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Econometricians often talk about a time series being integrated with order k, I(k). k being the minimum number of differences required to obtain a stationary time series.

What methods or statistical tests can be used to determine, given a level of confidence, the order of integration of a time series?

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There are a number of statistical tests (known as "unit root tests") for dealing with this problem. The most popular is probably the "Augmented Dickey-Fuller" (ADF) test, although the Phillips-Perron (PP) test and the KPSS test are also widely used.

Both the ADF and PP tests are based on a null hypothesis of a unit root (i.e., an I(1) series). The KPSS test is based on a null hypothesis of stationarity (i.e., an I(0) series). Consequently, the KPSS test can give quite different results from the ADF or PP tests.

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  • $\begingroup$ Is there a convenient way to determine the order of integration if beyond I(0) or I(1), e.g. I(2) etc.? $\endgroup$ – user45065 Feb 13 '17 at 13:24
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    $\begingroup$ Difference the data and apply the tests again. $\endgroup$ – Rob Hyndman Feb 13 '17 at 21:25
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Also, for some elaborate discussion (including bashing of ADF / PP / KPSS :) you might want to have a look at the book by Maddala and Kim:

http://www.amazon.com/Cointegration-Structural-Change-Themes-Econometrics/dp/0521587824

Quite extensive and not very easy to read sometimes, but a useful reference.

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