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

Andrei Velichko

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…