Order of integration

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For the technique for simplifying evaluation of integrals, see Order of integration (calculus).

Order of integration, denoted I(d), is a summary statistic for a time series. It reports the minimum number of differences required to obtain a covariance stationary series.

Integration of order zero[edit]

A time series is integrated of order 0 if it admits a moving average representation with

\sum_{k=0}^\infty \mid{b_k}^2\mid  <  \infty,

where b is the possibly infinite vector of moving average weights (coefficients or parameters). This implies that the autocovariance is decaying to 0 sufficiently quickly. This is a necessary, but not sufficient condition for a stationary process. Therefore, all stationary processes are I(0), but not all I(0) processes are stationary.

Integration of order d[edit]

A time series is integrated of order d if

(1-L)^d X_t \

is integrated of order 0, where L is the lag operator and 1-L is the first difference, i.e.

(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.

Constructing an integrated series[edit]

An I(d) process can be constructed by summing an I(d − 1) process:

  • Suppose X_t is I(d − 1)
  • Now construct a series Z_t = \sum_{k=0}^t X_k
  • Show that Z is I(d) by observing its first-differences are I(d − 1):
 \triangle Z_t = X_t,
X_t \sim I(d-1). \,

See also[edit]


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