Point Optimal Test for Cointegration with Unknown Variance-Covariance Matrix

This paper examines a point optimal invariant (POI) test for the null hypothesis of cointegration. Our test is different from Jansson's (2005) test in that we consider location invariance in wider directions and that we assume an unknown variance-covariance matrix for the error term, while it is assumed to be known in Jansson (2005). As the variance-covariance matrix is unknown in our paper, 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 POI test has the same asymptotic distribution as Jansson's (2005) test, which is a point optimal test among a class of location invariant tests. On the other hand, our LBIU test is shown to have a different characteristic from the locally best invariant test in Shin (1994). We also propose a modification of our tests to accommodate more general assumptions on the error term. Monte Carlo simulation is conducted to investigate the finite sample properties of the tests, and it is shown that our modified tests perform better in finite samples than either the Jansson or Shin tests.