Control variables provide an important means of controlling for endogeneity
in econometric models with nonseparable and/or multidimensional heterogeneity.
We allow for discrete instruments, giving identification results under a
variety of restrictions on the way the endogenous variable and the control
variables affect the outcome. We consider many structural objects of interest,
such as average or quantile treatment effects. We illustrate our results with
an empirical application to Engel curve estimation.Comment: 37 pages, 4 figure
