order of integration

A time series is integrated of order d if

:$(1-L)^d\; X\_t\; \backslash$

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

: $(1-L)\; X\_t\; =\; X\_t\; -\; X\_\{t-1\}\; =\; \backslash Delta\; X.$

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.

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 $Z\_t\; =\; \backslash sum\_\{k=0\}^t\; X\_k$
• Show that Z is I(d) by observing its first-differences are I(d − 1):

:: $\backslash Delta\; Z\_t\; =\; X\_t,$

: where

:: $X\_t\; \backslash sim\; I(d-1).\; \backslash ,$

• ARIMA
• ARMA
• Random walk
• Unit root test

{{More footnotes|date=December 2009}}

• Hamilton, James D. (1994) Time Series Analysis. Princeton University Press. p. 437. {{ISBN|0-691-04289-6}}.

Category:Time series