Rank-Based Tests of the Cointegrating Rank in Semiparametric Error Correction Models

This paper introduces rank-based tests for the cointegrating rank in an Error CorrectionModel with i.i.d. elliptical innovations. The tests are asymptotically distribution-free,and their validity does not depend on the actual distribution of the innovations. Thisresult holds despite the fact that, depending on the alternatives considered, the model exhibitsa non-standard Locally Asymptotically Brownian Functional (LABF) and LocallyAsymptotically Mixed Normal (LAMN) local structure—a structure which we completelycharacterize. Our tests, which have the general form of Lagrange multiplier tests, dependon a reference density that can freely be chosen, and thus is not restricted to be Gaussianas in traditional quasi-likelihood procedures. Moreover, appropriate choices of the referencedensity are achieving the semiparametric efficiency bounds. Simulations show thatour asymptotic analysis provides an accurate approximation to finite-sample behavior.Our results are based on an extension, of independent interest, of two abstract resultson the convergence of statistical experiments and the asymptotic linearity of statistics tothe context of, possibly non-stationary, time series