Order of integration

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

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

Integration of order zero

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

$\sum_{k=0}^\infty \mid{\gamma_k}^2\mid < \infty$

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.^{[citation needed]}

Integration of order P

A time series is integrated of order P if

$(1-L)^P X_t \$

is integrated of order 0, where L is the lag operator and (1 − L) is the first difference, ie: (1 − L)X_t = X_t − X_{t − 1} = ΔX.

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

Constructing an integrated series

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

• Suppose X_t is I(p-1)
• Now construct a series $Z_t = \sum_{k=0}^t X_k$

• Show that Z is I(p) by observing its first-differences are I(p-1):

$\triangle Z_t = \sum_{k=0}^t X_k - \sum_{k=0}^{t-1} X_k = X_t \sim I(p-1)$

See also

• ARIMA
• ARMA
• Random walk

References

• Hamilton, 1994 : "Time Series Analysis"