Time series and forecasting · insight-1229
Choose between a deterministic and a stochastic trend on purpose, because the choice sets interval width
- Evidence type
- documented fact
- Difficulty
- medium
- Roles
- Data scientist
- Industries
- aviation, energy, telecom
Summary: situation, action, reason, boundary
Situation
Annual passenger totals on a route have grown steadily for a decade, and two fitted models agree on the growth rate: a straight-line trend with ARMA errors, and an ARIMA model with one difference and drift.
Action
Fit both forms, compare point forecasts and interval coverage over rolling origins that include a growth slowdown, and default to the stochastic (differenced) trend for long horizons unless a physical or contractual reason fixes the slope.
Reason
A deterministic trend assumes the slope never changes, so its errors do not accumulate; a stochastic trend lets the slope drift and its intervals widen much faster, which reflects the extra uncertainty.
Boundary
Multi-year forecasts of volume, capacity or demand with a persistent upward or downward drift.
Symptom or risk
The two models give similar point forecasts but very different intervals, so picking the narrow one by default promises precision that the history does not support at long horizons.
Exceptions
At short horizons, or where a structural limit pins the growth rate, the two forms differ little.
Trade-offs
Wide intervals are honest but harder to plan against; narrow ones are easier to use and easier to be wrong with.
Pitfalls
Comparing the two models on point accuracy alone, which hides the entire difference between them.
Lessons and concepts
Claims and their sources
FPP3 contrasts a deterministic trend (a straight line plus stationary ARMA errors) with a stochastic trend (an ARIMA model with one difference); the stochastic version gives much wider prediction intervals even when the estimated growth is about the same, and the book recommends stochastic trends for longer horizons.