Order of integration


In statistics, the order of integration, denoted I, of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series.

Integration of order zero

A time series is integrated of order 0 if it admits a moving average representation with
where is the possibly infinite vector of moving average weights. 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, but not all I processes are stationary.

Integration of order ''d''

A time series is integrated of order d if
is a stationary process, where is the lag operator and is the first difference, i.e.
In other words, a process is integrated to order d if taking repeated differences d times yields a stationary process.

Constructing an integrated series

An I process can be constructed by summing an I process: