Time series and forecasting · insight-1216
Calibrate intervals separately at each horizon, since one-step residual spread is too narrow far ahead
- Evidence type
- practitioner heuristic
- Difficulty
- medium
- Roles
- Data scientist, ML engineer
- Industries
- cross-industry
Summary: situation, action, reason, boundary
Situation
A residual-based 80% interval built from one-step errors is reused for horizons 1 to 28.
Action
Estimate interval widths from h-step backtest errors (or a bootstrap of sample paths), report coverage per horizon, and widen or recalibrate the horizons that fall short.
Reason
Multi-step prediction intervals widen with the horizon; even for benchmark methods the standard deviation scales with a function of h.
Boundary
Residual-based and model-based intervals on multi-step forecasts.
Symptom or risk
Forecast error grows with the horizon, so far-ahead intervals are too narrow and the observed coverage falls well below 80% exactly where planning decisions are made.
Exceptions
Methods that already estimate horizon-specific spread (direct strategies trained per horizon with quantile loss).
Trade-offs
Per-horizon calibration needs enough origins at each horizon.
Pitfalls
Assuming the horizon effect is only a scale factor when error is also seasonal or heavy-tailed.
Example
Hypothetical random walk: using the one-step sd at h = 16 (true sd 4 times larger) gives about 25% actual coverage for a nominal 80% interval.
Lessons and concepts
Claims and their sources
Prediction intervals usually widen with the horizon, and the benchmark methods' multi-step forecast standard deviations (naive: square root of h times the residual standard deviation) grow with h.