This paper proposes some tests for parameter constancy in linear regressions. The tests use weighted empirical distribution functions of estimated residuals and are asymptotically distribution free. The local power analysis reveals that the proposed tests have nontrivial local power against a wide range of alternatives. In particular, the tests are capable of detecting error heterogeneity that is not necessarily manifested in the form of changing variances. The model allows for both dynamic and trending regressors. The residuals may be obtained based on any root-$n$ consistent estimator (under the null) of regression parameters. As an intermediate result, some weak convergence for (stochastically) weighted sequential empirical processes is established.