Enhancing Predictive Accuracy through the Analysis of Banking Time Series: A Case Study from the Amman Stock Exchange

Abstract

This empirical research endeavor seeks to enhance the accuracy of forecasting time series data in the banking sector by utilizing data from the Amman Stock Exchange (ASE). The study relied on daily closed price index data, spanning from October 2014 to December 2022, encompassing a total of 2048 observations. To attain statistically significant results, the research employs various mathematical techniques, including the non-linear spectral model, the maximum overlapping discrete wavelet transform (MODWT) based on the Coiflet function (C6), and the autoregressive integrated moving average (ARIMA) model. Notably, the study’s findings encompass the comprehensive explanation of all past events within the specified time frame, alongside the introduction of a novel forecasting model that amalgamates the most effective MODWT function (C6) with a tailored ARIMA model. Furthermore, this research underscores the effectiveness of MODWT in decomposing stock market data, particularly in identifying significant events characterized by high volatility, which thereby enhances forecasting accuracy. These results hold valuable implications for researchers and scientists across various domains, with a particular relevance to the fields of business and health sciences. The performance evaluation of the forecasting methodology is based on several mathematical criteria, including the mean absolute percentage error (MAPE), the mean absolute scaled error (MASE), and the root mean squared error (RMSE).

1. Introduction

Forecasting holds a paramount significance across a spectrum of life’s domains, particularly in the context of foreseeing future events amid the pervasive uncertainties that enshroud the global economy. The heightened importance of precision in statistical data forecasting assumes a pivotal role, empowering policymakers to make informed decisions, facilitate effective planning, address challenges confronting economic and societal progress, and enhance competitive capabilities. The banking sector in Jordan stands as a vital component within the financial landscape, playing a crucial role in financing the Jordanian economy, fostering economic expansion, facilitating the funding of infrastructural ventures, and nurturing collaborations between the public and private sectors, all while exhibiting a notable degree of financial efficiency.

By the conclusion of 2021, the banking sector in Jordan had achieved a remarkable milestone, with the total assets within the sector surpassing the threshold of 60 billion Jordanian dinars. This significant growth underscores the sector’s pivotal role in the country’s financial landscape. Moreover, the banking sector’s assets were found to be in a relatively strong position when compared to the overall economy, with an asset ratio equivalent to 194% of the gross domestic product (GDP). The magnitude of total deposits held within banks was equally impressive, amounting to 126% of the GDP. These figures demonstrate the sector’s substantial financial influence and its capacity to mobilize resources for the broader economic development of Jordan.

One notable aspect of the Jordanian banking sector is its sound financial health. The capital adequacy ratio, a critical measure of a bank’s financial strength, stood at a robust 18.3% by the end of 2020. This reflects the sector’s ability to maintain an adequate capital buffer to absorb unexpected losses and ensure the stability of the financial system. Additionally, the legal liquidity ratio reached a remarkable 136.5% during the same period, indicating that the sector has maintained a high degree of liquidity, which is essential for fulfilling its obligations and ensuring its resilience in the face of economic challenges. Furthermore, the banking sector in Jordan is not only demonstrating financial strength but is also actively embracing digital transformation as a key element of its strategy. This commitment to digitalization aligns with broader financial inclusion goals, as it opens up opportunities for more individuals and businesses to access and benefit from financial services. The sector’s forward-thinking approach is not only in response to changing technological landscapes, but also a proactive step to promote economic growth and foster inclusivity.

In conclusion, the Jordanian banking sector’s substantial asset base, its strong position within the economy, and its commitment to digital transformation and financial inclusion all contribute to its significance in the nation’s economic development. These developments not only reflect the sector’s past achievements, but also its potential to further drive growth and progress in Jordan’s financial and economic landscape.

The precision of forecasting financial statements within the banking sector holds a significant bearing on the overall performance of banks across various critical dimensions. These dimensions encompass liquidity, profitability, investment, employment, and the management of risks inherent in lending and providing financial facilities. As such, financial planners and decision makers within the banking industry are intrinsically motivated to strive for the utmost degree of accuracy in predicting the performance of banks. This pursuit of precision in forecasting financial outcomes is multifaceted and carries profound implications.

To begin, the accuracy of forecasting plays an instrumental role in assessing a bank’s liquidity position. Liquidity management is crucial for banks as it directly influences their capacity to meet short-term obligations, fund daily operations, and ensure the timely settlement of customer transactions. Accurate forecasts empower decision makers to preemptively allocate resources in order to maintain optimal liquidity levels, thus averting potential crises and optimizing the utilization of the available funds.

Profitability represents another pivotal facet of bank performance intricately linked to forecasting accuracy. The ability to project financial performance with precision aids in estimating profits, assessing income streams, and making informed decisions concerning investments and expansion strategies. Accurate financial forecasts enable banks to tailor their business strategies, allocate resources efficiently, and ultimately enhance their profitability, thus contributing to their sustained competitiveness. Furthermore, the judicious prediction of financial outcomes is invaluable for the prudent allocation of resources towards investments and employment. Banks rely on their forecasts to identify opportunities for growth, expansion, and investment in various financial products and services. By achieving a high degree of forecast accuracy, decision makers can confidently allocate capital to promising investment opportunities, thus fostering economic development and creating employment opportunities, which are vital for the overall economic prosperity of a nation.

Lastly, the effective management of risks is paramount for the stability and security of banks. Risks arise from lending and granting financial facilities, and a lack of accuracy in forecasting can expose banks to potential financial crises. Accurate financial projections provide insight into potential risks and allow decision makers to devise strategies to mitigate and hedge against these risks, therefore safeguarding the stability and soundness of the banking sector.

In conclusion, the pursuit of forecast accuracy within the banking sector is an endeavor with multifaceted implications. It underpins the effective management of liquidity, the enhancement of profitability, the prudent allocation of resources to investments and employment, and the mitigation of risks. By achieving a high degree of forecast accuracy, financial planners and decision makers not only elevate the competitiveness of banks, but also contribute to the stability and economic growth of the broader financial system. As such, the pursuit of accuracy in forecasting financial statements remains an integral element in the strategic decision-making process within the banking industry.

The early prediction of bank failure is of paramount importance as it serves as a crucial preventive measure to safeguard financial institutions from the perils of liquidation and bankruptcy. Consequently, an extensive body of literature has emerged, delving into the prediction of bank failures. This research has yielded invaluable insights into the dynamics of forecasting, as well as contributing to the enhancement of the understanding of the factors associated with the financial well-being of banks.

Cui (2011) stands as a notable contributor to this field, as their work is centered on the analysis of the bank failure rate and prediction using the autoregressive integrated moving average (ARIMA) model. The model selection, validation, and classification of bank failures based on total assets serve as key components of their study. Cui’s research endeavors to provide a comprehensive understanding of the factors that underlie the prediction of bank failures. Gnagne and Moran (2020) have offered a distinctive model for predicting the total monthly number of commercial bank failures in the United States. Their model is founded on macroeconomic variables, initially proposed by McCracken and Ng (2016). Their analysis has revealed a robust and pivotal connection between housing sector-related factors and the incidence of bank failures, highlighting the significance of the interplay between developments in this sector and the vulnerabilities within the banking industry. Adeyeye and Oloyede (2014) have made significant contributions by examining an improved linear discrimination model designed to forecast the bank failure rate in Nigeria. Their model combines principal component analysis (AE) and discriminant analysis (DA) to facilitate estimation. Impressively, their model achieved a high degree of accuracy in predicting the status of failing banks, suggesting that discriminant analysis can serve as an effective indicator of impending bank failures. This research underscores the potential for the early detection of issues within banks, thereby enabling timely corrective measures to prevent failure. Hasan and Sheakh (2020) have delved into an advanced forecasting technique tailored to the banking sector in Bangladesh. They have examined the application of the exponential smoothing method, the Holt method, and the Winter method, with a focus on predicting the number of customers and customer entities on various time scales, ranging from daily to yearly. This innovative approach not only assists in determining customer wait times, but also aids bank managers in optimizing the number of bank tellers, thereby reducing operational costs. Hasan and Sheakh’s research provides a practical tool for enhancing bank management and improving customer service.

In summary, the extensive body of research on the early prediction of bank failures encompasses a wide array of methodologies and insights. These studies collectively contribute to the advancement of knowledge in the field, providing valuable tools and strategies to aid financial institutions in preserving their stability and resilience in the face of potential economic challenges.

In the realm of forecasting bank failures, numerous studies have taken a distinct approach by harnessing the capabilities of machine learning methods. One such study, conducted by Agrapetidou et al. (2020), presented a machine learning methodology termed AutoML, specifically applied to forecast the failure of US banks during the period spanning 2007 to 2013. Referred to as “Just Add Data” (JAD), this model demonstrated exceptional predictive capabilities, with an area under the curve (AUC) score of 0.985. The outcomes of this research underscored the efficacy of JAD and the AutoML tool in enhancing the productivity of financial data analysts, while concurrently mitigating the risks associated with systematic statistical errors. This approach holds promise as a valuable asset for stakeholders involved in banking risk assessment and management.

On another front, several studies have steered their attention towards forecasting the efficiency of banks. Yaghoubi and Fazli (2022), for instance, delved into the assessment and prediction of bank efficiency by introducing a hybrid model that ingeniously amalgamates Monte Carlo simulation technology (MC) with the genetic algorithm (GA) and the imperial competitive algorithm (ICA). The primary objective of this hybrid model is to fine-tune the performance parameters of MC–GA and MC–ICA. The results of their investigation illuminated the model’s remarkable accuracy in forecasting efficiency, which has substantial implications for bank managers and regulators seeking to optimize their institutions’ performance. This innovative approach allows for more informed decision making, efficient resource allocation, and, ultimately, the enhancement of the overall banking landscape.

In conclusion, these studies represent two distinctive branches of research within the broader scope of banking forecasting. The first employs AutoML in the JAD model to predict bank failures, emphasizing the gains in productivity and error prevention it offers to financial data analysts. The second utilizes a hybrid model integrating MC, GA, and ICA to forecast bank efficiency, showcasing its ability to provide precise efficiency assessments. These approaches contribute to the refinement of predictive tools for risk assessment and management, as well as the optimization of bank performance. They exemplify the ongoing efforts to employ cutting-edge methodologies and technologies in the banking sector, ultimately enhancing its resilience and competitiveness in an ever-evolving financial landscape.

2. Literature Review

In the expansive realm of economic literature and statistical studies, a multitude of efforts have been directed towards the enhancement of predictive accuracy over time. These endeavors have yielded valuable insights, fostering an evolving understanding of data forecasting and its applications across diverse fields. This narrative provides a comprehensive overview of several notable contributions in this domain, encompassing various methodologies and their implications.

Hwang et al. (2012) addressed the limitations of conventional methods used to evaluate data-driven predictive model performance, specifically in the context of hydrological data-driven forecasting. Their study introduced the concept of the relative correlation coefficient (RCC) to gauge predictive efficiency concerning the naive model (representing an unchanged scenario). The findings accentuated the merits of RCC, indicating its superiority over traditional evaluation techniques by providing a metric for assessing model improvements relative to the naive model in data-driven forecasting.

Kriechbaumer et al. (2014) made significant strides in boosting the performance of autoregressive integrated moving average (ARIMA) models for forecasting metal prices. Their research entailed optimizing the choice of wavelet transform type, wave function, and the number of decomposition levels employed in multiple resolution analysis (MRA). The outcomes were striking, showcasing a substantial improvement in prediction accuracy, particularly in forecasting aluminum, copper, lead, and zinc prices. This achievement testified to the effectiveness of their model in surpassing classical ARIMA models. In a distinct line of inquiry, Hossain et al. (2021) explored statistical methods for prediction, focusing on both high-frequency and low-frequency empirical mode decomposition (EMD) components. They introduced two novel approaches, namely the EMD–ARIMA–EWMA and EMD–EWMA–ARIMA methods, which were subsequently compared with four other forecasting methods. The results underscored the superior predictive accuracy of the EMD–ARIMA–EWMA method when contrasted with basic ARIMA and EWMA techniques. This research offers a promising avenue for financial forecasting and decision making. Wang and Chaovalitwongse (2018) presented a comprehensive overview of forecasting methods, detailing their basic procedures, underlying assumptions, applications, and limitations. They accentuated the significance of deploying a blend of quantitative and qualitative techniques in order to evaluate prediction performance comprehensively, serving as a robust foundation for decision support. Tamuke et al. (2018) conducted an in-depth analysis of ARIMA and ARIMAX methodologies for predicting the headline consumer price index (HCPI) in Sierra Leone. Their research illuminated the suitability of both ARIMAX and ARIMA methodologies in providing accurate projections of HCPI trends, underscoring their potential for facilitating decision making in economic planning and policy development. Jordan and Messner (2019) explored the utilization of predict accuracy indicators and dynamic methods to enhance planning quality, with a focus on a practical case study of a manufacturing company. Their research underlined the critical role of organizational and market context in shaping the quality of performance measures, underscoring the importance of context-specific performance evaluation. Prayudani et al. (2019) delved into a method for measuring prediction accuracy using mean squared error (MSE) and mean absolute error percentage (MAPE) in conjunction with random K-nearest neighbor (RKNN). Their investigation yielded impressive results, exemplifying the potential for enhancing prediction accuracy through carefully selected error metrics and algorithmic approaches. Bandara et al. (2021) introduced a data-intensive forecasting approach to improve global forecasting models (GFM) accuracy, particularly in settings with limited data abundance. They harnessed time series augmentation techniques, including GRATIS, moving block bootstrap (MBB), and dynamic time warping barycentric averaging (DBA), to generate synthetic time series data. Their study advocated two strategies, the pooled and transfer learning strategy, with the pooled strategy being more suitable for smaller datasets and the transfer strategy excelling with more comprehensive data. Safarkhani and Moro (2021) made significant strides in developing an accurate ratin tool to predict customer subscription to long-term bank deposits. Their approach sought to reduce the complexity of data modeling while enhancing the classification algorithm performance. The research demonstrated improvements in classification accuracy when compared to the existing methods, emphasizing the efficacy of the proposed fuzzy algorithm and other modeling techniques. Koutsandreas et al. (2021) conducted a meticulous analysis of the measures of forecasting accuracy, exploring whether the relative order of forecasting methods changes across different accuracy metrics. Their research illuminated the minor differences between various error measures, both in terms of percentage, relative, and scale measures, providing valuable insights for accurate forecasting. Van Khanh et al. (2021) introduced a method to enhance the efficiency of the fuzzy time series prediction model by leveraging swarm optimization algorithms and a novel puzzle removal technique. Their findings demonstrated the superior efficiency of the proposed forecasting model for both first-order and high-order fuzzy time series predictions, setting the stage for enhanced predictive capabilities. Nystrom (2021) put forward a wavelet-enabled prediction methodology for lightning, using chaotic EFM data. The application of wavelet denoising to chaotic time series EFM was found to be promising in enhancing lightning prediction accuracy, particularly when compared to the basic persistence model. Additionally, the research highlighted an improved imputation method that generated prediction accuracy exceeding 95%, a noteworthy achievement in the field of meteorological forecasting. Gisavandani et al. (2022) delved into the impact of discrete wave transformations (DMODWTs) on machine learning (ML) models. Their research evaluated the performance of regression and classification models in estimating daily downtime and sediment loading (SSL). The study introduced innovative methods, such as TOPSIS and maximum interference DMODWT (MODMODWT), to enhance ML models. The findings underscored the potential of DMODWT in improving the accuracy of ML models, with the TOPSIS method and Taylor chart RT-MODMODWT-db4 being identified as the most effective models for estimating daily SSL.

Moving towards artificial intelligence tools in forecasting, a particular focus emerged on neural networks and their integration with wavelet transformations. Tealab et al. (2017) presented novel neural network models, including deep learning neural networks, with or without hybrid methodologies like fuzzy logic, to address the shortcomings of traditional neural networks. Their research aimed at improving predictive accuracy, particularly for nonlinear or dynamically changing time series data, with significant potential for performance enhancement. Mehdiyev et al. (2016) emphasized the evaluation of the prediction model performance, highlighting the importance of accuracy measures. The study undertook a comprehensive review of prediction technologies, including Bayesian networks, artificial neural networks, support vector machines (SVM), logistic regression, and others, accompanied by an in-depth analysis of accuracy metrics. This research provides a comprehensive reference for selecting the appropriate measures of accuracy to evaluate the performance of prediction models.

Bozic and Babić (2018) proposed a hybrid model that integrated wavelet transform, neural networks, and statistical time series models to enhance prediction accuracy. The research demonstrated the potential for noise reduction through the application of wavelet basis functions, ultimately generating more accurate results. Their work has implications for improving predictive.

This article is organized as follows: the mathematical models will be discussed in Section 2. Then, the research design and methodology are explained in Section 3. Section 4 will discuss the data set and the results of the study. Finally, Section 5 will conclude this research.

3. Mathematical Models

This section gives a background of the main concepts used in our study:

3.1. Wavelet Transform

The mathematical overcomplete discrete wavelet transform (MODWT) represents a powerful mathematical tool utilized for the transformation of original time series data into the time-scale domain. Its application holds particular allure when dealing with non-stationary data, such as those encountered in stock market analysis. MODWT can be broadly categorized into three forms: the discrete wavelet transform (DWT), the continuous wavelet transform (CWT), and the MODWT. While these different transformations share common features in terms of their underlying principles, they exhibit distinctions in their application.

DWT was introduced to overcome the disadvantages of Fourier transform (FT) since FT focuses on the domain only, whereas DWT focuses on the time-domain together. It is well known that most of the stock market data are non-stationary and, therefore, FT and the linear functions are an unusual choice for modeling, estimating, and predicting the stock market series (data). Moreover, DWT has some extra useful characteristics that make DWT suitable for non-stationary and nonlinear time series data which are as follows: zero mean and square-integrable to one, regularity, which is vital for smooth reconstructions, the existence related filters, orthogonality, and compact support (Al Wadi et al. 2011; Gençay et al. 2001).

One of the primary distinctions lies in the constraints imposed on the size of the input data. Specifically, the DWT necessitates that the number of observations in the data be a power of 2 (2^J). In contrast, MODWT is characterized by its remarkable flexibility, accommodating data of any size. This adaptability endows MODWT with a modern and versatile appeal, making it particularly suited to address contemporary data analysis challenges.

The foundation of MODWT can be traced back to the Fourier transform (FT), which is based on the utilization of sine and cosine functions. Notably, MODWT serves as an extension of the FT, augmenting its capabilities in several domains, including time series analysis and signal processing. An essential characteristic of MODWT is its adherence to the admissibility condition, as expounded upon by Jaber et al. in 2020. This condition serves as a crucial criterion, ensuring the mathematical soundness and applicability of MODWT in diverse contexts.

In summary, MODWT offers a versatile and contemporary approach to data analysis, with its applicability extending to non-stationary data, especially in the realm of stock market data. Its adaptability to data of varying sizes and its roots in the Fourier transform make MODWT a valuable tool for modern data analysis and signal processing endeavors. Moreover, its adherence to the admissibility condition enhances its mathematical rigor and utility in various applications. This article will delve further into the nuances and applications of MODWT, shedding light on its practical relevance in addressing complex data analysis challenges (Gençay et al. 2001).

$$C_{\phi }={\int }_0^{\infty }\phi \left(f\right)\vee \frac{\phi \left(t\right)}{f}df<\infty$$

(1)

where $\phi \left(f\right)$ is the FT and a function of frequency f, $\phi \left(t\right)$. The MODWT is used in many applications, such as image analysis and signal processing. It was introduced to overcome the problem of FT, especially when dealing with time, space, or frequency.

There are two types of MODWT, which are father wavelet, which describes the low-frequency (smooth coefficients) components, and mother wavelet, which describes the high-frequency (detailed coefficients) components, as shown in Equation (2), respectively, with j = 1, 2, 3, …, J in the J-level wavelet decomposition:

$$\begin{gathered} \varphi j,k=2^{\left(\frac{-j}{2}\right)}\varphi \left(t-\frac{2^{j}k}{2^j}\right) \\ \phi j,k=2^{\left(\frac{-j}{2}\right)}\phi \left(t-\frac{2^{j}k}{2^j}\right) \end{gathered}$$

(2)

where J denotes the maximum scale sustainable by the number of data points and the two types of wavelets stated above, namely father wavelets and mother wavelets, satisfying:

$$\int \varphi \left(t\right)dt=1\wedge \int \phi \left(t\right)dt=0$$

(3)

The general mathematical model is presented in Equation (4):

$$S_{j,k}=\int {\varphi }_{j,k}f\left(t\right)dt,d_{j,k}=\int {\phi }_{j,k}f\left(t\right)dt,$$

(4)

In more detail,

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

(5)

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

(6)

The MODWT is used to calculate the approximation coefficient in Equation (6), where $S_j\left(t\right)$ and $D_j\left(t\right)$ are introducing the smooth and detailed coefficients, respectively. The smooth coefficients contain the most important features of the original data, while the detailed coefficients are used to detect the main fluctuations of the original data. Generally, MODWT has popular transform functions (Gençay et al. 2001), namely Haar, Daubechies (d4), Coiflet (c6), least asymmetric (LA8), and best-localized (bl14). The number of main properties of these functions is as: MODWT functions are arbitrary regular, except for the Haar model. MODWT functions do not have explicit expressions, but for the Haar model. MODWT functions use real number. MODWT functions are characterized by orthogonality, compact support, an arbitrary number of zero moments, the existence of the scale function, orthogonal analysis, bio-orthogonal analysis, continuous/discrete transformation, exact reconstruction, and a fast algorithm.

3.2. Autoregressive Integrated Moving Average Model (ARIMA)

The auto-regressive moving average (ARMA) models are used in time series analysis to describe stationary time series. The ARMA model is a combination of a moving average (MA) model and an autoregressive (AR) model. A time series {$e_t$} is called a white noise (WN) process, and $\left\{Y_t\right\}$ is called the Gaussian process if for all t, $e_t$ is iid $N\left(0,{\sigma }^2\right)$. A time series {$Y_t$} is said to follow the ARMA(p,q) model if (Jaber et al. 2020):

$$Y_t=\mu +{\varphi }_1{Y}_{t-1}+{\varphi }_2{Y}_{t-2}\dots +{\varphi }_p{Y}_{t-p}+e_t-{\theta }_1{e}_{t-1}-{\theta }_2{e}_{t-2}\dots -{\theta }_q{e}_{t-q}$$

(7)

where $q$ and $p$ are non-negative integers, $p$ represents order of autoregressive part (AR), $q$ is defined as order of the first moving part (MA), and {$e_t$} is the white noise (WN) process. An extension of the ordinary ARMA model is the auto-regressive integrated moving-average model (ARIMA(p,d,q)), given by (Jaber et al. 2020):

$${\varphi }_p\left(B\right){\left(1-B\right)}^d{Y}_t={\theta }_0+{\theta }_q\left(B\right)e_t$$

(8)

where $p$, $d$, and $q$ denote the orders of auto-regression, integration (differencing), and moving average, respectively. When d = 0, the ARIMA model reduces to the ordinary ARMA model.

3.3. Accuracy Criteria

The authors use several types of accuracy criteria: The mean absolute percentage error (MAPE), the mean absolute scaled error (MASE), the root means squared error (RMSE), the Akaike information criterion (AIC, AICs), and the Bayesian information criterion (BIC). The MAPE criterion is also known as the mean absolute percentage deviation (MAPD) which measures a forecasting method’s prediction accuracy in statistics. It usually expresses accuracy as a percentage and is defined as $MAPE=\frac{\%100}{n}{\sum }_{t=1}^n\left|\frac{X_t-F_t}{X_t}\right|$, where $X_t$ is the actual value, and $F_t$ is the forecast value. The absolute value in this calculation is summed for every forecasted point in time and divided by the number of fitted points. Also, the MASE is recommended for determining the comparative accuracy of forecasts. It is estimated by $MASE=\frac{{\sum }_{t=1}^n\left|X_t-F_t\right|}{\frac{n}{n-1}{\sum }_{t=2}^n\left|X_t-X_{t-1}\right|}$, where the numerator is the forecast error for a given period, defined as the actual value ($X_t$) minus the forecast value ($F_t$) for that period, and the denominator is the mean absolute error which uses the actual value from the prior period as the forecast: $F_t=X_{t-1}$. Next, the RMSE is known also as the root-mean square deviation (RMSD) which is a frequently used measure of the differences between estimators. It measures the average error performed by the model in predicting the outcome for an observation. It is defined as the square root of the mean square error as: $RMSE=\sqrt{MSE}=\sqrt{\frac{{\sum }_{i=1}^N{\left(actualvalue-predictedvalue\right)}^2}{N}}$, where $N$ represents the number of observations. Another accuracy criterion is AIC, which is defined as $AIC=-2\ast log-likelihood+k\ast npar$, where $npar$ represents the number of parameters in the fitted model, k = 2, and n is the number of observations; AICc is a version of AIC corrected for small sample sizes. Finally, the BIC is $BIC=-2\ast log-likelihood+k\ast npar$, where $npar$ represents the number of parameters in the fitted model, k = log(n), and n is the number of observations (Jaber et al. 2020).

4. Research Design and Methodology

The research objectives can be succinctly delineated as follows: firstly, to model banking time series data through the utilization of the maximal overlap discrete wavelet transform (MODWT); secondly, to scrutinize fluctuations within historical data sourced from the Amman Stock Exchange (ASE), spanning the years 2011 to 2020; and finally, to enhance forecasting accuracy by integrating MODWT functions with a fitted autoregressive integrated moving average (ARIMA) model.

In greater detail, the employed methodology involves the decomposition of data using MODWT functions, specifically Haar, Daubechies (d4), Coiflet (c4), least symmetric (LA8), and best localized (Bl14), applying them to historical return data. Subsequently, the detailed coefficients are employed to identify fluctuations and analyze the historical volatility data. A fitted ARIMA model, utilizing the approximation coefficients, is then employed for forecasting. The proposed technique is subsequently compared with a direct ARIMA modeling approach.

The proposed method operates by transforming data into two sets: details series (DA1 (n)) and approximation series (CA1 (n)), which exhibit favorable characteristics for financial data due to its significant fluctuations. The filtering effect of MODWT contributes to the efficacy of these series, particularly in handling rough data patterns. The authors emphasize the repeated application of the wavelet process in instances of highly erratic data patterns. Pre-processing aims to minimize statistical criteria, such as root mean squared error (RMSE), between the signal before and after transformation, thus facilitating noise removal from the original data. The authors advocate for the adoption of MODWT twice during the pre-processing of training data to mitigate overfitting risks, in accordance with prior research (Gençay et al. 2001).

For a fair comparison, the authors allocate 90% of the dataset, encompassing both original and transformed data, to the proposed model, selecting the best model for further application. The chosen model is then tested against other proposed models using the remaining 10% of the data. This approach, while different from the conventional 90/10 data division, is justified by the substantial dataset at the authors’ disposal.

5. Empirical Results

Results and Discussion

This study undertakes an examination of the pattern exhibited by the selected dataset from the Amman Stock Exchange (ASE). The rationale for choosing this dataset is multifaceted; emerging markets, such as the ASE, provide a compelling historical context for the study of stock market volatility. The ASE, in particular, manifests noteworthy volatility, attributable to factors such as information imbalances, haphazard trading, and a lack of professionalism in financial analysis. Notably, investors from countries outside the Gulf Cooperation Council (GCC) are precluded from investing in Saudi stocks, rendering the ASE the largest stock exchange in the Middle East. Consequently, the banking time series data is subjected to decomposition using the maximal overlap discrete wavelet transform (MODWT) functions, specifically employing the C6 function. Subsequent to this decomposition, an analysis of fluctuations during the selected period ensues.

Firstly, the data statistical descriptive is as shown in the following

Table 1. Descriptive statistic of Amman Stock Market dataset.

	N	Mean	Std. Deviation	Skewness	Kurtosis	Minimum	Maximum
Closed Price Stock prices	2048	706.4	18.9	0.77	2.85	3106	4622

.

Figure 1 delineates the decomposition process applied to the dataset, utilizing the C6 function within the maximal overlap discrete wavelet transform (MODWT) model. This methodology proves instrumental in elucidating fluctuations, magnitudes, and phases inherent in the closing price data under consideration. The employment of MODWT (C6) on the historical data facilitates their decomposition into various resolution levels, thereby exposing their fundamental structure and yielding detailed coefficients at each decomposition level.

The MODWT mechanism is encapsulated by the equation X = TV1 + TW1, where X denotes the original signal. Subsequently, the first component comprises an approximation level (TV1), depicting the plot of approximation coefficients for the transformed data. The subsequent components, TW1, represent the details level, wherein TW1 serves as the plot of the first level of details coefficients, thereby elucidating fluctuations in the data.

Aligned with the Kingdom’s vision for 2030, initiatives aimed at market liberalization have been implemented, yielding positive outcomes such as capital stimulation, heightened liquidity, reduced capital costs, amplified total investment growth, increased support for small and medium-sized enterprises, and augmented market efficiency. These measures, integral to the economic reforms in the Kingdom of Saudi Arabia, seek to elevate the financial sector’s development and bolster its competitiveness, in accordance with the aforementioned vision.

Table 2. ARIMA–MODWT model forecasting results.

		ARIMA	ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
90% of data	Haar	(5,1,0) drift	0.821212346	43.0462	18.69768	0.01709041	0.4817221	0.018999024	0.7027236
	Dr	(5,1,0)	0.268337495	47.31613	19.53045	0.000866828	0.5078466	0.180533979	0.729835
	la8	(1,1,0) drift	0.009755415	112.10097	41.69464	0.056107239	1.140073	0.748944802	1.9988223
	bl14	(1,1,0) drift	0.002595647	141.60883	56.8405	0.085610701	1.5620789	0.84858781	2.703586
	c6	(4,1,0) drift	0.104025962	104.70962	40.477	0.038826902	1.0963204	0.742490165	2.0910647
	ARIMA-direct	(2,1,1) drift	0.003052208	22.00009	15.17782	0.000656064	0.3944505	0.003001232	0.8098273
		ME	RMSE	MAE	MPE	MAPE	MASE	ACF1
10% of data	Haar	67.84925	83.46813	71.32881	2.1870102	2.299683	0.911505	88.2912
	Dr	34.60203	61.12661	53.00363	1.1277554	1.727593	0.916336	67.11612
	la8	15.14149	51.82794	46.37243	0.4969902	1.521115	0.9145319	451.74601
	bl14	27.3507	57.28331	50.53291	0.8935353	1.651083	0.9138994	265.35401
	c6	12.06273	50.90685	45.67318	0.396421	1.499595	0.9168944	38.63534
	ARIMA-direct	32.53545	79.72908	54.90795	1.1094251	1.850167	0.9368992	40.16456

provides the outcomes derived from the proposed models applied to the initial 90% of the dataset. The ARIMA-direct model is attained by directly applying the pure ARIMA model to the original data. ARIMA–MODWT (Haar) combines ARIMA with the Haar function. ARIMA–MODWT (d4) integrates the ARIMA model with the Daubechies function (d4). ARIMA–MODWT (LA8) amalgamates the ARIMA model with the least square model (LA8) function. ARIMA–MODWT (Bl14) unites the ARIMA model with the best localized (Bl14) function. ARIMA–MODWT (C6) synergizes the ARIMA model with the Coiflet (C6) function. The authors conclude that, based on the comparison results, ARIMA–MODWT, utilizing the C6 function, emerges as the optimal approach, exhibiting minimal values for the root mean squared error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and mean absolute scaled error (MASE) when 10% of the observations are considered.

6. Conclusions

This article undertakes a comprehensive investigation into the volatility pattern inherent in the Amman Stock Exchange (ASE), encompassing modeling, study, and the enhancement of forecasting accuracy. The analytical focus is directed toward the entire closed price index dataset spanning the temporal domain from October 2014 to December 2022. The analytical framework employed herein rests upon a non-linear spectral model, specifically the maximum overlapping discrete wavelet transform (MODWT), utilizing the Coiflet function (C6), in conjunction with the autoregressive integrated moving average (ARIMA) model.

Throughout the article, particular attention is devoted to the examination of fluctuations and noteworthy events within the ASE during the specified period. These fluctuations and events are subjected to a detailed discussion, elucidating their implications and thus contributing to a holistic understanding of the observed volatility pattern. Furthermore, the article introduces a novel model, denoted as ARIMA–MODWT utilizing the C6 function. This model is systematically evaluated against alternative models, with the fitted ARIMA model serving as a benchmark. The comparative analysis is grounded in statistical criteria, encompassing metrics such as the mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE).

The findings of this comparative analysis substantiate the superiority of the proposed ARIMA-MODWT model employing the C6 function. Noteworthy improvements are evident in forecasting accuracy, as indicated by the aforementioned statistical metrics. This underscores the efficacy of MODWT in the decomposition of the dataset, thereby facilitating the identification and comprehension of significant events. The integration of MODWT contributes meaningfully to the forecasting capabilities, rendering the proposed model particularly robust.

The results of this research can be extended to future works by, for example, applying this method in other international stock markets. Moreover, many other types of data can be used, such as gold price, oil price, and others, which will be useful for investors and for decision making.

7. Study’s Limitations

In acknowledging the limitations of this study, it is essential to recognize the exclusive reliance on stock market data derived solely from the ASE. To advance the applicability and generalizability of the findings, future research endeavors may extend their purview to include datasets from diverse stock markets. Furthermore, the potential application of the proposed methodology to expansive datasets, particularly within the health sector, constitutes a promising avenue for future exploration. These considerations align with the scholarly imperative to continually refine and broaden the scope of empirical investigations, thereby contributing to the cumulative body of knowledge in this field.