order of integration

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.

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.

: $(1-L) X_t = X_t - X_ = \Delta X.$

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 $Z_t = \sum_^t X_k$
*Show that ''Z'' is ''I''(''d'') by observing its first-differences are ''I''(''d'' − 1):

:: $\Delta Z_t = X_t,$

: where

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

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.

Time series