  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 show that the test is asymptotically of
correct level, consistent and has power against local alternatives when the
independence condition is relaxed, under two main conditions. The first is a
slight reinforcement of the usual absence of correlation between the regressors
and the error term. The second is that the number of strata, defined by values
of the regressors not involved in the subvector test, is small compared to the
sample size. The latter implies that the vector of nuisance regressors is
discrete. Simulations and empirical illustrations suggest that the test has
good power in practice if, indeed, the number of strata is small compared to
the sample size.
