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 
may be computed from the first 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 
instead of the values of the time series 
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.