Parsimonious dependence structures are crucial in multivariate modelling as they offer better interpretation and may improve the quality of estimation. However, one must also be careful not to use an overly parsimonious model that results in underfitting. Model selection criteria like AIC and BIC do not typically provide inslight on the quality of model fit. We propose the use of adequacy-of-fit statistics for this purpose. Pairwise differences between empirical and model-based features are aggregated and the resulting statistic is compared against a cutoff value, the exceedance of which suggests model inadequacy. In this talk, we will explore the challenges in obtaining an appropriate cutoff and some pragmatic approaches we put forward. The techniques are applied to a financial data set with dependent returns.