Journal of Econometrics 110 (2002) 187 – 212 www.elsevier.com/locate/econbase A model of fractional cointegration, and tests for cointegration using the bootstrap  James Davidson ∗ Cardi Business School, Cardi University, Aberconway Building Colum Drive, Cardi CFI 3EU, UK Abstract The paper proposes a framework for modelling cointegration in fractionally integrated pro- cesses, and considers methods for testing the existence of cointegrating relationships using the parametric bootstrap. In these procedures, ARFIMA models are tted to the data, and the estimates used to simulate the null hypothesis of non-cointegration in a vector autoregressive modelling framework. The simulations are used to estimate p-values for alternative regression- based test statistics, including the F goodness-of-t statistic, the Durbin–Watson statistic and estimates of the residual d. The bootstrap distributions are economical to compute, being con- ditioned on the actual sample values of all but the dependent variable in the regression. The procedures are easily adapted to test stronger null hypotheses, such as statistical independence. The tests are not in general asymptotically pivotal, but implemented by the bootstrap, are shown to be consistent against alternatives with both stationary and nonstationary cointegrating resid- uals. As an example, the tests are applied to the series for UK consumption and disposable income. The power properties of the tests are studied by simulations of articial cointegrating relationships based on the sample data. The F test performs better in these experiments than the residual-based tests, although the Durbin–Watson in turn dominates the test based on the residual d. c  2002 Elsevier Science B.V. All rights reserved. JEL classication: C32; C15 Keywords: Bootstrap; Fractional integration; Cointegration 1. Introduction Cointegration methods are well established as a basis for testing relationships amongst nonstationary time series exhibiting stochastic trends. These methods invoke the ‘I (1)  Paper presented at the Cardi Conference on Long Memory and Nonlinear Time Series, July 9 –11, 2000. Research supported by the ESRC under award L138251025. ∗ Tel.: +44-2920-874558; fax: +44-2920-874419. E-mail address: davidsonje@cardi.ac.uk (J. Davidson). 0304-4076/02/$ - see front matter c  2002 Elsevier Science B.V. All rights reserved. PII: S0304-4076(02)00092-1