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.

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.

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

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 $(1-L{)}^0{X}_t=X_t$ is stationary.

Constructing an integrated series

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

• Suppose $X_t$ 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):

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

where

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

See also

• ARIMA
• ARMA
• Random walk
• Unit root test

References

• Hamilton, James D. (1994) Time Series Analysis. Princeton University Press. p. 437. Template:ISBN.