Order of integration

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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.

Contents

[edit] 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 sufficient, but not necessary condition for a stationary process. Therefore, all I(0) processes are stationary, but not all stationary processes are I(0). Most UK textbooks ignore this distinction.

[edit] Integration of order P

A time series is integrated of order P if:

  • (1-L)^P X_t \ is integrated of order 0

This says that a process is Integrated if taking repeated differences yields a stationary process.

(1-L) \ is the First difference operator, ie: (1-L) X_t = X_t - X_{t-1} = \Delta X \

For a discussion of this notation see Lag operator.

[edit] Constructing an integrated series

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

  • Suppose Xt is I(0)
  • Now construct a series Z_t = \sum_{k=0}^t X_k
  • Show that Z is I(1) by observing its first-differences are I(0):
\triangle Z_t = \sum_{k=0}^t X_k - \sum_{k=0}^{t-1} X_k = X_t \sim I(0)

[edit] See also

[edit] References

  • Hamilton, 1994 : "Time Series Analysis"
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