Interpreting Quantile Independence
How should one assess the credibility of assumptions weaker than statistical independence, like quantile independence? In the context of identifying causal effects of a treatment variable, we argue that such deviations should be chosen based on the form of selection on unobservables they allow. For quantile independence, we characterize this form of treatment selection. Specifically, we show that quantile independence is equivalent to a constraint on the average value of either a latent propensity score (for a binary treatment) or the cdf of treatment given the unobservables (for a continuous treatment). In both cases, this average value constraint requires a kind of non-monotonic treatment selection. Using these results, we show that several common treatment selection models are incompatible with quantile independence. We introduce a class of assumptions which weakens quantile independence by removing the average value constraint, and therefore allows for monotonic treatment selection. In a potential outcomes model with a binary treatment, we derive identified sets for the ATT and QTT under both classes of assumptions. In a numerical example we show that the average value constraint inherent in quantile independence has substantial identifying power. Our results suggest that researchers should carefully consider the credibility of this non-monotonicity property when using quantile independence to weaken full independence.
