On the Definitions of (Co‐)integration

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Abstract
Two problems exist in testing for (co‐)integration. One is that current definitions of fractional integration in the time domain can be incomplete. The other is that disregarding fractional orders of integration can cause incorrectly sized inference about cointegration. This paper completes the time‐domain definition of fractional integration, and defines cointegration in a general setting. As a by‐product of the latter, testing for cointegration in models without a pre‐specified functional form is shown to avoid a common inferential problem. Finally, we analyse the effects of using incomplete definitions of integration and/or cointegration. For the latter, incomplete I(1)‐null procedures make cointegration seem more likely than it actually is, while incomplete I(0)‐null procedures reject cointegration too often.

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