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
( 1 − L ) d X t {\displaystyle (1-L)^{d}X_{t}\ }
is a stationary process, where L {\displaystyle L} is the lag operator and 1 − L {\displaystyle 1-L} is the first difference, i.e.
( 1 − L ) X t = X t − X t − 1 = Δ X . {\displaystyle (1-L)X_{t}=X_{t}-X_{t-1}=\Delta X.}
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 ) 0 X t = X t {\displaystyle (1-L)^{0}X_{t}=X_{t}} is stationary.
Constructing an integrated series An I(d) process can be constructed by summing an I(d − 1) process:
Suppose X t {\displaystyle X_{t}} is I(d − 1) Now construct a series Z t = ∑ k = 0 t X k {\displaystyle Z_{t}=\sum _{k=0}^{t}X_{k}}
Show that Z is I(d) by observing its first-differences are I(d − 1):
Δ Z t = X t , {\displaystyle \Delta Z_{t}=X_{t},}
where
X t ∼ I ( d − 1 ) . {\displaystyle X_{t}\sim I(d-1).\,}
See also ARIMA ARMA Random walk Unit root test
References Hamilton, James D. (1994) Time Series Analysis. Princeton University Press. p. 437. ISBN 0-691-04289-6.
