Harman Patil (Editor)

Order of integration

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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.

Contents

Integration of order zero

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

∑ k = 0 ∞ ∣ b k ∣ 2 < ∞ ,

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

A time series is integrated of order d if

( 1 − L ) d X t  

is a stationary process, where L is the lag operator and 1 − L is the first difference, i.e.

( 1 − L ) X t = X t − X t − 1 = Δ 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

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 = ∑ k = 0 t X k
  • Show that Z is I(d) by observing its first-differences are I(d − 1):
    • where

    References

    Order of integration Wikipedia
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