  We propose a methodology for constructing confidence regions with partially
identified models of general form. The region is obtained by inverting a test
of internal consistency of the econometric structure. We develop a dilation
bootstrap methodology to deal with sampling uncertainty without reference to
the hypothesized economic structure. It requires bootstrapping the quantile
process for univariate data and a novel generalization of the latter to higher
dimensions. Once the dilation is chosen to control the confidence level, the
unknown true distribution of the observed data can be replaced by the known
empirical distribution and confidence regions can then be obtained as in
Galichon and Henry (2011) and Beresteanu, Molchanov and Molinari (2011).
