Adding covariates to bounds: what is the question?

Symbolic nonparametric bounds for partial identification of causal effects now have a long history in the causal literature. Sharp bounds, bounds that use all available information to make the range of values as narrow as possible, are often the goal. For this reason, many publications have focused on deriving sharp bounds, but the concept of sharp bounds is nuanced and can be misleading. In settings with ancillary covariates, the situation becomes more complex. We provide clear definitions for pointwise and uniform sharpness of covariate-conditional bounds, that we then use to prove some general and some specific to the instrumental variable (IV) setting results about the relationship between these two concepts. As we demonstrate, general conditions are much more difficult to determine and thus, we urge authors to be clear when including ancillary covariates in bounds via conditioning about the setting of interest and the assumptions made.