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

${\displaystyle \sum _{k=0}^{\infty }\mid {b_{k}}\mid ^{2}<\infty ,}$

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}\$

is a stationary process, 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.