On the Definitions of (Co-) Integration

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.