Beyond One-Step Ahead Predictions#

Stage 6 · 🏗️ Building & Forecasting Models · Lesson 17 of 18 · advanced

◀ Previous · SARIMA Models: Seasonal ARIMA · Next · Exponential Smoothing Models ▶ · ↑ Section

Important

✨ AI-generated content. This page was written with the assistance of an AI language model and is provided as a learning aid. Despite careful review, it may still contain mistakes, omissions, or out-of-date information. Whether you are new to the topic, a team lead, or a senior practitioner, treat it as a starting point rather than an authoritative reference: read it critically and independently verify anything you act on (code, commands, figures, and factual claims) against official documentation and primary sources before relying on it.

Forecasting further#

A one-step forecast is just the start. Usually you want the value \(h\) steps ahead — the h-step forecast — which is the conditional expectation \(\hat{x}_{n+h} = \mathbb{E}[x_{n+h} \mid x_1, \dots, x_n]\): the mean of the future given everything observed so far.

Building it up#

The practical route is recursive (“plug-in”): apply the one-step predictor, then feed its output back in as if it were observed, and step forward again. Each future value the model needs but has not seen is replaced by its own forecast, so errors from early steps propagate into later ones.

Uncertainty grows#

The crucial consequence: forecast uncertainty widens with the horizon. Every step adds a fresh shock whose variance accumulates, so the prediction intervals fan out the further you look — a forecast for next month is far tighter than one for next year. Reporting the interval, not just the point, is essential. (get_forecast(steps=h) in statsmodels returns both.)

Two limiting cases#

How the intervals grow depends on the model. For a stationary ARMA, forecasts revert to the mean and the interval width levels off at the unconditional variance — the process forgets its current state. For an ARIMA with differencing, there is no mean to revert to, so the intervals keep widening without bound, random-walk style. Either way, small parameter errors compound over long horizons.

Hint