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

[math]\displaystyle{ (1-L)^d X_t \ }[/math]

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

[math]\displaystyle{ (1-L) X_t = X_t - X_{t-1} = \Delta X. }[/math]

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 [math]\displaystyle{ (1-L)^0 X_t = X_t }[/math] is stationary.

Constructing an integrated series

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

• Suppose [math]\displaystyle{ X_t }[/math] is I(d − 1)
• Now construct a series [math]\displaystyle{ Z_t = \sum_{k=0}^t X_k }[/math]
• Show that Z is I(d) by observing its first-differences are I(d − 1):

[math]\displaystyle{ \Delta Z_t = X_t, }[/math]

where

[math]\displaystyle{ X_t \sim I(d-1). \, }[/math]

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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