Forecasting Model Accuracy Measures

Formula

Let

• $n$: number of observations
• $t$: time index
• $X_t$: actual value at time $t$
• ${\hat{X}}_t$: forecasted value at time $t$
• $X_t-{\hat{X}}_t$: forecast error at time $t$

Measure	Formula	Comment
MAD/MAE (Mean Absolute Deviation/Error)	$\frac{1}{n}{\sum }_{t=1}^n|X_t-{\hat{X}}_t|$	Mean absolute deviation of forecast errors
MSD/MSE (Mean Squared Deviation/Error)	$\frac{1}{n}{\sum }_{t=1}^n(X_t-{\hat{X}}_t{)}^2$	Penalizes large errors more
RMSE (Root Mean Squared Error)	$\sqrt{\frac{1}{n}{\sum }_{t=1}^n(X_t-{\hat{X}}_t{)}^2}$	Same units as $X_t$, interpretable form of MSD
MAPE (Mean Absolute Percentage Error)	$\frac{1}{n}{\sum }_{t=1}^n\left|\frac{X_t-{\hat{X}}_t}{X_t}\right|\times 100\%$	Scale-independent
MPE (Mean Percentage Error)	$\frac{1}{n}{\sum }_{t=1}^n\left(\frac{X_t-{\hat{X}}_t}{X_t}\right)\times 100\%$	Measures forecast bias