There exist a variety of reasons for the failure to find a uniquecointegrating relationship between economic time series where onewould normally be expected on the basis of economic theory. Amongthese are the testing procedure, the span of the data set, the choiceof lag length in generating the test statistic, the presence ofstructural breaks, and the presence of cointegration only beyond somethreshold. We propose the concept of regime-sensitive cointegrationwhereby the underlying series need not be cointegrated at all times.We show that cointegration can be switched off when a commonstochastic trend is added. Alternatively, cointegration can beswitched on or off because series normally believed to contain a unitactually do not. This implies that a linear combination of suchvariables need not be cointegrated. To illustrate the conceptempirically, we test the hypothesis of interest-rate parity, andrelated hypotheses, using daily Eurorates for the United States andCanada.