Order of integration

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In statistics, the order of integration, denoted I(d), 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[edit]

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 (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.

Integration of order d[edit]

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[edit]

An I(d) process can be constructed by summing an I(d − 1) process:

  • Suppose is I(d − 1)
  • Now construct a series
  • Show that Z is I(d) by observing its first-differences are I(d − 1):

See also[edit]


  • Hamilton, James D. (1994) Time Series Analysis. Princeton University Press. p. 437. ISBN 0-691-04289-6.