Calculates the mean absolute scaled error (MASE) between the forecast and the eventual outcomes.
Syntax
MASE(X, F, M)
 X
 is the eventual outcome time series sample data (a onedimensional array of cells e.g. row or column).
 F
 is the forecast time series data (a onedimensional array of cells e.g. row or column).
 M
 is the seasonal period (for nonseasonal time series, set M=1 (default) or leave it blank).
Remarks
 The time series is homogeneous or equally spaced.
 The time series X and F must be of identical size
 The time series X or F may include observations with missing values (e.g. #N/A or blank).
 Observations with missing values in Y or F are excluded from the MASE calculation.
 The absolute scaled error is defined as follows:
$$ q_t = \left\{\begin{array}{l} \frac{\left  e_t \right }{\frac{1}{n1}\times\sum_{t=2}^n \left  y_i  y_{i1}\right } \\ \frac{\left e_t \right }{\frac{1}{nM}\times\sum_{t=M+1}^n \left  y_i  y_{iM}\right } \end{array} \right. \begin{array}{l} \{y_i\}\,\mathrm{is \, nonseasonal} \\ \{y_i\}\,\mathrm{is \, seasonal} \end{array} $$
Where:
 $\{y_t\}$ is the actual observations time series
 $\{e_t\}$ is the forecast error for a given period.
 The absolute scaled error is generally the same as the absolute error divided (i.e. scaled) by the Mean Absolute error (MAE) of the naïve benchmark model.
$$ q_t=\left\{\begin{array}{l} \frac{\left e_t \right }{\mathrm{MAE^*}} \\ \frac{\left e_t \right }{\mathrm{MAE_M^*}} \end{array} \right. \begin{array}{l} \{y_i\}\,\mathrm{is \, nonseasonal} \\ \{y_i\}\,\mathrm{is \, seasonal} \end{array} $$
Where
 $\mathrm{MAE^*}$ is the mean absolute error of the naïve (aka naive 1) benchmark forecast model calculated insample.
 $\mathrm{MAE_M^*}$ is the insample mean absolute error of the seasonal naïve (aka naive 2) benchmark forecasting model calculated insample.
 The mean absolute scaled error (MASE) is calculated according to:
$${\displaystyle {\mathrm{MASE}=\frac{1}{N}\sum_{i=1}^N q_i=\frac{\mathrm{MAE}}{\mathrm{MAE^*}}}}$$  The MASE measure is symmetrical and resistant to outliers.
 Division by zero occurs only in one trivial case where all the values in the input time series are equal (i.e. constant).
 $\mathrm{MASE} \gt 1$ implies that actual forecast does worse than a naïve benchmark forecasting method calculated insample.
 $\mathrm{MASE} \lt 1\quad$ implies that actual forecast performance better than a naïve method
 We can use the MASE values for comparing different forecasting methods. The lower the MASE value, the lower the relative absolute forecast error, the better the method.
 The MASE function is available starting with version 1.65 HAMMOCK.
Examples
Example 1:


Formula  Description (Result) 

=MASE($B$3:$B$21,$C$3:$C$21,1)  MASE (9.83%) 
Files Examples
Related Links
References
 R.J. Hyndman, A.B. Koehler, "Another look at measures of forecast accuracy", International Journal of Forecasting, 22 (2006), pp. 679688
 James Douglas Hamilton; Time Series Analysis; Princeton University Press; 1st edition(Jan 11, 1994), ISBN: 691042896
 Tsay, Ruey S.; Analysis of Financial Time Series; John Wiley & SONS; 2nd edition(Aug 30, 2005), ISBN: 0471690740
 D. S.G. Pollock; Handbook of Time Series Analysis, Signal Processing, and Dynamics; Academic Press; Har/Cdr edition(Nov 17, 1999), ISBN: 125609906
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