In stead of assuming the starting observations as zero, the cointegration test statistic derived in this paper takes the initial conditions into consideration. The statistic helps us understand how the initial conditions affect the test and helps us recognize whether the cointegration relationship exists in the data generation process or not. Beside, with this statistic derived with the concept of discriminant function and residual based test, we can simulate our own critical value table in according to what starting observations we have in hand, what significant level we want and what value of rho we meet under H_1.

Nested by linear cointegration first provided in Granger (1981), the definition of nonlinear cointegration is presented in this paper. Sequentially, a nonlinear cointegrated economic system is introduced. What we mainly study is testing no nonlinear cointegration against nonlinear cointegration by residual-based test, which is ready for detecting stochastic trend in nonlinear autoregression models. We construct cointegrating regression along with smooth transition components from smooth transition autoregression model. Some properties are analyzed and discussed during the estimation procedure for cointegrating regression, including description of transition variable. Autoregression of order one is considered as the model of estimated residuals for residual-based test, from which the teststatistic is obtained. Critical values and asymptotic distribution of the test statistic that we request for different cointegrating regressions with different sample sizes are derived based on Monte Carlo simulation. The proposed theoretical methods and models are illustrated by an empirical example, comparing the results with linear cointegration application in Hamilton (1994). It is concluded that there exists nonlinear cointegration in our system in the final results.