Naive

226. Naive

226.1. Naïve Forecast

The simplest possible forecast: next value equals last value.

{\hat{𝑥}}_{𝑡, 𝑡 + 1} = 𝑥_{𝑡}

For a $ℎ$ -step-ahead forecast: ${\hat{𝑥}}_{𝑡, 𝑡 + ℎ} = 𝑥_{𝑡}$ — flat extrapolation of the last observation.

226.1.1. 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:

𝑥_{𝑡} = 𝑥_{𝑡 - 1} + 𝜀_{𝑡}, 𝜀_{𝑡} \sim iid (0, 𝜎^{2})

Examples of approximate random walks:

Daily stock prices (efficient-market hypothesis)
Daily exchange rates
High-frequency series with no seasonality or trend

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.

226.1.2. Variants

Variant	Formula
Naïve	${\hat{𝑥}}_{𝑡, 𝑡 + ℎ} = 𝑥_{𝑡}$
Seasonal naïve	${\hat{𝑥}}_{𝑡, 𝑡 + ℎ} = 𝑥_{𝑡 + ℎ - 𝑚}$ — repeat the value from one period ago, where $𝑚$ is the seasonal period
Drift	${\hat{𝑥}}_{𝑡, 𝑡 + ℎ} = 𝑥_{𝑡} + ℎ \cdot (𝑥_{𝑡} - 𝑥_{1}) / (𝑡 - 1)$ — 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.

226.1.3. Naïve as a benchmark

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

MASE = \frac{MAE of model}{MAE of naïve}

$MASE < 1$ : the model beats naïve. $MASE \geq 1$ : 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):

$𝑡$	1	2	3	4	5
$𝑥_{𝑡}$	100	102	101	104	103

Step 1 — naïve 1-step forecast at $𝑡 = 5$

{\hat{𝑥}}_{5, 6} = 𝑥_{5} = 103

Step 2 — naïve $ℎ$ -step forecasts

{\hat{𝑥}}_{5, 6} = {\hat{𝑥}}_{5, 7} = {\hat{𝑥}}_{5, 8} = \dots = 103

Flat at 103 forever.

Step 3 — drift variant

Average historical step: $(𝑥_{5} - 𝑥_{1}) / (5 - 1) = (103 - 100) / 4 = 0.75$ .

{\hat{𝑥}}_{5, 6} = 103 + 1 \cdot 0.75 = 103.75

{\hat{𝑥}}_{5, 10} = 103 + 5 \cdot 0.75 = 106.75

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.