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Differencing (of Time Series)

Differencing (of Time Series):

Differencing of a time series Math image in discrete time Math image is the transformation of the series Math image to a new time series Math image where the values Math image are the differences between consecutive values of Math image . This procedure may be applied consecutively more than once, giving rise to the “first differences”, “second differences”, etc.

The first differences Math image of a time series Math image are described by the following expression:

the second differences Math image may be computed from the first differences Math image according to the expression

The general expression for the differences of order Math image is given by the recursive formula

where the top index means the order of the difference.

The reason to use differences Math image instead of the values of the time series Math image 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.

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