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
References Hamilton, James D. (1994) Time Series Analysis. Princeton University Press. p. 437. .