  We consider a residuals-based distributionally robust optimization model,
where the underlying uncertainty depends on both covariate information and our
decisions. We adopt regression models to learn the latent decision dependency
and construct a nominal distribution (thereby ambiguity sets) around the
learned model using empirical residuals from the regressions. Ambiguity sets
can be formed via the Wasserstein distance, a sample robust approach, or with
the same support as the nominal empirical distribution (e.g., phi-divergences),
where both the nominal distribution and the radii of the ambiguity sets could
be decision- and covariate-dependent. We provide conditions under which desired
statistical properties, such as asymptotic optimality, rates of convergence,
and finite sample guarantees, are satisfied. Via cross-validation, we devise
data-driven approaches to find the best radii for different ambiguity sets,
which can be decision-(in)dependent and covariate-(in)dependent. Through
numerical experiments, we illustrate the effectiveness of our approach and the
benefits of integrating decision dependency into a residuals-based DRO
framework.
