Differencing (of Time Series):
Differencing of a time series in discrete time
is the transformation of the series
to a new time series
where the values
are the differences between consecutive values of
. This procedure may be applied consecutively more than once, giving rise to the "first differences", "second differences", etc.
The first differences of a time series
are described by the following expression:
the second differences
according to the expression
The general expression for the differences of order is given by the recursive formula
where the top index means the order of the difference.
The reason to use differences
itself is that the differences of a broad class of nonstationary time series are stationary time series . Thus, the differencing procedure makes it possible to apply analytical tools and theoretical results developed for stationary time series to nonstationary time series.
The differencing procedure, combined with the ARMA model for stationary time series, gives rise to the ARIMA model for nonstationary time series.