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 stationary series.

Integration of order zero [link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{k=0}^\infty \mid{\gamma_k}^2\mid < \infty

which means that the autocovariance is decaying to 0 sufficiently fast. 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.^{[citation needed]}

[edit] Integration of order d

A time series is integrated of order d if

Failed to parse (Missing texvc executable; please see math/README to configure.): (1-L)^d X_t \

is integrated of order 0, where Failed to parse (Missing texvc executable; please see math/README to configure.): L

is the lag operator and Failed to parse (Missing texvc executable; please see math/README to configure.): (1-L) 
is the first difference, ie: Failed to parse (Missing texvc executable; please see math/README to configure.): (1-L) X_t = X_t - X_{t-1} = \Delta X 

.

In other words, a process is integrated to order d if taking repeated differences d times yields a stationary process.

Constructing an integrated series [link]

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

• Suppose Failed to parse (Missing texvc executable; please see math/README to configure.): X_t

is I(d-1)

• Now construct a series Failed to parse (Missing texvc executable; please see math/README to configure.): Z_t = \sum_{k=0}^t X_k

• Show that Z is I(d) by observing its first-differences are I(d-1):

Failed to parse (Missing texvc executable; please see math/README to configure.): \triangle Z_t = \sum_{k=0}^t X_k - \sum_{k=0}^{t-1} X_k = X_t \sim I(d-1)

See also [link]

• ARIMA
• ARMA
• Random walk

This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (December 2009)

References [link]

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