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| {{For|the technique for simplifying evaluation of integrals|Order of integration (calculus)}}
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| '''Order of integration''', denoted ''I''(''d''), is a [[summary statistics|summary statistic]] for a [[time series]]. It reports the minimum number of differences required to obtain a covariance [[stationary series]].
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| == Integration of order zero ==
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| A [[time series]] is integrated of order 0 if it admits a [[moving average representation]] with
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| :<math>\sum_{k=0}^\infty \mid{b_k}^2\mid < \infty,</math>
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| where <math>b</math> is the possibly infinite vector of moving average weights (coefficients or parameters). This implies that the autocovariance is decaying to 0 sufficiently quickly. This is a necessary, but not sufficient condition for a [[stationary process]]. Therefore, all stationary processes are I(0), but not all I(0) processes are stationary.{{Citation needed|date=December 2009}}
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| == Integration of order ''d'' ==
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| A [[time series]] is integrated of order ''d'' if
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| :<math>(1-L)^d X_t \ </math>
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| is integrated of order 0, where <math>L</math> is the [[lag operator]] and <math>1-L </math> is the first difference, i.e.
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| : <math>(1-L) X_t = X_t - X_{t-1} = \Delta X. </math>
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| In other words, a process is integrated to order ''d'' if taking repeated differences ''d'' times yields a stationary process.
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| == Constructing an integrated series ==
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| An ''I''(''d'') process can be constructed by summing an ''I''(''d'' − 1) process:
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| *Suppose <math>X_t </math> is ''I''(''d'' − 1)
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| *Now construct a series <math>Z_t = \sum_{k=0}^t X_k</math>
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| *Show that ''Z'' is ''I''(''d'') by observing its first-differences are ''I''(''d'' − 1):
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| :: <math> \triangle Z_t = (1-L) X_t,</math> | |
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| : where
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| :: <math>X_t \sim I(d-1). \,</math>
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| == See also ==
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| *[[ARIMA]]
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| *[[Autoregressive–moving-average model|ARMA]]
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| *[[Random walk]]
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| {{More footnotes|date=December 2009}}
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| == References ==
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| * Hamilton, James D. (1994) ''Time Series Analysis.'' Princeton University Press. p. 437. ISBN 0-691-04289-6.
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| [[Category:Time series analysis]]
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