Fully modified estimation of seasonally cointegrated processes

Résumé

We extend the framework of the fully modified, ordinary least squares (OLS) estimator introduced by Phillips and Hansen (1990) to the case of seasonally cointegrated processes at a given frequency. First we extend a weak convergence result of sample covariance matrices (Phillips, 1988) to the case of seasonal unit roots. Using a complex number framework, we then show that we can take into account the constraints that exist in a situation of seasonal cointegration as illustrated in Gregoir (1999a) and derive estimates of the cointegration vectors that allow for asymptotic normal inference. This allows us to propose a test whose null hypothesis is the existence of seasonal cointegration. A Monte Carlo exercise investigates the finite sample properties of this test procedure. The paper closes with the analysis of situations in which there exist more than one frequency at which seasonal cointegration can be observed.