When can we learn from biased samples? We study regression when outcomes are observed only after passing through selection filters that depend on both covariates and outcomes themselves, a ubiquitous challenge spanning clinical trials with patient dropout, labor markets with self-selection, and auctions with strategic entry. Ignoring such selection yields systematically biased conclusions with real-world consequences. This challenge has a long history in econometrics and statistics, starting with Heckman's seminal two-stage model and followed by numerous generalizations. While these works provide various sufficient conditions for identification, a complete characterization of when such regression is possible has remained elusive. In this work, we provide a characterization for when regression is possible in the presence of sample selection bias. Our results establish the minimal assumptions required on the functional forms of selection processes under which regression remains possible, which are particularly relevant in modern settings where selection mechanisms are increasingly complex and opaque. As a corollary of our characterization, we show that there are settings where the regression function can be identified even when the selection filter itself cannot. This observation already goes beyond the ``estimate selection filter, then debias regression'' paradigm that is followed by virtually all existing approaches. Under natural strengthenings of our identification conditions, we also establish finite-sample estimation guarantees with explicit convergence rates and provide oracle-efficient algorithms. This yields the first general-purpose estimation method for this broad class of selection problems. Finally, we explore the implications of our results for several well-studied econometric settings with complex selection mechanisms such as auctions with entry costs and labor markets.

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