In statistics, the mean signed difference, deviation, or error (MSD) is a sample statistic that summarises how well a set of estimates
θ
^
i
match the quantities
θ
i
that they are supposed to estimate. It is one of a number of statistics that can be used to assess an estimation procedure, and it would often be used in conjunction with a sample version of the mean square error.
For example, suppose a linear regression model has been estimated over a sample of data, and is then used to extrapolate predictions of the dependent variable out of sample after the out-of-sample data points have become available. Then
θ
i
would be the i-th out-of-sample value of the dependent variable, and
θ
^
i
would be its predicted value. The mean signed deviation is the average value of
θ
^
i
−
θ
i
.
The mean signed difference is derived from a set of n pairs,
(
θ
^
i
,
θ
i
)
, where
θ
^
i
is an estimate of the parameter
θ
in a case where it is known that
θ
=
θ
i
. In many applications, all the quantities
θ
i
will share a common value. When applied to forecasting in a time series analysis context, a forecasting procedure might be evaluated using the mean signed difference, with
θ
^
i
being the predicted value of a series at a given lead time and
θ
i
being the value of the series eventually observed for that time-point. The mean signed difference is defined to be
MSD
(
θ
^
)
=
∑
i
=
1
n
θ
i
^
−
θ
i
n
.