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In statistics, the order of integration, denoted ''I''(''d''), of a
time series In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. E ...
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 In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. E ...
is integrated of order ''d'' if :(1-L)^d X_t \ is a stationary process, where L is the
lag operator In time series analysis, the lag operator (L) or backshift operator (B) operates on an element of a time series to produce the previous element. For example, given some time series :X= \ then : L X_t = X_ for all t > 1 or similarly in term ...
and 1-L is the first difference, i.e. : (1-L) X_t = X_t - X_ = \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 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 Arima, officially The Royal Chartered Borough of Arima is the easternmost and second largest in area of the three boroughs of Trinidad and Tobago. It is geographically adjacent to Sangre Grande and Arouca at the south central foothills of ...
* ARMA * Random walk *
Unit root test In statistics, a unit root test tests whether a time series variable is non-stationary and possesses a unit root. The null hypothesis is generally defined as the presence of a unit root and the alternative hypothesis is either stationarity, tre ...


References

* Hamilton, James D. (1994) ''Time Series Analysis.'' Princeton University Press. p. 437. {{ISBN, 0-691-04289-6. Time series