  We consider parametric tests for multidimensional ergodic diffusions based on
high frequency data. We propose two-step testing method for diffusion
parameters and drift parameters. To construct test statistics of the tests, we
utilize the adaptive estimator and provide three types of test statistics:
likelihood ratio type test, Wald type test and Rao's score type test. It is
proved that these test statistics converge in distribution to the chi-squared
distribution under null hypothesis and have consistency of the tests against
alternatives. Moreover, these test statistics converge in distribution to the
non-central chi-squared distribution under local alternatives. We also give
some simulation studies of the behavior of the three types of test statistics.
