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.

Integration of order zero

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

$\sum_{k=0}^\infty \mid{b_k}\mid^2 < \infty,$

where $b$ 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.

Integration of order d

A time series is integrated of order d if

$(1-L)^d X_t \$

converges to Brownian as t tends to infinity, where $L$ is the lag operator and $1-L$ is the first difference, i.e.

$(1-L) X_t = X_t - X_{t-1} = \Delta X.$

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

Constructing an integrated series

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

• Suppose $X_t$ is I(d − 1)
• Now construct a series $Z_t = \sum_{k=0}^t X_k$
• Show that Z is I(d) by observing its first-differences are I(d − 1):

$\triangle Z_t = X_t,$

where

$X_t \sim I(d-1). \,$

See also

• ARIMA
• ARMA
• Random walk

References

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