Order of integration

Order of integration, denoted I(d), is a summary statistic for a time series. It reports the minimum number of differences required to obtain a covariance stationary series.

A time series is integrated of order 0 if it admits a moving average representation with

where ${\displaystyle b}$ is the possibly infinite vector of moving average weights (coefficients or parameters). This implies that the autocovariance is decaying to 0 sufficiently quickly. This is a necessary, but not sufficient condition for a stationary process. Therefore, all stationary processes are I(0), but not all I(0) processes are stationary.

A time series is integrated of order d if

is a stationary process, where ${\displaystyle L}$ is the lag operator and ${\displaystyle 1-L}$ is the first difference, i.e.

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