Order of integration

For the technique for simplifying evaluation of integrals, see Order of integration (calculus).

Order of integration, denoted I(d), is a summary statistic for a time series. It reports the minimum number of differences required to obtain a covariance stationary series.

Integration of order zero

A time series is integrated of order 0 if it admits a moving average representation with

where is the possibly infinite vector of moving average weights (coefficients or parameters). This implies that the autocovariance is decaying to 0 sufficiently quickly. This is a necessary, but not sufficient condition for a stationary process. Therefore, all stationary processes are I(0), but not all I(0) processes are stationary.

Integration of order d

A time series is integrated of order d if

is a stationary process, where is the lag operator and is the first difference, i.e.

In other words, a process is integrated to order d if taking repeated differences d times yields a stationary process.

Constructing an integrated series

An I(d) process can be constructed by summing an I(d  1) process:

where

See also

References

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