Spectral Consistency Regularization for Long-Horizon Probabilistic Forecasting with Temporal Fusion Transformers

Abstract

Long-horizon probabilistic forecasting in financial markets is challenging because errors accumulate, distributions drift, and regimes shift, often yielding unstable trajectories and miscalibrated intervals. Although the Temporal Fusion Transformer is effective for probabilistic forecasting, its step-wise training objective does not explicitly enforce long-term structural consistency. We propose the Spectral Temporal Fusion Transformer, which augments Temporal Fusion Transformer training with a spectral-consistency regularizer applied to the cumulative median forecast path, without changing the model architecture or inference procedure. The regularizer matches low-frequency Fourier magnitude spectra between predicted and observed cumulative paths, promoting stable long-term dynamics while preserving local flexibility. Experiments on six US equity assets from 2010 to 2023 across horizons of 60, 120, and 250 steps and multiple training cutoffs show the largest gains in the most volatile regime, achieving about 38 percent mean absolute error reduction at horizon 120, while remaining comparable at earlier cutoffs. These results suggest that low-frequency spectral regularization is a simple mechanism for stabilizing long-horizon probabilistic forecasts.