the capabilities of the online mode. The online ECM is a fast and efficient algorithm for dynamic cluster estimation and center identification in the input data space. It is a distance-based connectionist clustering method that does not require optimization. The ECM represents cluster centers using evolved nodes (rule nodes) and ensures that the maximum distance (MaxDist) between an example point and the cluster center is below a displaced threshold (Dthr) value, influencing the estimated number of clusters. The distance, between two vectors x and y, determines a general Euclidean distance defined as follows:

$$‖x-y‖=\sqrt{{\displaystyle \frac{\sum _{i=1}^q{|x_i-y_i|}^2}{q}}}$$

(3)

where x, y$\in R$, and $i=1, 2, 3,\dots ,q$.

The ECM algorithm can be described as follows: Step 0: To create the first cluster, $C_1^0$, in the ECM algorithm, the initial position of $C_1^0$ is taken from the input stream as the initial cluster center, ${Cc}_1^0$. A predefined value is set for the cluster radius, ${Ru}_1$. Step 1: If all examples of the data stream have been processed, the algorithm is finished. Otherwise, the current input example, $x_i$, is selected, and the distances between this example and all n already created cluster centers, ${Cc}_j$, are calculated using the Euclidean distance formula, denoted as $D_{ij}=‖x_i-{Cc}_j‖, j=1,2,\dots ,n$. Step 2: If there exists a value of $D_{ij}$, that is equal to or less than at least one of the radii, ${Ru}_j$, it indicates that the current value of $x_i$ belongs to a cluster $C_m$ with the minimum distance $D_{im}=‖x_i-{Cc}_m‖=min\left(‖x_i-{Cc}_j‖\right)$. Subject to the constraint $D_{ij}\le {Ru}_j$,$j=1,2,\dots ,n$. (The algorithm returns to Step 1 when a distance value is equal to or less than any radius; otherwise, it proceeds to the next step.) Step 3: Cluster $C_a$ (with center ${Cc}_a$ and cluster radius ${Ru}_a$) is found from all $n$ existing cluster centers by calculating the values $S_{ij}=D_{ij}+{Ru}_j$, and then choosing the cluster center ${Cc}_a$ with the minimum value $S_{ia}$: $S_{ia}=D_{ij}+{Ru}_j=min\left(S_{ij}\right), j=1,2,\dots ,n$. Step 4: If the value of $S_{ia}$ is greater than 2 times the Dthr, it indicates that the example $x_i$ does not belong to any existing clusters. In such cases, a new cluster is created following the same procedure as described in Step 0, and the algorithm returns to Step 1. Step 5: When the value of $S_{ia}$ is less than twice the value of Dthr, updating the cluster $C_a$ through updating the position of its center (${Cc}_a$) and increasing the value of its radius ${Ru}_a.$ The updated radius ${Ru}_a^{new}$ = $S_{ia}/2$ and the new center ${Cc}_a^{new}$ is located at the point on the line connecting the $x_i$ and ${Cc}_a$, and the distance from the new center ${Cc}_a^{new}$ to the point $x_i$ is equal to ${Ru}_a^{new}$. The algorithm returns to Step 1. By following this approach, the maximum distance from any cluster center to the examples belonging to that cluster remains within the Dthr. However, it is important to note that the algorithm does not retain any information about the examples that have been processed.

The offline version of the ECM, known as ECMc, incorporates an optimization procedure to refine the cluster centers obtained from the application of the ECM. The ECMc segments a dataset consisting of p vector $x_i$, i = 1, 2, …, p, into n clusters $C_j, j=1,2,\dots ,n,$ identifying a cluster center in each cluster to minimize an objective function reliant on a distance measure within specified constraints. Utilizing the standard Euclidean distance as the measure among the corresponding cluster center ${Cc}_j$ and an example vector, $x_i$, in cluster j, the objective function is represented by Equation (4):

$$J=\sum _{j=1}^n\left(\sum _{x_i\in C_j}‖x_i-{Cc}_j‖\right)$$

(4)

$$\mathrm{with} {\sum }_{x_i\in C_j}‖x_i-{Cc}_j‖\le Dthr$$

(5)

The partitioned clusters are typically defined by a $p\times n$ binary membership matrix U, where the element $u_{ij}$ is 1 if the i-th data point $x_i$ belongs to cluster j, and 0 otherwise. Once the cluster centers ${Cc}_j$ are fixed, the minimizing $u_{ij}$ for (4) and (5) is derived as follows:

$$\begin{gathered} \mathrm{If} ‖x_i-{Cc}_j‖\le ‖x_i-{Cc}_k‖, \mathrm{for} \mathrm{each} j\ne k; \\ {u}_{ij}=1, else u_{ij}=0 \end{gathered}$$

(6)

For a batch-mode operation, the method determines the cluster centers ${Cc}_j$ and the membership matrix U iteratively using the following steps: Step 1: Initialize the cluster center, ${Cc}_j$, j = 1, 2, …, n, using the result obtained from the ECM clustering process. Step 2: Determine the membership matrix, U, using Equation (6). Step 3: Utilize the constrained minimization method with Equations (4) and (5) to obtain new cluster centers. Step 4: Calculate the objective function based on Equation (5). If the result is below a specified tolerance value, or the improvement compared to the previous iteration is below a threshold, or the iteration number of the minimization operations exceeds a specified limit, the algorithm stops. Otherwise, it returns to Step 2.

3.3. Dynamic Evolving Neural Fuzzy Inference System

In 2002, (Kasabov and Song 2002) introduced a method known as the dynamic evolving neural fuzzy inference system (DENFIS). The DENFIS offers a powerful and adaptive approach for time series prediction, combining the strengths of ANNs, fuzzy logic, and evolving clustering methods. The DENFIS framework consists of several key components, including evolving clustering methods (such as evolving clustering method (ECM)), membership function generation, rule generation, parameter estimation, and fuzzy inference. These components work together to create an adaptive and efficient system for modeling and predicting dynamic time series (Kasabov and Song 2002).

The DENFIS plays a crucial role in adaptive online and offline learning, as well as in dynamic time series prediction applications. The online learning mode of the DENFIS enables it to process data in a sequential manner, making it well suited for real-time applications. It can efficiently update its internal structure, such as cluster centers and membership functions, to adapt to changing data patterns and maintain model accuracy. On the other hand, DENFIS can also be trained in an offline mode, where it processes the entire dataset at once. This mode is useful when the entire dataset is available in advance, and there is no requirement for real-time updates (Kasabov and Song 2002).

The output of DENFISs is determined by a fuzzy inference system, which dynamically chooses a subset of m-most activated fuzzy rules from a fuzzy rule set. A DENFIS offers two approaches for constructing the fuzzy rule set: an online model and an offline model. In the online model, the fuzzy rule set is dynamically created on the fly using a first-order Takagi–Sugeno–Kang (TSK)-type fuzzy rule set. This allows for the rules to adapt and evolve as new data are encountered during the learning process. In contrast, the offline model allows for the creation of a first-order TSK-type fuzzy rule set or an expanded high-order one. Fuzzy rules can be inserted into DENFIS before or during the learning process, and they can also be extracted during or after the learning process. Regardless of the learning mode, both the online and offline DENFIS models make use of the ECM to determine cluster centers and radii, which are crucial for generating the fuzzy rules. The ECM ensures the effective adaptation and performance of a DENFIS, making it a versatile and powerful tool for handling dynamic time series prediction tasks (Kasabov and Song 2002).

However, this method, primarily used for regression tasks, involves several key steps. The process starts by employing the evolving clustering method (ECM) to identify cluster centers in the input space. The ECM is a distance-based clustering approach that utilizes the Dthr to determine the number of clusters created. The clustering process begins by selecting the first instance from the training data as a cluster center with an initial radius of zero. Subsequent instances are then used to adjust the cluster centers and radii using specific ECM mechanisms. This iterative process continues until all training data instances have been evaluated, resulting in the acquisition of all cluster centers. Once the cluster centers are determined, the next step involves updating the parameters in the consequent part of the TSK model, assuming that the antecedent part obtained from ECM remains fixed. It is important to note that while the ECM can function effectively as an online clustering method, in this case, it is employed in an offline mode (Kasabov and Song 2002).

The TSK-type fuzzy inference engine is utilized by both the online and offline models of DENFISs for their operations. In the case where the consequent functions consist of crisp constants ($f_i(x_1, x_2,\dots ,x_q)=C_i.i=1,2,\dots ,m$), the system is referred to as a zero-order TSK-type fuzzy inference system. If the consequent functions are represented by linear functions ($f_i(x_1, x_2,\dots ,x_q)$), the system is known as a first-order TSK-type fuzzy inference system. On the other hand, if the consequent functions are non-linear, the system is categorized as a high-order TSK fuzzy inference system. The weighted average of each rule’s output is defined as follows:

$$y^0={\displaystyle \frac{\sum _{i=1}^m{\omega }_i{f}_i{x}^0}{\sum _{i=1}^m{\omega }_i}}$$

(7)

where $x^0=\left[x_1^0,x_2^0,\dots ,x_q^0\right]$ refers to the input vector, $y^0$ indicates the output of the system, and ${\omega }_i=\prod _{j=1}^q{\mu }_{R_{ij}}\left(x_j^0\right);i=1,2,\dots ,m,j=1,2,\dots ,q.$ The online model of DENFISs utilizes first-order TSK-type fuzzy rules, where the consequences are represented by linear functions. These linear functions in the consequences can be created and updated using the linear least-square estimator (LSE). Each of the linear functions can be expressed as follows:

$$y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\dots +{\beta }_q{x}_q.$$

(8)

where $\beta ={[b_0{b}_1{b}_2\dots b_{\mathrm{q}}]}^T$ are calculated as the coefficients. In the DENFIS models, we use a weighted LSE method as follows:

$$b_{\omega }={{(A}^{T}WA)}^{-1}{A}^{T}Wy$$

(9)

where

$$A=\left(\begin{array}{ccccc}1& x_{11}& x_{12}& \dots & x_q\\ 1& x_{11}& x_{12}& \dots & x_q\\ .& .& .& \dots & .\\ .& .& .& \dots & .\\ .& .& .& \dots & .\\ 1& x_{p1}& x_{p2}& \dots & x_{pq}\end{array}\right)W=\left(\begin{array}{ccccc}{\omega }_1& 0& .& \dots & 0\\ 0& {\omega }_2& .& \dots & 0\\ .& .& .& \dots & .\\ .& .& .& \dots & .\\ .& .& .& \dots & .\\ 0& .& .& \dots & {\omega }_p\end{array}\right)$$

$$y={[y_1,y_2,\dots ,y_p]}^{\mathrm{T}}$$

Let the k-th row vector of matrix A defined in (9) be $a_k^T=[1 x_{k1} x_{k2} \dots x_{kq}]$ and the k-th element of y be $y_k$. Equation (10) is the formula of a weighted recursive LSE with a forgetting factor ($\lambda$) defined as follows:

$$\begin{gathered} b_{k+1}=b_k+{\omega }_{k+1}{P}_{k+1}{a}_{k+1 }(y_{k+1}-a_{k+1}^T{b}_k) \\ \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \\ {P}_{k+1}={\displaystyle \frac{1}{\lambda }}\left(P_k-{\displaystyle \frac{{\omega }_{k+1}{P}_k{a}_{k+1 }{a}_{k+1}^T{P}_k)}{\lambda +a_{k+1 }{a}_{k+1}^T{P}_k}}\right), k=n, n+1,\dots p-1 \end{gathered}$$

(10)

Figure 2 shows the flowchart of the DENFIS. This algorithm can be described as follows: Layer 1: The primary function of this layer is to cluster input vectors and identify the cluster center. The cluster center, along with the distance threshold, plays a crucial role in calculating membership degrees. The calculation of membership degrees requires three parameters, which are determined based on the cluster center and distance threshold. The formula for a triangular membership function, denoted as (11), represents the general form of this function.

$$\mu (x)=mf(x,a,b,c)=\left\{\begin{array}{c}0, x\le a\\ {\displaystyle \frac{x-a}{b-a}},a\le x\le b\\ {\displaystyle \frac{c-x}{c-b}}, b\le x\le c\\ 0,c\le x\end{array}\right\}$$

(11)

Here, x is the input to be fuzzified, a, b, c are the parameters of a triangular membership, b is the center for the cluster along the x dimension, and a = b − d $\times Dthr$ and c = b + d $\times Dthr$, where d varies from 1.2 to 2. Layer 2: In this layer, the output of each node is computed by taking the product (Π) of the corresponding nodes from all the inputs in the preceding layer. Layer 3: In this layer, the output from the previous layer undergoes normalization (N) to ensure that the values are within a specific range or scale. Layer 4: The calculation of each rule output is performed. The DENFIS utilizes a first-order TSK-type inference system. The rule output of TSK is determined using Equation (8), where the βs represent the rule consequents, and their values are obtained through the application of the least-square estimator (LSE) method. Layer 5: The primary function of this layer is to compute the final output, which is obtained as the weighted sum of the outputs from each rule.

Several types of accuracy criteria are used to evaluate the performance of the MODWT models and the DENFIS models. These include the Mean Absolute Percentage Error (MAPE), the Mean Percentage Error (MPE), the Mean Error (ME), the Mean Absolute Error (MAE), and the Root Mean Squared Error (RMSE). The MAPE criterion, also known as Mean Absolute Percentage Deviation (MAPD), is a statistical measure that assesses the accuracy of a forecasting method’s predictions. It is commonly expressed as a percentage and represents the deviation between predicted and actual values.

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{1}{n}}{\sum }_{t=1}^n\left|{\displaystyle \frac{X_t-F_t}{X_t}}\right|,$$

(12)

$$\mathrm{M}\mathrm{P}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}{\sum }_{\mathrm{t}=1}^{\mathrm{n}}{\displaystyle \frac{{\mathrm{X}}_{\mathrm{t}}-{\mathrm{F}}_{\mathrm{t}}}{{\mathrm{X}}_{\mathrm{t}}}},$$

(13)

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}{\sum }_{\mathrm{t}=1}^{\mathrm{n}}|{\mathrm{X}}_{\mathrm{t}}-{\mathrm{F}}_{\mathrm{t}}|,$$

(14)

$$\mathrm{M}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}{\sum }_{\mathrm{t}=1}^{\mathrm{n}}{(\mathrm{X}}_{\mathrm{t}}-{\mathrm{F}}_{\mathrm{t}}),$$

(15)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{t}=1}^{\mathrm{n}}{{(\mathrm{X}}_{\mathrm{t}}-{\mathrm{F}}_{\mathrm{t}})}^2}$$

(16)

Here, $X_t$ is the actual value, $F_t$ is the forecasted value, and $n$ is the sample size.

4. Results

4.1. Data Description

The research collected daily data spanning from 1 January 2006 to 30 April 2024, to investigate factors impacting the Tadawul stock market in Saudi Arabia. The sample size of observations is 4609. Data were sourced from authoritative entities such as the Saudi Central Bank (SAMA), the General Authority for Statistics, and the Saudi Exchange Market. The time series dataset examined the primary output of the study, which was the LSCP. The LSCS can be expressed as $\sigma =\sqrt{{\displaystyle \frac{\sum _{i=1}^n{(p_i-\overline{p})}^2}{n-1}}}$, where $p$ is the closing stock price. The LSCP variable was influenced by various factors, including the lagged one of the natural LCP, LCL, Loil, and the interbank rates (IB). The dataset collected information on these variables on a daily basis. The dataset was analyzed using R programming language (version 4.3.1) to examine volatility patterns in the Tadawul.

Table 1. Descriptive statistics of input and output factors.

Factors	Mean	SD	Minimum	Maximum
Close prices	8491.209	2365.031	4130.010	20,634.860
Oil prices	77.811	24.194	19.330	146.080
Cost-of-living index	115.090	12.280	97.000	143.000
Interbank rate	0.023	0.017	0.010	0.060

presents descriptive statistics for these variables, encompassing both input and output factors. The close prices have a mean of 8491.209, with a standard deviation (SD) of 2365.031. The minimum and maximum values are 4130.010 and 20,634.860, respectively. The oil prices have a mean of 77.811, SD of 24.194, and a range from 19.330 to 146.080. The cost-of-living index has a mean of 115.090, SD of 12.280, and ranges from 97.000 to 143.000. Finally, the interbank rate has a mean of 0.023, SD of 0.017, and varies from 0.010 to 0.060. These statistics provide insights into the average values, variability, and range of each factor.

4.2. Endogeneity Issues

In order to choose appropriate variables for the DENFIS model, several steps can be taken, including addressing autocorrelation, partial autocorrelation, multicollinearity, conducting OLS, fixed effects, and random effects, and performing causality tests.

4.2.1. Autocorrelation and Partial Autocorrelation

Table 2. Autocorrelation and partial autocorrelation for LCP.

Lag	AC	PAC	Q-Stat	Prob.
1	0.997	0.997	4587.2	0.000
2	0.994	−0.065	9147.4	0.000
3	0.991	0.022	13682	0.000
4	0.988	−0.035	18189	0.000
5	0.985	−0.013	22668	0.000
6	0.982	−0.037	27117	0.000
7	0.978	0.021	31537	0.000
8	0.975	0.034	35931	0.000
9	0.972	−0.008	40298	0.000
10	0.969	−0.012	44638	0.000
11	0.966	0.007	48950	0.000
12	0.963	−0.038	53234	0.000

presents the values of AC, PAC, Q-statistic, and probability values for different lag values of the LCP. For lag 1, there is an extremely high positive value of AC = 0.997, indicating a strong correlation between the current value of LCP and its value at the previous time step. This implies that the previous value of LCP is highly influential in determining its current value. Also, the value of PAC = 0.997 shows a very strong direct influence of the lagged LCP on the current value. The Q-statistic value of 4587.2 and a probability value of 0.000 suggest a significant departure from randomness, highlighting the statistical significance of the relationship between LCP and its lagged value at lag 1. Based on these findings, selecting lag 1 in our model is appropriate as it captures a strong and significant correlation between the current value of LCP and its immediate previous value. Incorporating lagged LCP as a predictor in the model at this specific lag can enhance our understanding and predictive capabilities related to the variable LCP.

4.2.2. Multicollinearity Test

The correlation coefficients in

Table 3. The correlation between the output and input variables.

Factors	LSCP	LCP	Loil	LCL	IB
LSCP	1	−0.528	0.236	0.346	−0.119
LCP		1	0.235	−0.440	0.495
Loil			1	0.150	−0.009
LCL				1	−0.455
IB					1

provide insights into the relationships between the output and input variables. The variables include LSCP, LCP, Loil, LCL, and IB. The correlation coefficients refer to understanding the nature and direction of the relationships between these factors. In addition, correlations below 0.5 are considered weak, while correlations above 0.5 are considered strong. Of note, the number 1 in each diagonal cell indicates a perfect correlation of each variable with itself, which is expected. Moving to the off-diagonal cells, the correlation coefficient of −0.528 between LSCP and LCP suggests a strong negative correlation, meaning that as LCP increases, LSCP tends to decrease. Similarly, the coefficient of 0.236 between LSCP and Loil indicates a weak positive correlation, implying that as Loil increases, there is a slight tendency for LSCP to increase. The coefficient of 0.346 between LSCP and LCL signifies a weak positive correlation, suggesting that as LCL increases, there is a slight tendency for LSCP to increase. The correlation coefficient of −0.119 between LSCP and IB indicates a weak negative correlation, suggesting that there is a slight tendency for LSCP to decrease as IB increases.

Further, the correlation coefficient of 0.235 between LCP and Loil suggests a weak positive correlation, indicating that as LCP increases, there is a slight tendency for Loil to increase. The coefficient of −0.440 between LCP and LCL signifies a weak negative correlation, indicating that as LCP increases, LCL tends to decrease. These interpretations provide insights into the relationships between the variables, shedding light on their associations. The coefficient of 0.495 between LCP and IB indicates a weak positive correlation, suggesting that as LCP increases, there is a strong tendency for IB to increase as well. The correlation coefficient of 0.150 between Loil and LCL signifies a weak positive correlation, indicating that as Loil increases, there is a slight tendency for LCL to increase. The correlation coefficient of −0.009 between Loil and IB suggests a very weak correlation, indicating that there is almost no relationship between the two variables. Lastly, the coefficient of −0.455 between LCL and IB signifies a weak negative correlation, suggesting that as LCL increases, there is a tendency for IB to decrease. Indeed, the weak correlations between the input variables indicate the absence of multicollinearity. The low correlation coefficients suggest that the variables have limited linear associations with each other, reducing the risk of redundancy or duplication of information. This implies that the input variables provide distinct and independent contributions to the output variable, enhancing the reliability and interpretability of the model.

4.2.3. Granger Causality Tests

Table 4. Granger causality tests.

Null Hypothesis	Res. Df	Df	F-Statistics	p-Value	Results
LCP ≥ LSCP	4605	−1	3.2464	0.0716 ***	No
Loil ≥ LSCP	4606	−1	13.772	0.0002 ***	Yes
LCL ≥ LSCP	4606	−1	9.9177	0.0016 ***	Yes
IB ≥ LSCP	4606	−1	9.2962	0.0023 ***	Yes
LSCP ≥ LCP	4605	−1	4.0653	0.0438 **	Yes
LSCP ≥ Loil	4606	−1	0.3071	0.5795	No
LSCP ≥ LCL	4606	−1	2.0473	0.1525	No
LSCP ≥ IB	4606	−1	4.1641	0.0414 **	Yes

Signif. codes: ‘***’ 0.01 and ‘**’ 0.05.

presents the findings of the Granger causality tests conducted to assess the forecasting capabilities of one input time series variable on another output time series variable. The results of these tests indicate the presence of a bidirectional causal relationship between the input variables and the output variable. These findings suggest that the input variables have the potential to be valuable predictors for forecasting the output variable in the stock market. The tests examine the null hypothesis that one variable does not Granger-cause another variable. The table provides the number of observations, the residual degrees of freedom (Res. Df), the change in the degrees of freedom (Df), F-statistics, and p-values for each test. Starting with the causal relationship of LCP on LSCP, the test indicates that the null hypothesis is not rejected because the p-value is greater than 5%. This suggests that there is insufficient evidence to claim a causal relationship between LCP and LSCP. Moving on to the causality tests for Loil, LCL, and IB on LSCP, the results show statistically significant evidence of causality in all three cases. The p-values for Loil, LCL, and IB are 0.0002, 0.0016, and 0.0023, respectively. These low p-values (less than 5%) indicate that there is strong evidence to support the claim that Loil, LCL, and IB Granger-cause LSCP.

Regarding the reverse causal relationships, the tests examine the causality of LSCP on LCP, Loil, LCL, and IB. The results reveal that LSCP Granger-causes LCP (p-value = 0.04383) and IB (p-value = 0.04135) with statistically significant evidence. However, the tests do not provide sufficient evidence to support the existence of causal relationships between LSCP and Loil (p-value = 0.5795) or LCL (p-value = 0.1525). In summary, based on the Granger causality tests, there is evidence to suggest that Loil, LCL, and IB have a causal influence on LSCP. Additionally, LSCP is found to Granger-cause LCP and IB. However, there is insufficient evidence to establish causality between LSCP and Loil or LCL.

4.2.4. Multiple Regression

In

Table 5. Regression analysis (OLS, fixed effect, and random effect).

Variables	OLS		Fixed Effect		Random Effect
	Coefficient	Std. Error	Coefficient	Std. Error	Coefficient	Std. Error
Intercept	11.729758 ***	0.209158			17.960823 ***	0.661746
LCP	−0.719012 ***	0.01349	−1.057274 ***	−46.9247	−1.032565 ***	0.022257
Loil	0.304516 ***	0.008836	0.485177 ***	28.0788	0.475032 ***	0.017096
LCL	0.256233 ***	0.030949	−0.51185 ***	−3.5653	−0.575235 ***	1.28 × 10−1
IB	4.394997 ***	0.188673	−4.739217 ***	−9.9826	−3.818316 ***	0.453456
Observations	4609		4609		4609
R-square	0.4792		0.32843		0.38694
Adjusted R-square	0.4788		0.3252		0.38641
F-statistic/Chisq	1059 ***		560.56 ***		2201.01 ***

Signif. codes: ‘***’ 0.01.

, the regression analysis is conducted to examine the relationship between the output variable LSCP and various input variables using ordinary least square (OLS), fixed effect, and random effect methods. The table presents the results for each input variable, including their coefficients and corresponding standard errors. In the OLSs, the intercept coefficient is 11.729758, with a standard error of 0.209158, indicating the expected value of LSCP when all input variables are zero. LCP has a coefficient of −0.719012, with a standard error of 0.01349, suggesting that an increase in LCP is associated with a decrease in LSCP, and this relationship is statistically significant at a level of less than 5%. On the other hand, Loil has a coefficient of 0.304516 (standard error = 0.008836), and LCL has a coefficient of 0.256233 (standard error = 0.030949), both indicating that higher values of Loil and LCL correspond to increased LSCP, and these relationships are also statistically significant at a level of less than 5%. The coefficient for IB is 4.394997, with a standard error of 0.188673, indicating that an increase in IB is associated with a higher LSCP, and this relationship is statistically significant at a level of less than 5%. If the number of observations is 4609, and the R-square value is 0.4792, then the input variables explain 47.92% of the LSCP. The adjusted R-square value is 47.88%, and the F-statistic is 1059, both suggesting overall statistical significance at a level of less than 5%.

In the fixed effect regression analysis presented in Table 5, the coefficients and standard errors of the variables are as follows: The LCP coefficient is −1.057274 with a standard error of 46.9247, and it is statistically significant at a level of less than 5%. The coefficient for Loil is 0.485177 with a standard error of 28.0788, and it is also statistically significant at a level of less than 5%. The coefficient for LCL is −0.51185 with a standard error of −3.5653, and it is also statistically significant at a level of less than 5%. The coefficient for IB is −4.739217 with a standard error of −9.9826, and it is also statistically significant at a level of less than 5%. The analysis is based on a total of 4609 observations. If the R-square value is 0.32843, then approximately 32.843% of the variance in the output variable LSCP is explained by the input variables. The adjusted R-square value is 32.52%. The F-statistic value is 560.56 with statistical significance at a level of less than 5%.

In the random effect regression analysis presented, the coefficients and standard errors of the variables are as follows: The intercept coefficient is 17.960823 with a standard error of 0.661746 and is statistically significant at a level of less than 5%. The coefficient for LCP is −1.032565 with a standard error of 0.022257 and is also statistically significant at a level of less than 5%. The coefficient for Loil is 0.475032 with a standard error of 0.017096 and is statistically significant at a level of less than 5%. The coefficient for LCL is −0.575235 with a standard error of 0.128 and is statistically significant at a level of less than 5%. The coefficient for IB is −3.818316 with a standard error of 0.453456 and is statistically significant at a level of less than 5%. The analysis is based on 4609 observations. The R-square value is 0.38694, indicating that approximately 38.694% of the variance in the dependent variable is explained by the independent variables. The adjusted R-square value is 38.64%. The chi-square value is 2201.01 and suggests an overall statistical significance at a level of less than 5%.

In the OLS, fixed effect, and random effect models, the LCP has a statistically significant negative effect on the LSCP. This implies that as the stock close price in the previous period decreases, the volatility of stock close prices tends to increase. This relationship holds across all three models, indicating a robust finding. Furthermore, the Loil has a statistically significant positive effect on LSCP, suggesting that higher oil prices are associated with increased volatility in stock close prices. These relationships hold consistently across all models. In the OLS model, the LCL and the IB both have a statistically significant positive effect on the LSCP at a significance level of less than 5%. So, the higher levels of the cost of living and interbank rate are associated with increased volatility in stock close prices. However, in the fixed effect and random effect models, both LCL and IB have a statistically significant negative effect on LSCP. This suggests that in these models, higher levels of the cost of living and interbank rate are associated with decreased volatility in close prices. The differences in results between the OLS model and the fixed effect/random effect models can be explained by the consideration of time-specific effects. By applying fixed effect and random effect models across multiple years, we can account for unobserved heterogeneity and omitted variables that may influence the relationship between LCL, IB, and LSCP. These models provide a more comprehensive analysis of the data and offer a deeper understanding of the associations at play. These findings highlight the importance of considering different models when examining the relationship between these variables and volatility.

The Hausman test, with a chi-squared statistic of 56.51 and four degrees of freedom, yields an extremely small p-value of 1.568 × 10⁻¹¹ (less than 5%). This significant p-value provides strong evidence to reject the null hypothesis and indicates a clear difference between the fixed effect and random effect models. Based on the results, we find that the performance of the fixed effect model is preferred over the random effect model in terms of efficiency and consistency for analyzing the panel data. This indicates reliance on the fixed effects model to account for individual-specific heterogeneity and capture time-invariant factors that may influence the relationships within the panel data.

4.3. Discussion

This study focuses on analyzing the data obtained from the closing prices of Tadawul. The selection of Tadawul for this analysis is driven by several reasons. Firstly, studying the history of stock market volatility in emerging economies is of great interest. Tadawul serves as an intriguing example of how significant volatility can arise due to factors such as informational imbalances, irrational trading behaviors, and inexperienced financial analysis. Furthermore, it is important to note that investors from countries outside the Gulf Cooperation Council (GCC) are currently allowed to invest in Tadawul. This change in regulations has opened up opportunities for international investors to participate in the Saudi Arabian securities market. The inclusion of foreign investors from various regions adds a new dimension to the dynamics of Tadawul. It introduces different investment strategies, perspectives, and capital flows that can influence market volatility. This increased participation from international investors brings forth a broader spectrum of factors that may impact the volatility patterns observed in Tadawul. Figure 3 illustrates the Haar wavelet decomposition of the Tadawul stock market closing price data. The plot below represents the original closing price data, while the subsequent plots display the decomposition process. The top plot shows the approximation coefficient, representing the low-fluctuated component of the data. The middle plots show the detail coefficients, representing the high fluctuated components at different scales or resolutions. The decomposition is carried out using the WT mechanism, where the original signal is referred to as $X={TV}_1+{TW}_1$. The next component consists of the approximation coefficients (${TV}_1$), showing the transformed data plot. The subsequent parts, denoted as ${TW}_1$, represent the level of detail, specifically the first level of detail coefficients. This level of detail helps explain the fluctuations observed in the data.

Tadawul has been significantly affected by various crises between 2006 and 2024. Notable among these is the global financial crisis of 2008, which led to a sharp decline in stock prices and increased volatility. The Arab Spring unrest in 2011, along with oil price volatility from 2014 to 2016, also had substantial impacts on Tadawul, resulting in heightened market fluctuations. Additionally, the COVID-19 pandemic in 2020–2022 caused significant disruptions and uncertainty, leading to volatility and declining stock prices. Moreover, Tadawul was also impacted by the Russian war against Ukraine, which began on 24 February 2022. The geopolitical tensions and uncertainties surrounding this conflict had repercussions on global financial markets, including Tadawul. The heightened geopolitical risks and market volatility influenced investor sentiment and contributed to fluctuations in stock prices. Furthermore, the Gaza war, which occurred on 7 October 2023, also had an impact on Tadawul. The escalation of tensions and military conflict in the region led to increased uncertainty and geopolitical risks. Such events can create a ripple effect in financial markets, including the Tadawul, as investors react to the evolving situation and adjust their investment strategies accordingly, potentially resulting in heightened market volatility during this period.

Table 6. WT functions of LSCP for 90% of dataset.

WT Function	ARIMA	ME	RMSE	MAE	MPE	MAPE
MODWT-Haar	(1, 2, 0)	0.000224189	0.03639061	0.001692966	0.001486559	0.02771326
MODWT-d4	(1, 1, 0) with drift	0.000399738	0.04741871	0.002278591	0.012942063	0.04236988
MODWT-la8	(1, 1, 0) with drift	3.71768 × 10−7	0.08567493	0.006146197	0.009557341	0.0893773
MODWT-bl14	(0, 1, 4) with drift	0.003037199	0.09548655	0.009468093	0.04904537	0.13277981
MODWT-c6	(1, 1, 0) with drift	4.18276 × 10−6	0.08487614	0.00617998	0.009307658	0.08971793

presents the results obtained from applying the proposed models using the first 90% of the dataset. The original LSCP dataset is provided by the MODWT models, with MODWT-Haar identified as the most favorable model based on comparison. It achieved the lowest values of 0.03639061, 0.001692966, 0.001486559, and 0.02771326 for RMSE, MAE, MPE, and MAPE, respectively. Note that the inputs in the DENFIS model are the values LCP, Loil, LCL, and IB, and the MODWT-Haar model utilized LSCP as the output variable.

Table 7. WT function of LCSP for 10% of dataset.

MODWT + DENFIS	ME	RMSE	MAE	MPE	MAPE
MODWT-Haar-DENFIS	1.838045611	2.400346293	2.083737559	39.01628681	41.99472697
MODWT-d4-DENFIS	2.390995893	2.72102146	2.496370191	51.05089676	52.34976187
MODWT-la8-DENFIS	2.32034751	2.660603756	2.454813829	48.96148489	50.60665145
MODWT-bl14-DENFIS	2.175358767	2.647173806	2.381527291	47.2883517	49.81025274
MODWT-c6-DENFIS	2.347182939	2.67846319	2.462737804	49.65928578	51.08222501
DENFIS + ARIMA direct	2.379833197	2.689491235	2.458738011	50.29882928	51.26072766
DENFIS	2.344125068	2.691874344	2.486763422	49.85705367	51.60197546

displays the results obtained from forecasting the WT models with a DENFIS using the remaining 10% of the original and transformed data as a means of validating our findings. The proposed models used in the training phase were employed once again. The best-performing model in terms of forecast accuracy is MODWT-Haar-DENFIS, as it demonstrates the lowest values for ME, RMSE, MAE, MPE, and MAPE. Consistent with the training phase, LSCP serves as the output variable, while LCP, Loil, LCL, and IB are used as input variables to construct MODWT + DENFIS models.

5. Conclusions

In this study, a hybrid model called MODWT-Haar-DENFIS is proposed and applied to forecast the volatility risk of the close price in Tadawul. The selection of input values for the model is based on various statistical analyses, including AC, PAC, correlation, multiple regressions, fixed effects, random effects, OLSs, and the Granger causality test. Through these analyses, the input values chosen for the model are LCP, Loil, LCL, and BI. The autocorrelation and partial autocorrelation analysis reveals a highly significant value for LCP at lag 1, emphasizing the importance of including lagged LCP in the model for improved prediction capabilities. The correlation analysis reveals a negative correlation between LCP and LSCP (−0.528) and weak positive correlations between LSCP and Loil and LCL. LCP shows weak positive correlations with Loil and IB, while LCL is weakly negatively correlated with LCP and IB. Granger causality tests show bidirectional causal relationships between input variables (Loil, LCL, and IB) and the output variable (LSCP). There is evidence of significant causality from Loil, LCL, and IB to LSCP, while LSCP influences LCP and IB. However, there is no significant evidence of causality between LSCP and Loil or LCL.

Regression analysis reveals that in the OLS model, LCP has a significant negative effect on LSCP, while Loil has a significant positive effect. In the fixed effect model, LCL and IB have significant positive effects on LSCP, while in the random effect model, LCL and IB have significant negative effects. The Hausman test favors the use of the fixed effect model over the random effect model. Fixed effect and random effect models were utilized to analyze the data across years. In the results, MODWT (Haar) showed the best performance with the lowest values for RMSE, MAE, MPE, and MAPE, using 90% of the dataset. The DENFIS model used LCP, Loil, LCL, and IB as input variables, while MODWT (Haar) used LSCP as the output variable. Validating the findings with the remaining 10% of the data confirmed that MODWT-Haar-DENFIS achieved the highest forecast accuracy.

This study on improving volatility prediction of Tadawul using MODWT-Haar-DENFIS has limitations that include the use of a limited set of input variables, namely cost of living, internal bank rate, and oil prices, without considering other macroeconomic variables. Additionally, the study focuses solely on the Tadawul dataset, restricting the generalizability of the findings to other stock exchange markets. To overcome these limitations, future research should incorporate a wider range of macroeconomic variables and expand the experiments to include data from prominent stock exchange markets such as NYSE, NASDAQ, SSE, and HKSE to provide a more comprehensive understanding of stock market volatility prediction. In addition, we will include a qualitative analysis of investor behavior to further enrich our study and provide a holistic view of the factors influencing stock market volatility. By integrating qualitative insights into our modeling approach, we aim to capture the nuances of investor sentiment, emotions, and decision-making processes more effectively.