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.

Integration of order d

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.

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

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.

Constructing an integrated series

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):

${\displaystyle \mathrm{\Delta }{Z}_t=X_t,}$

where

${\displaystyle X_t\sim I(d-1).}$

See also

• ARIMA
• ARMA
• Random walk
• Unit root test

References

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

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