Order of integration

For the technique for simplifying evaluation of integrals, see Order of integration (calculus).

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.

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

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):

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

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. ISBN 0-691-04289-6.