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

  • Hamilton, James D. (1994) Time Series Analysis. Princeton University Press. p. 437. ISBN 0-691-04289-6.