We consider the multivariate linear model for multilevel data where units are nested within a hierarchy of clusters. We propose permutation procedures to test for variance components at any given level. The tests are moment based and require no distributional assumptions except finite second moments. Moreover, they are based on a statistic with a closed-form expression which is an advantage for a permutation test since it has to be evaluated many times. Problems of convergence of likelihood-based statistics are thus avoided. We introduce the R package mvctm, which implements the tests. It can perform tests based on the original observations and score-based tests using ranks and signs. A simulation study shows that the new tests maintain the desired type I error, except for the sign-based test with univariate data. It also compares their power. The results suggest that the tests based on the original observations and the rank-based tests are very competitive. With univariate data, the former one is even more powerful than a likelihood ratio test based on a mixture of chi-squared distributions. The proposed tests are illustrated using the PISA data.