Diagnostic Testing for Cointegration

We develop a sequence of tests for specifying the cointegrating rank of, possibly fractional, multiple time series. Memory parameters of observables are treated as unknown, as are those of possible cointegrating errors. The individual test statistics have standard null asymptotics, and are related to Hausman speci…cation test statistics: when the memory parameter is common to several series, an estimate of this parameter based on the assumption of no cointegration achieves an e¢ ciency improvement over estimates based on individual series, whereas if the series are cointegrated the former estimate is generally inconsistent. However, a computationally simpler but asymptotically equivalent approach, which avoids explicit computation of the "e¢ cient" estimate, is instead pursued here. Two versions of it are initially proposed, followed by one that robusti…es to possible inequality between memory parameters of observables. Throughout, a semiparametric approach is pursued, modelling serial dependence only at frequencies near the origin, with the goal of validity under Corresponding author: Department of Economics, London School of Economics, Houghton Street, London WC2A 2AE, UK; tel. +44-20-7955-7516; fax: +44-20-7955-6592. Email address: [email protected]