We develop a new permutation test for inference on a subvector of coefficients in linear models. The test is exact when the regressors and the error terms are independent. Then, we study its asymptotic properties under two main conditions. The first is the usual absence of correlation between the regressors and the error term. The second is that the average number of strata, defined by values of the regressors not involved in the subvector test, is small compared to the sample size. Under these conditions, the test is consistent and has power against local alternatives. Simulations and an empirical illustration suggest that the test has good power in practice. (with Xavier D’Haultfœuille (CREST)