# MSE - Mean Squared Error

Calculates the mean squared error (MSE) between the forecast and the eventual outcomes.

## Syntax

MSE(X, F)
X
is the eventual outcome time series sample data (a one-dimensional array of cells (e.g. row or column).
F
is the forecast time series data (a one-dimensional array of cells (e.g. row or column).

## Remarks

1. The time series is homogeneous or equally spaced.
2. The time series X and F must be of identical size
3. The time series X or F may include observations with missing values (e.g. #N/A or blank).
4. Observations with missing values in Y or F are excluded from the MSE calculation.
5. The mean squared error (MSE) is defined as follows:

$${\mathrm{MSE}=\frac{1}{N}\times\sum_{t=1}^N \left(y_t- f_t \right )^2 = \frac{\mathrm{SSE}}{N}}$$

Where:
• $y_t$ is the actual outcome value at time t.
• $f_t$ is the forecast value at time t.
• $\mathrm{SSE}$ is the sum of squared errors.
6. The MSE provides a quadratic loss function as it squares and subsequently averages the various errors, which gives considerably more weight to large errors (outliers) than smaller ones
7. MSE is most useful when concerned about large errors than smaller ones.
8. The MSE equation is similar to the statistical measure of variance ($\sigma^2$), which allows us to measure the uncertainty around our most likely forecast - $f_T$. In other words, the MSE can be viewed as the variance of the forecast error.
9. The main drawback of MSE is the scale dependency. If the forecast task includes objects with different sales or magnitudes then the MSE measure cannot be applied.
10. The MSE function is available starting with version 1.65 HAMMOCK.

## Examples

Example 1:

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A B C
Date Data Forecast
2008-01-01 -2.9 -2.95
2008-02-01 -2.83 -2.7
2008-03-01 -0.95 -1.00
2008-04-01 -0.88 -0.68
2008-05-01 1.21 1.50
2008-06-01 -1.67 -1.00
2008-07-01 0.83 0.90
2008-08-01 -0.27 -0.37
2008-09-01 1.36 1.26
2008-10-01 -0.34 -0.54
2008-11-01 0.48 0.58
2008-12-01 -2.83 -2.13
2009-01-01 -0.95 -0.75
2009-02-01 -0.88 -0.89
2009-03-01 1.21 1.25
2009-04-01 -1.67 -1.65
2009-05-01 -2.99 -3.20
2009-06-01 1.24 1.29
2009-07-01 0.64 0.60

Formula Description (Result)
=MSE($B$3:$B$21,$C$3:$C$21) MSE (0.065795)