|Vine copulas are flexible multivariate dependence models, which are built up from a set of bivariate copulas in different hierarchical levels. However, vine copulas have a computational complexity that is increasing quadratically in the number of variables. This complexity can be reduced by focusing on the sub-class of truncated vine copulas, which use only a limited number of hierarchical levels. We propose a new approach to select the adequate number of levels, such that the vine copula is still sufficiently flexible to provide a good fit to given data. The approach is based on fit indices, as used for structural equation models, to measure the goodness of a fitted truncated model. To select such truncated models, we propose methods to effectively explore the search space of truncated vine copulas, so that we are able to improve over previous greedy sequential approaches that optimized over one tree of the vine at each step. This new selection approach is evaluated in a simulation study as well as in two applications to data sets of financial returns. (C) 2015 Elsevier Inc. All rights reserved.