Order of integration

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 (i.e., a time series whose mean and autocovariance remain constant over time).

The order of integration is a key concept in time series analysis, particularly when dealing with non-stationary data that exhibits trends or other forms of non-stationarity.

A time series is integrated of order d if

${\displaystyle (1-L)^{d}X_{t}\ }$

is a stationary process, where ${\displaystyle L}$ is the lag operator and ${\displaystyle 1-L}$ is the first difference, i.e.

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

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

In particular, if a series is integrated of order 0, then ${\displaystyle (1-L)^{0}X_{t}=X_{t}}$ is stationary.

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

• Suppose ${\displaystyle X_{t}}$ is I(d − 1)
• Now construct a series ${\displaystyle Z_{t}=\sum _{k=0}^{t}X_{k}}$
• Show that Z is I(d) by observing its first-differences are I(d − 1):

${\displaystyle \Delta Z_{t}=X_{t},}$

where

${\displaystyle X_{t}\sim I(d-1).\,}$

• ARIMA
• ARMA
• Random walk
• Unit root test

This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (December 2009) (Learn how and when to remove this message)

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