Developments in maximum likelihood unit root tests. The exact maximum likelihood estimate provides a test statistic for the unit root test that is more powerful than the usual least-squares approach. In this article, a new derivation is given for the asymptotic distribution of this test statistic that is simpler and more direct than the previous method. The response surface regression method is used to obtain a fast algorithm that computes accurate finite-sample critical values. This algorithm is available in the R package mleur that is available on CRAN. The empirical power of the new test is shown to be much better than the usual test not only in the normal case but also for innovations generated from an infinite variance stable distribution as well as for innovations generated from a GARCH(1,1) process.