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This paper investigates a point optimal invariant (POI) test for the null hypothesis of cointegration assuming the unknown variance-covariance matrix of the error term. Since the variance-covariance matrix is unknown, we consider the POI test among a class of tests that are invariant to scale change as well as location shift in the dependent variable. As a special case of the POI test, we also derive the locally best invariant and unbiased (LBIU) test. We find that our LBIU and POI tests have different characteristic from the locally best invariant test in Shin (1994) and the POI test in Jansson (2005), both of which consider only location invariance. We show that the difference comes from the distributional difference between the maximal invariants. We also propose to modify our tests to accommodate general assumptions on the error term. Monte Carlo simulations are conducted to investigate the finite sample properties of the tests, and it is shown that our modified tests perform better than the Jansson's and Shin's tests in finite samples.