Order of integration

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). \, == See also ==