Order of integration
From HandWiki
Revision as of 15:34, 8 May 2022 by imported>Steve Marsio (change)
Short description: 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.
Integration of order d
A time series is integrated of order d if
- [math]\displaystyle{ (1-L)^d X_t \ }[/math]
is a stationary process, where [math]\displaystyle{ L }[/math] is the lag operator and [math]\displaystyle{ 1-L }[/math] is the first difference, i.e.
- [math]\displaystyle{ (1-L) X_t = X_t - X_{t-1} = \Delta X. }[/math]
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 [math]\displaystyle{ (1-L)^0 X_t = X_t }[/math] is stationary.
Constructing an integrated series
An I(d) process can be constructed by summing an I(d − 1) process:
- Suppose [math]\displaystyle{ X_t }[/math] is I(d − 1)
- Now construct a series [math]\displaystyle{ Z_t = \sum_{k=0}^t X_k }[/math]
- Show that Z is I(d) by observing its first-differences are I(d − 1):
- [math]\displaystyle{ \Delta Z_t = X_t, }[/math]
- where
- [math]\displaystyle{ X_t \sim I(d-1). \, }[/math]
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.
Original source: https://en.wikipedia.org/wiki/Order of integration.
Read more |