Order of integration explained

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 - 1) process:

• Suppose

X _{t} is I(d - 1)

• Now construct a series

Z _{t} = \sum ^{t} _{k =0} 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. .