  We study partially linear models when the outcome of interest and some of the
covariates are observed in two different datasets that cannot be linked. This
type of data combination problem arises very frequently in empirical
microeconomics. Using recent tools from optimal transport theory, we derive a
constructive characterization of the sharp identified set. We then build on
this result and develop a novel inference method that exploits the specific
geometric properties of the identified set. Our method exhibits good
performances in finite samples, while remaining very tractable. We apply our
approach to study intergenerational income mobility over the period 1850-1930
in the United States. Our method allows us to relax the exclusion restrictions
used in earlier work, while delivering confidence regions that are informative.
