This paper compares and generalizes some testing procedures for structural change in the context of cointegrated regression models. The Lagrange Multiplier (LM) tests proposod by Hansen (1992) are generalized to testing for partial structural change. An exponential average LM test is also suggested following the idea of Andrews and Ploberger (1992). In particular, an optimal test for cointegration is developed. We also propose a new cointegration test which is robust to a possible one-time discrete jump in the intercept. We tabulate the asymptotic critical values for the above tests and conduct a small Monte Carlo simulation to investigate their finite sample performance.