A Regime-Gated Hybrid Forecasting Framework: Coupling Autoregressive Distributed Lags, Smooth-Transition Dynamics, and Kolmogorov–Arnold Residual Learning

We propose and evaluate ARDL–STR–KAN, a three-stage hybrid forecasting framework that decomposes a financial time series into a linear conditional-mean component, a regime-dependent transition mechanism, and a nonlinear residual correction learned by a Kolmogorov–Arnold Network (KAN). The linear backbone is an Autoregressive Distributed Lag (ARDL) model estimated by conditional maximum likelihood; a logistic smooth-transition (STR) gate, with its transition variable selected by an exact Teräsvirta-type F-test, modulates the contribution of the nonlinear stage; and a KAN with learnable B-spline activations approximates the structured nonlinearity remaining in the ARDL residuals. The framework is demonstrated on 25 years of daily NASDAQ Composite data (N = 6,243, 2000–2024) with the 10-year Treasury yield (DGS10) and the volatility index (VIX) as exogenous drivers. On a chronological 80/20 holdout the hybrid attains an out-of-sample RMSE of 0.01063 against 0.01138 for the linear ARDL baseline, a 6.59% reduction with R² = 0.9973, and a Harvey–Leybourne–Newbold-corrected Diebold–Mariano statistic of 2.798 (p = 0.0052) confirms that the gain is statistically significant rather than an artefact of in-sample overfitting. Beyond equities, we outline the framework's extension to energy and currency markets, indicating its applicability across diverse macro-financial series. We give the full mathematical derivation, the estimation procedure, the statistical-consistency rationale for the additive decomposition, and a computational discussion of the lag order, the transition mechanism, and the KAN architecture.