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

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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

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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

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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.