The autoregressive conditional heteroscedasticity (ARCH) family of models has grown to encompass a wide range of specifications, each of which is designed to enhance the ability of the model to capture the characteristics of the data. In this paper, the ability of a number of model selection criteria to correctly identify the data generating process in simulated data is established. The results of this study suggest that the Hannan-Quinn and stochastic complexity criteria provide a superior level of performance for ARCH and generalized ARCH (GARCH) processes compared to the more commonly used criteria. Where leverage and/or power effects are present, however, none of the procedures considered perform well. A new LM based test for the presence of nonlinearity and power effects is introduced and tested.