The Knowable Future: Mapping the Decay of Past-Future Mutual Information Across Forecast Horizons

The ability to assess ex-ante whether a time series is likely to be accurately forecast is important for forecasting practice because it informs the degree of modelling effort warranted. We define forecastability as a property of a time series (given a declared information set), and measure horizon-specific forecastability as the reduction in uncertainty provided by the past, using auto-mutual information (AMI) at lag h. AMI is estimated from training data using a k-nearest-neighbour estimator and evaluated against out-of-sample forecast error (sMAPE) on a filtered, balanced sample of 1,350 M4 series across six sampling frequencies. Seasonal Naive, ETS, and N-BEATS are used as probes of out-of-sample forecast performance. Training-only AMI provides a frequency-conditional diagnostic for forecast difficulty: for Hourly, Weekly, Quarterly, and Yearly series, AMI exhibits consistently negative rank correlation with sMAPE across probes. Under N-BEATS, the correlation is strongest for Hourly (p= -0.52) and Weekly (p= -0.51), with Quarterly (p= -0.42) and Yearly (p = -0.36) also substantial. Monthly is probe-dependent (Seasonal Naive p= -0.12; ETS p = -0.26; N-BEATS p = -0.24). Daily shows notably weaker AMI-sMAPE correlation under this protocol, suggesting limited ability to discriminate between series despite the presence of temporal dependence. The findings support within-frequency triage and effort allocation based on measurable signal content prior to forecasting, rather than between-frequency comparisons of difficulty.