Fractional Long-Memory Dynamics and Residual Machine Learning for Medium-Horizon Agricultural Commodity Price Forecasting

This paper develops a hybrid fractional-order framework for medium-horizon forecasting of wheat, corn, and soybean prices, combining a Caputo fractional differential equation with exogenous macro-climatic drivers (weather, crude oil, exchange rate, inflation) and a machine learning residual-correction layer. Existence, uniqueness, and Ulam–Hyers stability are established for both Caputo and Atangana–Baleanu formulations via Banach fixed-point theory, with numerical illustration through a fractional Adams–Bashforth–Moulton predictor–corrector scheme. Formal unit-root tests (ADF, KPSS, Phillips–Perron) confirm I(1) behaviour in log-price levels and stationarity in first differences. Residuals are corrected using XGBoost and two feedforward neural networks (MLP-A, MLP-B), producing a family of hybrid forecasting models. Using 25 years of monthly FRED data (January 2000–December 2024), multi-method long-memory diagnostics (Hurst exponent, Lo’s modified R/S test, DFA, local Whittle estimation) confirm near unit-root fractional integration in log-price levels (H∈[0.985,1.044]), with estimated fractional orders α^∈{0.737,0.884,0.451} for wheat, corn, and soybean. Under a strict rolling-origin protocol (17 origins, horizons h∈{1,5,6,12} months), conventional benchmarks remain competitive at h=1, but their MAPE degrades to 16.6–31.9% at h=12, while Frac+XGBoost error stays flat at 3.9–14.5%. This horizon-robust advantage holds across three market regimes (COVID-19 pandemic shock, 2021–2022 super-cycle, 2023–2024 normalisation) and is confirmed by Holm–Bonferroni-corrected Diebold–Mariano tests (p<0.001 at h=12). The model’s structural advantage emerges at h≥5 months, supporting procurement planning, food-security buffer stocks, and import budgeting.