Beyond RMSE and Granger Gain: Multi-Horizon Forecast-Error Entropy for Nonlinear Predictive Coupling

Abstract

We investigate whether the past of an auxiliary time series provides predictive information about a target beyond that contained in the target's own history. We compare matched out-of-sample target-only and target-plus-source forecasts and introduce the forecast-error distribution entropy (FEDE) predictive-causality framework, where predictive causality is understood in the Granger sense: auxiliary history is informative when it improves prediction beyond the target's own past. FEDE quantifies how the full signed forecast-error distribution changes across prediction horizons. Controlled nonlinear benchmarks hide source-target dependence behind nonlinear transformations not supplied to the predictors. LightGBM recovers these hidden channels, whereas Ridge does not. Across four equal-power nonlinearities and four noise levels, the mean response-sensitivity statistic for FEDE remains 29.00, 27.47, 23.22, and 28.67, compared with 14.09, 10.68, 12.67, and 12.71 for mean log-RMSE gain. Directional and source controls separate the true source-to-target channel from the reverse direction and from independent periodic, quasiperiodic, chaotic, and autoregressive auxiliaries. Chronological Granger-style comparisons show strong model-form dependence: a degree-2 polynomial model is exceptionally sensitive to the matched quadratic channel but becomes weak or oppositely aligned for other nonlinearities, while basic convergent cross mapping is weak or directionally inconsistent in this stochastic, partly non-invertible setting. FEDE remains positively aligned across all 108 tested combinations of window length, history length, and maximum forecast horizon. Together, these results establish FEDE as a sensitive and robust horizon-resolved tool that extracts predictive information beyond conventional error-magnitude measures when the nonlinear interaction form is unknown. Code available in Zenodo.