Differences

Differences
command computes the differenced series for selected time series. Differencing
can help stabilize the mean of a time series by removing changes in the level,
and so eliminating trend and seasonality (Hyndman and Athanasopoulos, 2014). A
differenced series has *k* fewer values than the original series, where *k*
is the *differencing order*.

# How To

Run: Statistics→Time Series → Differences...

Select one or multiple variables with time series.

If the Remove Mean option is checked** the sample mean
is first subtracted from the series before the differencing.**

Optionally, change the lag for differencing (Differencing lag option) or use the Differences of order option to apply differencing more than one time.

For example, to
get the second-order differences set the Repeat
Differencing Operation ** (differencing order)** option to two
(2) and the Differencing lag option to one (1).

# Results

The differenced series of order *n *is computed for
each input time series.

The difference operator is defined as , where is the lag
operator defined as . Then the
series of first differences can be written as follow: ; and the differences
of order *n, *produced by the command, can be written as follow:

# References

Hyndman, R. J., Athanasopoulos, G. (2014). Forecasting: principles and practice, OTexts: Melbourne, Australia.

Enders, W. (2004). "Stationary Time-Series Models". Applied Econometric Time Series (Second ed.). New York: Wiley.