# Interpretation of an integrable time series of an order zero

What is the interpretation/intuition behind a time series integrable of order 0? I am reading something on cointegration and this is not yet clear to me.

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Commonly, a time series is said to be $I(0)$ if the time series itself is stationary (no need to differentiate it to obtain stationarity).

The Wikipedia page you mention says that not all $I(0)$ time series are stationary. I didn't know this and I think that indeed many authors do not make the difference. The paper from Engle and Granger (1987) says that all $I(0)$ are stationary.

Two times series $X^1_t$ and $X^2_t$ are said to be cointegrated if $\exists n,d>0$ and $\beta \in \mathbb{R}$ so that

• $X^1_t \sim I(n)$
• $X^2_t \sim I(n)$
• $X^1_t+\beta X^2_t \sim I(n-d)$

See Engle, Granger (1987) http://www.ntuzov.com/Nik_Site/Niks_files/Research/papers/stat_arb/EG_1987.pdf

If you reach $n-d=0$ indeed your linear combination of time series is stationary.

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the wikipedia page is a bit strange. The condition for $I(0)$ presented is satisfied for all processes which admit MA representation as in MA link. For all practical purposes $I(0)$ is synonim for stationarity. –  mpiktas Apr 7 '11 at 8:20