A procedure is developed to test whether conditional variances are constant over time in the context of generalized autoregressive conditional heteroscedasticity (GARCH) models with possible GARCH-in-mean effects. The approach is based on the quasilikelihood function, leaving the true distribution of model disturbances parametrically unspecified. The presence of possible nuisance parameters in the conditional mean is dealt with by using a pivotal bound and Monte Carlo resampling techniques to obtain a level-exact test procedure. Simulation experiments reveal that the permutation-based, quasilikelihood ratio test has very attractive power properties in comparison with omnibus Lagrange multiplier tests. An empirical application of the new procedure finds overwhelming evidence of GARCH effects in Fama-French portfolio returns, even when conditioning on the market risk factor.