This paper assesses the performance of linear and nonlinear causality tests in the presence of multivariate conditional heteroskedasticity, exogenous volatility regressors, and additive volatility outliers. Monte Carlo simulations show that tests based on the least squares covariance matrix estimator can frequently lead to finding spurious Granger causality. The degree of oversizing tends to increase with the sample size and is substantially larger for the nonlinear test. On the other hand, heteroskedasticity-robust tests which are based on the Fixed Design Wild Bootstrap perform adequately in terms of size and power. Consequently, reliable causality in mean tests can be conducted without the need to specify a conditional variance function. As an empirical application, we re-examine the return-volume relationship.