{{Short description|Summary statistic}} {{For|the technique for simplifying evaluation of integrals|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.
== Integration of order ''d'' ==
A time series is integrated of order ''d'' if
:<math>(1-L)^d X_t \ </math>
is a stationary process, where <math>L</math> is the lag operator and <math>1-L </math> is the first difference, i.e.
: <math>(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>(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>X_t </math> is ''I''(''d'' − 1) *Now construct a series <math>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> \Delta Z_t = X_t,</math>
: where
:: <math>X_t \sim I(d-1). \,</math>
== See also == *ARIMA *ARMA *Random walk *Unit root test
{{More footnotes|date=December 2009}}
== References == * Hamilton, James D. (1994) ''Time Series Analysis.'' Princeton University Press. p. 437. {{ISBN|0-691-04289-6}}.
Category:Time series