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

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