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Order of integration

Summary statistic

Summary statistic

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 \

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} = \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 is stationary.

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 = \sum_{k=0}^t X_k
  • Show that Z is I(d) by observing its first-differences are I(d − 1):

:: \Delta Z_t = X_t,

: where

:: X_t \sim I(d-1). ,

References

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