A Hybrid GARCH-BiLSTM-KAN Model for Crude Oil Price Forecasting: Capturing Volatility, Temporal Dependencies, and Nonlinear Dynamics

Summary

Here, we present a hybrid GARCH-BiLSTM-KAN model for forecasting crude oil prices. The model integrates volatility estimation, bidirectional temporal learning, and nonlinear refinement to enhance prediction accuracy, offering a robust tool for energy market participants.

Abstract

Crude oil prices, as a cornerstone of global energy markets, exhibit intricate dynamics---including volatility clustering, asymmetric temporal dependencies, and nonlinear responses to geopolitical, economic, and supply-demand shocks---posing formidable challenges to accurate forecasting. Existing models often struggle to simultaneously capture these multifaceted characteristics, limiting their predictive robustness. To address this, this study proposes a novel hybrid framework which synergistically integrates three complementary components: (1) the Generalized Autoregressive Conditional Heteroskedasticity model to quantify time-varying volatility and address clustering effects; (2) the Bidirectional Long Short-Term Memory network to model bidirectional temporal relationships, capturing both historical and future contextual influences on price movements; and (3) the Kolmogorov-Arnold Network to refine nonlinear patterns through univariate basis functions, enhancing the mapping of complex high-dimensional dependencies beyond the capabilities of traditional neural networks. Empirical validation is conducted using 39 years of daily West Texas Intermediate crude oil prices (1986-2025), a dataset encompassing critical events such as the 2008 financial crisis, 2020 COVID-19 pandemic, and 2022 geopolitical tensions, ensuring robustness across diverse market conditions. The proposed model is rigorously compared against benchmark models, including traditional volatility models, standalone deep learning architectures, and other hybrid models. Results demonstrate that the proposed hybrid achieves superior performance with the lowest root mean squared error, mean absolute error, and the highest coefficient of determination. Statistical tests confirm the significance of its outperformance, highlighting the synergistic value of integrating volatility modeling, bidirectional sequence learning, and advanced nonlinear refinement. This research advances energy economics by providing a robust forecasting tool, with implications for policymakers in strategic energy planning, energy firms in risk hedging, and financial institutions in derivative pricing and portfolio optimization.

Introduction

Energy commodities, particularly crude oil, serve as fundamental drivers of global economic systems, influencing production costs, inflation rates, and international trade balances. The complex pricing dynamics of crude oil reflect intricate interactions among geological constraints, technological innovations in extraction, geopolitical tensions, and macroeconomic policies. Since the oil crises of the 1970s, understanding and forecasting oil price movements has emerged as a critical research domain intersecting energy economics, financial engineering, and computational intelligence^1,2. The inherent volatility of oil markets, compounded by periodic supply disruptions and demand shocks, creates substantial challenges for diverse stakeholders, including national governments, energy corporations, and financial institutions.

Crude oil, as a pivotal strategic resource, plays an indispensable role in global energy security, economic stability, and industrial development³. Its price fluctuations exert far-reaching impacts on key macroeconomic indicators, including inflation, employment, and trade balances⁴, while simultaneously influencing micro-level decision-making across industries ranging from transportation to manufacturing⁵. However, crude oil prices exhibit complex dynamic characteristics including volatility clustering (periods of high volatility followed by high volatility), nonlinear temporal dependencies, and heightened sensitivity to geopolitical events, economic policies, and supply-demand shocks^6,7. These multifaceted characteristics pose significant challenges to accurate forecasting, establishing it as a long-standing focus in energy economics and financial research.

Accurate crude oil price forecasting is essential for multiple stakeholders with distinct operational requirements. For governments and policymakers, it informs critical decisions regarding energy policy formulation, strategic reserves management, and inflation control mechanisms⁸. For energy companies, it supports strategic investment decisions, risk hedging strategies, and production planning optimization⁵. For financial market participants, it guides the precise pricing of oil-related derivatives and enhances portfolio optimization techniques⁹. Conversely, inaccurate forecasts may precipitate market distortions, encourage excessive speculation, or lead to suboptimal policy responses⁸. Consequently, developing robust forecasting models remains an urgent research priority with significant practical implications.

Crude oil prices are influenced by a complex interplay of multidimensional factors, including: (1) supply-demand fundamentals such as OPEC production quotas, shale oil extraction costs, and global energy transition trends¹⁰; (2) financialization effects through speculative activities in commodity markets that amplify price volatility¹¹; and (3) geopolitical and macroeconomic shocks including conflicts in oil-producing regions, economic recessions, and abrupt policy shifts¹². These diverse factors generate price dynamics that systematically violate the fundamental assumptions of traditional linear models, particularly homoskedasticity and stationarity^13,14. Volatility clustering, for instance, implies that past price fluctuations contain valuable information about future volatility-a phenomenon first formalized by Engle (1982)¹⁵ through the Autoregressive Conditional Heteroskedasticity (ARCH) model. Additionally, nonlinear relationships among fundamental drivers and bidirectional temporal dependencies further complicate the forecasting process⁶, with extreme events such as the 2020 negative WTI prices demonstrating severe nonlinear anomalies that challenge conventional modeling approaches¹⁶.

Despite substantial progress in the field, three critical research gaps have yet to be adequately addressed in the existing literature. First, there remains insufficient integration between volatility modeling and sequence learning. Current hybrid approaches often treat volatility estimation and temporal learning as separate components, failing to encode time-varying risk measures as structured inputs into sequential networks. This disjuncture limits the ability of models to adapt temporal representations to dynamically changing market risk conditions. Second, bidirectional dependencies in temporal modeling are largely overlooked. Most LSTM-based hybrid models rely on unidirectional architectures, which cannot capture how future expectations and forward-looking information retroactively influence current price formation-a crucial mechanism in expectation-driven commodity markets. Finally, deep architectures exhibit limited nonlinear refinement capabilities. Conventional activation functions in neural networks struggle to approximate discontinuous market behaviors or account for extreme anomalies, such as the 2020 negative WTI prices triggered by storage capacity constraints^17,18.

This study makes three key contributions: methodologically, it introduces a novel hybrid framework that synergistically integrates GARCH-based volatility modeling, bidirectional temporal learning via BiLSTM, and advanced nonlinear refinement with Kolmogorov-Arnold Networks (KAN) to capture the complex characteristics of crude oil prices; empirically, it constitutes a significant advancement by demonstrating, through rigorous evaluation on 39 years of daily WTI prices spanning diverse market regimes, superior forecasting performance against a wide range of benchmarks, with the robustness of these improvements confirmed by statistical significance tests; and practically, it offers substantial utility by providing a robust tool for energy market participants, with direct applications in risk management for energy firms, strategic planning for policymakers, and derivative pricing for financial institutions.

This study is guided by three central research questions. It first investigates how volatility modeling can be effectively integrated with deep learning architectures to enhance the accuracy of crude oil price forecasts. Furthermore, it examines the extent to which bidirectional temporal learning captures asymmetric dependencies in oil price dynamics compared to conventional unidirectional approaches. Finally, the study assesses whether the Kolmogorov-Arnold Network (KAN) provides superior nonlinear refinement over traditional activation functions within hybrid forecasting models.

The evolution of crude oil price forecasting methodologies reflects broader trends in time series analysis and financial econometrics. Early approaches predominantly relied on linear statistical models, with Autoregressive Integrated Moving Average (ARIMA) models and their variants forming the foundational framework^19,20. While effective for capturing linear temporal dependencies, these models proved inadequate for addressing the heteroskedasticity and volatility clustering characteristic of financial time series.

To model time-varying volatility, Engle (1982)¹⁵ introduced the Autoregressive Conditional Heteroskedasticity (ARCH) model, where conditional variance is modeled as a function of past squared residuals. Bollerslev (1986)²¹extended this through the Generalized ARCH (GARCH) model, allowing conditional variance to depend on both past squared residuals and past conditional variances. The GARCH(1,1) variant, with its parsimonious specification, has become a workhorse for volatility modeling in finance²². Further developments included exponential GARCH (EGARCH) to capture leverage effects²³ and fractionally integrated GARCH for long memory processes²⁴. However, these models struggle to capture complex nonlinear relationships and long-range dependencies²⁵.

With advances in computational power and algorithmic sophistication, machine learning models emerged as powerful alternatives for capturing nonlinear patterns in financial data^26,27. Support Vector Regression (SVR)²⁸ and Random Forests demonstrated improved performance over linear models for certain forecasting horizons^29,30,31.

The interplay between energy markets, environmental policy, and forecasting methodologies constitutes a critical area of research. The real-world impact of energy dynamics is profound, as evidenced by studies quantifying the household welfare loss stemming from energy price crises, highlighting the socio-economic urgency of accurate energy market analysis³¹. Within the policy realm, the dynamic relationship between carbon trading systems and financial markets, such as the low-carbon stock market, further reveals the intricate connectivity between regulatory mechanisms and economic performance³². To enhance the operational efficiency and integration of renewable energy, advanced forecasting techniques are paramount. Recent advancements include adaptive spatiotemporal graph models for predicting wind power generation at the farm-cluster level, which effectively capture complex spatial and temporal dependencies³³. Concurrently, the drive for model transparency in this field has led to the application of interpretable AI techniques, such as the LIME algorithm, to demystify 'black-box' wind power forecasts and build trust in their outputs³⁴. Collectively, these studies underscore a multidisciplinary approach that combines economic analysis, market linkage research, and cutting-edge, explainable forecasting models.

Deep learning models, particularly Recurrent Neural Networks (RNNs) and their variants, have shown remarkable success in sequential data modeling. Long Short-Term Memory (LSTM) networks, with their gated mechanisms addressing the vanishing gradient problem, have demonstrated exceptional performance in oil price forecasting^35,36. Recent innovations include multi-headed variational neighbour search-tuned RNNs for gasoline and crude oil prediction, decomposition-aided LSTM frameworks with SHAPley value explanation for bitcoin forecasting, and optimization-enhanced LSTM variants tuned by improved seagull optimization, salp swarm algorithms with disputation operators, and enhanced Harris hawks optimization for crude oil price forecasting.

Bidirectional LSTM (BiLSTM) architectures further enhance modeling capability by processing sequences in both temporal directions, capturing how both historical patterns and future expectations influence current price formation³⁷. This bidirectional modeling proves particularly valuable in commodity markets where forward-looking information significantly impacts spot prices.

Recognizing the complementary strengths of statistical and machine learning approaches, researchers have developed hybrid models that integrate multiple methodologies³⁸. GARCH-LSTM combinations represent one prominent strand, where GARCH-derived volatility estimates enrich the feature space for LSTM temporal learning³⁹. Similarly, CNN-LSTM models use convolutional neural networks to extract local features from time series before processing with LSTM⁴⁰. These hybrids typically outperform standalone models but often retain limitations in capturing bidirectional dependencies and refining complex nonlinearities.

Recent studies have explored various sophisticated hybrid approaches. The application of hybrid forecasting models to capture the complex linear and nonlinear characteristics of crude oil prices has become a prominent research direction. For instance, one study demonstrated the efficacy of combining predictions from multiple individual time series models to construct a hybrid framework for forecasting day-ahead Brent crude oil prices⁴¹. Advancing this approach, another study proposed a novel hybrid technique that integrates the linear ARIMA model with the nonlinear Long Short-Term Memory (LSTM) network, aiming to simultaneously capture intrinsic price patterns and dynamic nonlinear fluctuations⁴². These works collectively underscore the superior performance of hybrid strategies and establish a solid foundation for subsequent model development.

The recent introduction of Kolmogorov-Arnold Networks (KAN) by Liu et al. (2024)⁴³ represents a paradigm shift in neural network design, replacing fixed activation functions with learnable spline-based basis functions. Drawing on the Kolmogorov-Arnold representation theorem, KANs decompose high-dimensional functions into combinations of univariate functions, offering superior accuracy and interpretability compared to traditional Multi-Layer Perceptrons (MLPs).

While KANs have shown promising results in scientific computing and physics-informed machine learning, their application in financial time series forecasting remains nascent. Preliminary applications demonstrate KAN's potential in capturing complex nonlinear patterns that elude conventional architectures. Compared to recent hybrid modeling approaches, KAN-based frameworks offer distinct advantages in interpretability through visualizable basis functions and adaptive refinement of nonlinear mappings.

The integration of KAN specifically addresses the limitation of traditional activation functions (e.g., ReLU, tanh) in approximating discontinuous jumps and extreme market anomalies, such as the 2020 negative WTI prices. By employing cubic B-spline basis functions, KAN enables fine-grained refinement of temporal features, capturing residual nonlinearities that significantly impact forecasting accuracy during market turbulence.

Our review identifies three critical gaps that motivate the current study. First, existing hybrids insufficiently integrate volatility modeling with sequence learning, treating them as separate rather than complementary components. Second, bidirectional temporal dependencies remain underexplored despite their importance in expectation-driven commodity markets. Third, nonlinear refinement capabilities are limited by traditional activation functions, particularly during extreme market events⁴⁴. The GARCH-BiLSTM-KAN framework proposed in this study systematically addresses these gaps through synergistic integration of complementary modeling paradigms.

Protocol

This section elaborates on the methodological framework of the proposed GARCH-BiLSTM-KAN hybrid model, which integrates the strengths of GARCH for volatility modeling, BiLSTM for bidirectional temporal dependency capture, and KAN for nonlinear refinement. The architecture is designed to address the multifaceted characteristics of financial time series, including volatility clustering, temporal asymmetry, and complex nonlinear patterns. Figure 1 shows the overall framework flowchart of the GARCH-BiLSTM-KAN hybrid model.

GARCH for volatility estimation

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, proposed by Bollerslev (1986)²¹, serves as the foundational component for volatility estimation in this hybrid framework. Financial time series, such as stock returns, exhibit volatility clustering-periods of high volatility followed by high volatility and low volatility followed by low volatility-which violates the homoskedasticity assumption of traditional linear models. GARCH models capture this phenomenon by modeling the conditional variance as a function of past squared residuals and past conditional variances.

Mathematical formalization

For a given time series of returns r_t, the GARCH (p, q) specification consists of two equations: the mean equation and the variance equation. The mean equation is defined as:

where μ is the constant mean return, and  is the error term at time t, which follows a conditional normal distribution with F_t-1 representing the information set up to time t - 1.

The conditional variance , which measures the volatility, is specified in the GARCH (p, q) variance equation as:

where ω > 0 is the constant term, α_i ≥ 0  are the ARCH coefficients capturing the impact of past squared residuals (news about volatility from the recent past), and β_j ≥ 0  are the GARCH coefficients representing the persistence of volatility. To ensure the positivity and stationarity of the conditional variance, the parameters must satisfy ω > 0, α_i ≥ 0, β_j ≥ 0, and .

In this hybrid model, the GARCH (1,1) variant was employed, which is parsimonious and widely documented to effectively capture volatility dynamics in financial data^21,22.The estimated conditional volatility from GARCH (1,1) serves as a critical input to the subsequent BiLSTM and KAN components, providing a structured measure of historical volatility that complements the raw return series.

BiLSTM for bidirectional temporal learning

While GARCH models excel at volatility estimation, they are limited in capturing complex temporal dependencies, especially those involving long-range interactions and bidirectional relationships. To address this, a Bidirectional Long Short-Term Memory (BiLSTM) network is integrated, an extension of the LSTM architecture⁴⁵, which is designed to model sequential data by preserving information from both past and future time steps.

Architecture specification

LSTM networks overcome the vanishing gradient problem of traditional Recurrent Neural Networks (RNNs) through a gated cell structure, enabling them to learn long-term dependencies. Each LSTM cell contains three key gates: the forget gate, input gate, and output gate, which regulate the flow of information into and out of the cell state C_t.

The mathematical formulation of each gate is defined as:

Forget Gate: Determines which information to discard from the cell state:

Input Gate: Controls the update of the cell state with new information:

Cell State Update: Combines the forget gate output and input gate output

Output Gate: Determines the hidden state based on the cell state:
 

where σ is the sigmoid activation function, denotes element-wise multiplication, x_t is the input at time t, h_{t-1} is the hidden state from the previous time step, W and b and are weight matrices and bias vectors, respectively.

Bidirectional processing:

The Bidirectional Long Short-Term Memory Network (BiLSTM) architecture consists of two parallel LSTM networks: a forward LSTM that processes the sequence from past to future (t = 1 to t = T) and a backward LSTM that processes it from future to past (t = T to t = 1). The hidden states from both directions are concatenated at each time step to form the final output, capturing both historical and future contextual information:

In this model, the BiLSTM is configured with 32 hidden units in each direction (64 after concatenation) and 2 layers, employing a dropout rate of 0.2 between layers to prevent overfitting. The network processes sequences of 19-time steps derived from the 20-day lookback window. The BiLSTM is configured with 32 hidden units in each direction (64 after concatenation) and 2 layers, trained using the Adam optimizer with a learning rate of 0.01. This allows the BiLSTM to learn temporal patterns in both returns and volatility, with the bidirectional design enabling it to capture asymmetric dependencies. The output of the BiLSTM, , is a high-dimensional representation of bidirectional temporal features, which is fed into the KAN layer for further refinement. Figure 2 shows the schematic diagram of the hidden state fusion process in BiLSTM.

KAN for nonlinear refinement

Despite the strength of BiLSTM in modeling temporal dynamics, financial time series often exhibit highly nonlinear relationships that are challenging to capture with standard neural network architectures. To address this, a Kolmogorov-Arnold Network (KAN)⁴³, a novel neural network paradigm that leverages the Kolmogorov-Arnold representation theorem to model complex nonlinear functions through a combination of univariate functions and linear operations is incorporated.

Mathematical foundation

KANs differ from traditional feedforward neural networks by replacing the linear transformations followed by activation functions in each layer with a set of univariate basis functions applied to individual input dimensions, followed by a linear combination. The Kolmogorov-Arnold representation theorem states that any multivariate continuous function can be represented as a composition of univariate functions:

Implementation specification:

In this specific implementation, a KAN layer transforms an input vector  = [z₁,z₂,...,z_d ] into an output vector y = [y₁,y₂,...,y_m ]. The transformation for each output neuron k is defined by:

where Φ_{k,i}: R  R are the learnable univariate basis functions. This study implements these functions Φ(x) as cubic B-splines for their smoothness and expressive power, as defined in

where B_j; are the third-order B-spline basis functions, c_j are the trainable spline coefficients, and G is the number of basis functions, determined by the grid size and spline order.

Within the forecasting framework, the KAN layer receives the final hidden state from the preceding Bidirectional LSTM layer as its input. The configured KAN has an architecture of [64, 1], meaning it accepts the 64-dimensional BiLSTM output and produces a single scalar value as the final forecast. The spline functions Φ_k,i are defined using a grid of 5 equidistant knots. To mitigate overfitting, an L1 regularization term with a coefficient of λ = 0.001 is applied to the spline coefficients during training. Furthermore, the spline grid is updated every 100 optimization steps to adaptively refine the function approximations based on the evolving loss landscape.

In this framework, the KAN receives the output of the BiLSTM layer, , as input. The KAN has a width structure of [64,1] and uses cubic B-spline basis functions with 5 knots for Φ_k,i, chosen for their flexibility in approximating smooth nonlinear functions and interpretability⁴³. The output of the KAN is denoted by , represents the refined prediction after accounting for both temporal dependencies and nonlinear patterns. Figure 3 shows the schematic diagram of the KAN (Kolmogorov-Arnold Network) model structure.

Hybrid model integration

The GARCH-BiLSTM-KAN hybrid model is integrated in a sequential manner, where each component's output serves as input to the next, culminating in a final prediction. Table 1 summarizes all critical parameters for model replication. The parameters include GARCH order, BiLSTM architecture details, KAN configuration, and training hyperparameters. All parameter values reported in the text correspond exactly to those implemented in the code.

All model parameters are consistent across mathematical formulation, textual description, and code implementation: This ensures complete reproducibility and alignment between theoretical formulation and practical implementation. Table 1 summarizes the key parameters of each component.

GARCH preprocessing: The raw return series r_t is first input to the GARCH(1,1) model to estimate the conditional volatility . This step transforms the univariate return series into a bivariate sequence , enriching the input with volatility information.

BiLSTM feature extraction: The bivariate sequence is fed into the BiLSTM network, which processes it in both forward and backward directions to generate bidirectional temporal features . The network uses a lookback window of 20 to construct input sequences, with 32 hidden units per direction and 2 layers, trained via the Adam optimizer.

KAN refinement: The BiLSTM features  are input to the KAN, which applies cubic B-spline basis functions (with 5 knots) to each feature dimension, followed by a linear combination to produce the final prediction y_t. The KAN has a width structure of [64,1] and uses the same Adam optimizer.

The sequential integration is theoretically motivated by the 'hierarchical feature refinement' principle: GARCH first decomposes raw returns into predictable volatility patterns, reducing noise for subsequent layers. BiLSTM then encodes bidirectional dependencies-forward LSTMs capture supply-driven trends, while backward LSTMs model demand expectations. Finally, KAN refines these temporal features using cubic B-splines, which outperform ReLU in approximating non-smooth functions. This pipeline ensures that volatility, temporal asymmetry, and nonlinearity are modeled as interdependent rather than isolated phenomena.

Training and loss function: The entire model is trained end-to-end for 100 epochs using a mean squared error (MSE) loss function, defined as:

where y_t  is the true value and is the model's prediction. Three evaluation metrics are used to assess performance:

Root Mean squared Error (RMSE):

Mean Absolute Error (MAE):

Coefficient of Determination (R²):

The model is trained on 80% of the data (7877 samples) and tested on the remaining 20% (1969 samples), with price data normalized using MinMaxScaler (feature range = (-1, 1)) to facilitate training.

The integration of GARCH, BiLSTM, and KAN is motivated by their complementary strengths: GARCH provides a statistically grounded measure of volatility, BiLSTM captures bidirectional temporal dependencies in both returns and volatility, and KAN refines these features by modeling residual nonlinearities. This hierarchical approach ensures that the model leverages both parametric (GARCH) and nonparametric (BiLSTM, KAN) techniques, making it robust to the diverse characteristics of financial time series.

Experimental procedure overview

The experimental procedure was meticulously designed as a systematic pipeline to ensure the reproducibility and robustness of the findings. The process commenced with data acquisition, where daily West Texas Intermediate (WTI) crude oil price data spanning from 1986 to 2025 was sourced from the U.S. Energy Information Administration (EIA) database. Subsequently, the raw price data underwent a comprehensive preprocessing stage. To achieve stationarity, the raw prices were transformed into logarithmic returns. These return series were then normalized to the range of [-1, 1] using Min-Max scaling to facilitate stable and efficient model training.

Following preprocessing, the dataset was split into training and testing sets following a temporal order, with no shuffling, to preserve the chronological structure of the time series. To prevent any data leakage, all volatility modeling was strictly confined to the training data.

Specifically, for the initial model training and evaluation with the static 80-20 split, the GARCH(1,1) model was fitted exclusively on the training set (data from 1986 to 2016). The estimated parameters from this training period were then used to generate the conditional volatility series for both the training and test sets. This ensures that the volatility inputs for the BiLSTM and KAN components during testing are based solely on information available up to the time of prediction, without incorporating any future data from the test set.

Furthermore, for the rolling window validation introduced in the results section, this principle was rigorously upheld. For each rolling window, the GARCH(1,1) model was re-estimated from scratch using only the data within that specific training window. The resulting volatility estimates were then used as inputs for training the subsequent BiLSTM-KAN network and for forecasting the corresponding test window. This recursive re-estimation mimics a real-world forecasting scenario and guarantees that no future information is leaked at any step of the evaluation process.

For model input, a feature engineering step was conducted. This involved aligning the derived volatility series with the price returns to form bivariate input sequences, which were structured using a 20-day lookback window to capture temporal dependencies. During the model training phase, all neural network architectures were trained for 100 epochs. The optimization was performed using the Adam optimizer with a learning rate of 0.01, and the Mean Squared Error (MSE) served as the loss function to guide the learning process.

The evaluation of the models was carried out on a held-out test set, comprising 20% of the total data. Performance was quantified using a suite of metrics, including Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and the coefficient of determination (R²). To further substantiate the findings, a series of robustness tests were implemented, including ablation studies to assess component importance, rolling window validation to test temporal.

CODE AND DATA AVAILABILITY

The daily WTI crude oil price data used in this study are publicly available from the U.S. Energy Information Administration (EIA) database (https://www.eia.gov/dnav/pet/hist/RWTCd.htm) and have been validated using historical records from the Federal Reserve Economic Data (FRED) database (https://fred.stlouisfed.org/series/DCOILWTICO). All raw data, processed datasets, and the complete analysis code that support the findings of this study have been deposited in the Zenodo repository and are publicly available at DOI: 10.5281/zenodo.17614060. The provided source code encompasses the full implementation of the proposed GARCH-BiLSTM-KAN hybrid model, all benchmark models used for comparison, and the scripts necessary to replicate the entire experimental pipeline, including data preprocessing, model training, evaluation, and the generation of all figures and tables presented in this paper. To guarantee the reproducibility of the results, the repository includes detailed documentation for environment setup. Furthermore, this study employed a fixed random seed (42) across all experiments and utilized a consistent set of hyperparameters for model training. All neural networks were optimized using the Adam optimizer with a learning rate of 0.01 and a batch size of 32, trained for 100 epochs. This rigorous standardization ensures that the reported outcomes can be reliably replicated.

Results

Data Description:

The analysis in this study relies on daily price data of West Texas Intermediate (WTI) crude oil, a benchmark for global oil markets due to its liquidity and widespread use in pricing agreements. The dataset spans a 39-year period from January 2, 1986, to March 10, 2025, encompassing 9,866 daily observations. This time frame is carefully chosen to capture diverse market conditions, including periods of economic expansion, recession, geopolitical crises, and e...

Discussion

The empirical findings presented above underscore the superior performance of the GARCH-BiLSTM-KAN hybrid model in forecasting WTI crude oil prices, outperforming both traditional volatility models and alternative hybrid architectures across key metrics. This section discusses the theoretical and practical implications of these results, contextualizes them within existing literature, and identifies potential avenues for further inquiry.

The outperformance of the GARCH-BiLSTM-KAN model stems fr...

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
Computing Workstation	Standard research computing environment	Configuration: NVIDIA GPU, 32GB RAM, multi-core CPU	Used for all model training, validation, and experimental procedures.
Daily Brent Crude Oil Spot Price	U.S. Energy Information Administration (EIA)	Zenodo Repository: DOI 10.5281/zenodo.17614060	Served as an independent dataset for validating model generalizability.
Daily West Texas Intermediate (WTI) Crude Oil Spot Price	U.S. Energy Information Administration (EIA)	Zenodo Repository: DOI 10.5281/zenodo.17614060	Used as the primary dataset for model development and evaluation.
GARCH-BiLSTM-KAN Model Implementation	This study	Zenodo Repository: DOI 10.5281/zenodo.17614060	Complete source code for the proposed hybrid model and all benchmark models.
Key Python Libraries (NumPy, Pandas, Scikit-learn, Matplotlib)	Open-source community	Versions as specified in repository requirements.txt	Used for data processing, statistical analysis, and visualization.
Python Programming Language	Python Software Foundation	Version 3.9; https://www.python.org/	Main programming language for implementing all models and analyses.
PyTorch Library	PyTorch Foundation	Version 2.0; https://pytorch.org/	Primary deep learning framework for implementing BiLSTM and KAN components.