We consider test statistics for autoregressive conditional heteroskedasticity (ARCH) in the residuals from a possibly nonlinear and dynamic multivariate regression model. Our approach is based on estimation of the multivariate spectral density of squared and cross-residuals. We advocate a simple wavelet-based spectral density estimator, which is a particularly suitable analytic tool when the spectral density exhibits peaks or kinks that may arise from strong cross-dependence, seasonal patterns and other forms of periodic behaviors. In several circumstances, the spectral density may have peaks at various frequencies, such as seasonal frequencies, and the wavelet method may capture them effectively. Compared to kernel-based test statistics for multivariate ARCH effects, the weighting scheme offered by the new wavelet-based test statistics differs in several important aspects. An asymptotic analysis under the null hypothesis of no ARCH effects shows that the new wavelet-based test statistic converges in distribution to a convenient standard normal distribution. Under fixed alternatives, the consistency of the wavelet-based test statistics is established in a class of static regression models with uncorrelated but dependent errors. In a Monte Carlo study comparisons are made under various alternatives between: the proposed wavelet-based test statistics; the kernel-based test statistics for ARCH effects of Duchesne and Lalancette (2003); and the test statistic of Hosking (1980) adapted for multivariate ARCH effects.