208. Naive

208.1. Naïve Forecast

The simplest possible forecast: next value equals last value.

For a -step-ahead forecast: — flat extrapolation of the last observation.

208.1.1. When it’s the right choice

The naïve forecast is optimal for a random walk — a series where each step is the previous value plus mean-zero noise:

Examples of approximate random walks:

For these, no model can beat naïve because there’s no exploitable structure. Naïve is the textbook baseline: any forecaster should outperform it on series with structure, and not outperforming it is a flag that the data is essentially a random walk.

208.1.2. Variants

Variant Formula
Naïve
Seasonal naïve — repeat the value from one period ago, where is the seasonal period
Drift — naïve + average historical drift

Seasonal naïve is the surprisingly-good baseline for highly seasonal series (retail, weather, energy demand). Drift is the equivalent baseline for series with a clear trend.

208.1.3. Naïve as a benchmark

Always evaluate forecasts relative to naïve. Define the MASE (mean absolute scaled error):

: the model beats naïve. : it doesn’t — try a different model or accept that the series is a random walk.

The same logic applies to seasonal series with seasonal naïve as the denominator.

Example: Naïve forecast and the random-walk test

Given (last 5 days of a series):

1 2 3 4 5
100 102 101 104 103

Step 1 — naïve 1-step forecast at

Step 2 — naïve -step forecasts

Flat at 103 forever.

Step 3 — drift variant

Average historical step: .

Step 4 — interpret

Since the series wandered up and down between 100 and 104 with no clear trend, naïve’s flat 103 is probably about as good as you can do. If a forecaster predicts something fancy and gets MASE > 1, the series is essentially noise — switch to naïve, save the effort.