Categorical distribution

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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

$\sum _{k=0}^{\infty }\mid {b_{k}}^{2}\mid <\infty ,$

where $b$ 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.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

Integration of order d

A time series is integrated of order d if

$(1-L)^{d}X_{t}\$

is integrated of order 0, where $L$ is the lag operator and $1-L$ is the first difference, i.e.

$(1-L)X_{t}=X_{t}-X_{t-1}=\Delta X.$

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 $X_{t}$ is I(d − 1)
• Now construct a series $Z_{t}=\sum _{k=0}^{t}X_{k}$

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

$\triangle Z_{t}=(1-L)X_{t},$

where

$X_{t}\sim I(d-1).\,$

See also

• ARIMA
• ARMA
• Random walk

References

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