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When estimating a regression model, we might have data where some labels aremissing, or our data might be biased by a selection mechanism. When theresponse or selection mechanism is ignorable (i.e., independent of the responsevariable given the features) one can use off-the-shelf regression methods; inthe nonignorable case one typically has to adjust for bias. We observe thatprivileged information (i.e. information that is only available duringtraining) might render a nonignorable selection mechanism ignorable, and werefer to this scenario as Privilegedly Missing at Random (PMAR). We propose anovel imputation-based regression method, named repeated regression, that issuitable for PMAR. We also consider an importance weighted regression method,and a doubly robust combination of the two. The proposed methods are easy toimplement with most popular out-of-the-box regression algorithms. Weempirically assess the performance of the proposed methods with extensivesimulated experiments and on a synthetically augmented real-world dataset. Weconclude that repeated regression can appropriately correct for bias, and canhave considerable advantage over weighted regression, especially whenextrapolating to regions of the feature space where response is never observed.

Correcting for Selection Bias and Missing Response in Regression using Privileged Information