We study the problem of predictive inference with outcomes missing not at random, when instrumental variables are available. While conformal prediction and weighted conformal prediction provide valid distribution-free inference when outcomes are missing completely at random and missing at random, respectively, inference is generally not possible without additional assumptions when outcomes are missing not at random. In this work, we consider a setting where we remain within the distribution-free framework but instead have access to an instrumental variable (IV). We explore the theoretical possibility of IV predictive inference and establish that it is characterized by a parameter that we refer to as the identification index. We derive an impossibility result for the case where the identification index is not sufficiently large, showing that any distribution-free approach fails to provide informative inference in this case. For the case where the identification index is sufficiently large, we introduce a practical method that achieves informative predictive inference. Our method leverages modern semiparametric efficiency theory which involves deriving the efficient influence function of the identifying functional for sharp IV bounds of the quantiles of the missing outcome, delivering a double robustness property in the coverage guarantee of the proposed framework for IV-based predictive inference. We illustrate the theoretical results and the performance of the proposed procedure through simulations and two real-data applications---FES wages and NHANES III self-reported heights.

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