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

is a stationary process, where is the lag operator and is the first difference, i.e.

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 is stationary.

Constructing an integrated series

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

• Suppose is I(d − 1)
• Now construct a series
• Show that Z is I(d) by observing its first-differences are I(d − 1):

where

See also

• ARIMA
• ARMA
• Random walk
• Unit root test

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References

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