The goal of this project is to develop a new statistical theory of nonlinear co-integration in nonlinear vector autoregressive (VAR) models and to extend the main findings by testing the Changliunit root hypothesis in univariate nonlinear time series models contained in Sandberg (2004) and He and Sandberg (2003, 2004) to the case of multivariate nonlinear models. In addition, we will apply the theory of nonlinear co-integration in NVAR models to data (for example to a system of macroeconomic time series) and to examine economical aspects such as long-run equilibriums and nonlinear common features. The project is divided into the following five subprojects. (1) Models and representation theorems - A family of NVAR models containing nonlinear co-integrations are introduced. (2) Testing and specification procedures -Testing linear co-integration against nonlinear co-integration in a nonlinear VAR model under the assumption of unit roots. (3) Estimation of nonlinear co-integrating vectors - To obtain estimates of the nonlinear co-integrating vectors we shall apply and generalize the existing techniques proposed in Johansen (1988, 1991), i.e. reduced-rank regression and canonical correlation analysis. (4) Evaluation - We have to verify that the underlying assumptions made in NVAR models with nonlinear co-integrations are satisfied. (5) Applications - To apply nonlinear co-integration in NVAR models having a triangular or common nonlinear feature representation to, say, the OECD unemployment data.