Existing specification tests for conditional heteroskedasticity are derived under the assumption that the density of the innovation, or standardized error, is Gaussian, despite the fact that many recent empirical studies provide evidence that this density is not Gaussian. We obtain specification tests for conditional heteroskedasticity under the assumption that the innovation density is a member of a general family of densities. Our test statistics maximize asymptotic local power and weighted average power criteria for the general family of densities. We establish both first order and second order theory for our procedures. Monte Carlo simulations indicate that asymptotic power gains are achievable in finite samples. We apply the tests to shock futures data sampled at high frequency and find evidence of conditional heteroskedasticity in the residuals from a GARCH(1,1) model, indicating that the standard (1,1) specification is not adequate.