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.

See also the short course Time Series Forecasting .