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.


 * $$(1-L) X_t = X_t - X_{t-1} = \Delta X. $$

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 &minus; 1) process:
 * Suppose $$X_t $$ is I(d &minus; 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 &minus; 1):


 * $$ \Delta Z_t = X_t,$$


 * where


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