Order of integration
Integration of order zero
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
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:
- Suppose is I(d − 1)
- Now construct a series
- Show that Z is I(d) by observing its first-differences are I(d − 1):
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