Order of integration

Short description: Summary statistic

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.

## Contents

* 1 Integration of order d
* 2 Constructing an integrated series
* 3 See also
* 4 References

## Integration of order d

A time series is integrated of order d if

(1−L)dXt

is a stationary process, where L is the lag operator and 1−L is the first difference, i.e.

(1−L)Xt=Xt−Xt−1=ΔX.

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)0Xt=Xt is stationary.

## Constructing an integrated series

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

* Suppose Xt is I(d − 1)
* Now construct a series Zt=∑k=0tXk
* Show that Z is I(d) by observing its first-differences are I(d − 1):

ΔZt=Xt,

where

Xt∼I(d−1).

## 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.

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