MPE

170. MPE

170.1. MPE (Mean Percent Error)

Signed-percentage version of MD. Captures relative bias — whether the forecast over- or under-shoots in percentage terms.

MPE = \frac{1}{𝑛} \sum_{𝑡 = 1}^{𝑛} \frac{𝑒_{𝑡}}{𝑥_{𝑡}} = \frac{1}{𝑛} \sum_{𝑡 = 1}^{𝑛} \frac{𝑥_{𝑡} - {\hat{𝑥}}_{𝑡}}{𝑥_{𝑡}}

$MPE \approx 0$ : forecast is unbiased on a percentage scale
$MPE > 0$ : under-prediction (actuals exceed forecast)
$MPE < 0$ : over-prediction

Caveats.

Positive and negative errors cancel — same warning as MD. Pair with MAPE to separate bias from accuracy.
Undefined / unstable when any $𝑥_{𝑡} = 0$ or $𝑥_{𝑡}$ is small relative to the error.