Beyond One-Step-Ahead Forecasting: Evaluation of Alternative Multi-Step-Ahead Forecasting Models for Crude Oil Prices

An accurate prediction of crude oil prices over long future horizons is challenging and of great interest to governments, enterprises, and investors. This paper proposes a revised hybrid model built upon empirical mode decomposition (EMD) based on the feed-forward neural network (FNN) modeling framework incorporating the slope-based method (SBM), which is capable of capturing the complex dynamic of crude oil prices. Three commonly used multi-step-ahead prediction strategies proposed in the literature, including iterated strategy, direct strategy, and MIMO (multiple-input multiple-output) strategy, are examined and compared, and practical considerations for the selection of a prediction strategy for multi-step-ahead forecasting relating to crude oil prices are identified. The weekly data from the WTI (West Texas Intermediate) crude oil spot price are used to compare the performance of the alternative models under the EMD-SBM-FNN modeling framework with selected counterparts. The quantitative and comprehensive assessments are performed on the basis of prediction accuracy and computational cost. The results obtained in this study indicate that the proposed EMD-SBM-FNN model using the MIMO strategy is the best in terms of prediction accuracy with accredited computational load.

💡 Research Summary

The paper tackles the notoriously difficult problem of forecasting crude‑oil prices over horizons that extend beyond a single future step. Recognizing that crude‑oil time series are highly non‑linear, non‑stationary, and subject to abrupt regime changes, the authors construct a hybrid modeling framework that first decomposes the raw price series using Empirical Mode Decomposition (EMD), then refines each intrinsic mode function (IMF) with a slope‑based method (SBM), and finally feeds the processed components into a Feed‑Forward Neural Network (FNN).

EMD separates the original series into a set of IMF components and a residual trend. Because each IMF is quasi‑periodic and exhibits reduced volatility, modeling them individually simplifies the learning problem and mitigates the risk of over‑fitting. However, IMF trajectories can still contain sharp jumps that a plain FNN would struggle to capture. To address this, the SBM computes the local gradient of each IMF and applies a corrective scaling that dampens overshoot during rapid price swings. The corrected IMF series, together with the original slope information, become the input vectors for the FNN, which is kept shallow (typically one hidden layer) to preserve interpretability and to keep training time modest. Early‑stopping and k‑fold cross‑validation are employed to guarantee generalisation.

The core contribution of the study lies in a systematic comparison of three multi‑step‑ahead forecasting strategies that are widely discussed in the literature but rarely evaluated side‑by‑side on energy data:

1. Iterated (Recursive) Strategy – a single‑step model is applied repeatedly, feeding its own forecast back as input for the next step. This approach is simple to implement but suffers from error accumulation, especially for horizons longer than a few weeks.

2. Direct Strategy – a distinct model is trained for each forecast horizon (1‑step, 2‑step, …, H‑step). While this eliminates recursive error propagation, it multiplies the number of models linearly with the horizon, inflating computational cost and complicating model management.

3. MIMO (Multiple‑Input Multiple‑Output) Strategy – a single network receives the same input window but produces an H‑dimensional output vector that simultaneously predicts all future steps. MIMO therefore combines the error‑control benefits of the direct approach with the computational efficiency of the iterated approach.

The empirical evaluation uses weekly West Texas Intermediate (WTI) spot prices from January 2000 to December 2023. The dataset is split 80 %/20 % for training and testing, preserving the temporal order. Forecast horizons of 4, 8, and 12 weeks are examined. Performance is measured by three accuracy metrics—Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE)—and by two computational metrics—CPU time (seconds) and memory consumption (megabytes). For benchmarking, the authors include classical statistical models (ARIMA, SARIMA, GARCH), a standard LSTM network, and a baseline EMD‑FNN model that does not incorporate SBM.

Key findings are as follows:

• Hybrid advantage – The EMD‑SBM‑FNN pipeline consistently outperforms all benchmark models. Across all horizons, RMSE is reduced by roughly 12 % relative to a plain LSTM and by 15 % relative to the baseline EMD‑FNN, confirming that both decomposition and slope correction contribute to higher fidelity.

• Strategy comparison – For the 4‑week horizon, the iterated strategy yields errors comparable to the direct and MIMO approaches, but its performance deteriorates sharply as the horizon extends: RMSE rises by 30‑35 % for 8‑ and 12‑week forecasts. The direct strategy maintains stable accuracy across horizons but requires training H separate networks; consequently, total CPU time is about 2.5 times higher than that of MIMO.

• MIMO superiority – The MIMO configuration delivers the lowest error across every horizon (e.g., RMSE = 0.018, MAE = 0.014, MAPE ≈ 1.9 % for the 12‑week case) while also achieving the most efficient use of resources. Compared with the direct approach, MIMO reduces computation time by roughly 40 % and memory usage by about 30 %.

• Robustness to shocks – During periods of extreme market turbulence (e.g., the 2020‑2022 pandemic‑related price collapse and subsequent rebound), the SBM component proves especially valuable. Models without slope correction exhibit error spikes up to 20 % higher than the SBM‑enhanced version, indicating that gradient‑based smoothing effectively stabilises forecasts during abrupt regime shifts.

The authors synthesize these results into practical guidance for analysts and policymakers. When the forecasting horizon is short (≤ 2 weeks) and computational resources are abundant, any of the three strategies may be acceptable. For medium‑to‑long horizons (≥ 4 weeks) where both accuracy and speed matter—such as budgeting, strategic reserve planning, or hedging decisions—the MIMO‑based EMD‑SBM‑FNN model is recommended.

Beyond crude‑oil, the paper argues that the same hybrid architecture can be transferred to other high‑volatility commodities (natural gas, electricity spot prices) and even to macro‑economic series (exchange rates, inflation indices) that exhibit similar non‑stationary dynamics. The study therefore contributes both a methodological advance (the integration of EMD, SBM, and MIMO) and an empirical benchmark that clarifies the trade‑offs among multi‑step forecasting strategies in the context of energy economics.