Multivariate linear regression and randomization-based inference are two essential methods in statistics and econometrics. Nevertheless, the problem of producing a randomized test for the value of a single regression coefficient that is exactly valid when errors are exchangeable, and which is asymptotically valid for the best linear predictor, has remained elusive. In this paper, we produce a test that is exactly valid with exchangeable errors and which allows for general covariate designs; covariates may be continuous as well as discrete and may be correlated. The test is asymptotically valid when the errors are not exchangeable, in particular in the presence of conditional heteroskedasticity.