The evolution of a time series is often influenced not only by its past values but also by external covariates, giving rise to forecasting with exogenous variables. Nonlinear and delayed effects between covariates and the target series make this task particularly challenging. We propose a forecasting framework based on Ridge regression with sliding-window signature features, combining the computational efficiency of (semi-)parametric models with the flexibility of nonparametric approaches. These features capture complex dependencies between the target and the exogenous variables without relying on prior assumptions about the form of these relationships or the influence of past observations. We establish theoretical guarantees in the discrete-time setting, including universality of approximation and stationarity for the sliding-window transform. To ease implementation, we introduce a novel sequential algorithm that exploits window overlap for efficient signature updates. The method is evaluated on both synthetic and real electricity demand data. Results show that signature features effectively encode temporal and nonlinear dependencies, yielding accurate forecasts competitive with those based on expert knowledge and recent neural network approaches.

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