Tests for non-correlation of two cointegrated ARMA time series

In multivariate time series modelling, we are often led to investigate the existence of a relationship between two time series. Here, we generalize the procedure proposed by Haugh (1976 ) and extended by El Himdi and Roy (1997 ) for multivariate stationary ARMA time series to the case of cointegrated (or partially nonstationary) ARMA series. The main contribution consists in showing that, in the case of two uncorrelated cointegrated time series, an arbitrary vector of residual cross-correlation matrices asymptotically follows the same distribution as the corresponding vector of cross-correlation matrices between the two innovation series. The estimation method from which the residuals are obtained can be the conditional maximum likelihood method as discussed in Yap and Reinsel (1995 ) or some other which has the same convergence rate. From this result, it follows that the considered test statistics, which are based on residual cross-correlation matrices, asymptotically follow χ² distributions. The finite sample properties, under the null hypothesis, of the test statistics are studied by simulation.