Does the Heckman correction with an exclusion restriction provide causal inference? I think I might be getting instrumental variables estimation and the Heckman correction with an exclusion restriction confused.  I know that instrumental variables estimation is way to show causal relationships.  I also know that the Heckman correction helps with self-selection, but does the Heckman correction with an exclusion restriction help with both self-selection and causality?
 A: This indeed can be confusing, because the term "selection bias" pops up in two problems which are clearly different:


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*People "select" a treatment $D$ based on a variable that influences an outcome $Y$, so that $E[Y|D = 1] - E[Y|D = 0] \neq E[Y|do(D = 1)] - E[Y|do(D = 0)]$, because of unblocked back-door paths

*People "select" into the sample (S = 1) based on variables that somehow correlate with outcome $Y$ or treatment $D$ or both. In this case, $E[Y|S = 1] \neq E[Y]$, and $E[Y|D = 1, S = 1] - E[Y|D = 0, S = 1] \neq E[Y|D = 1] - E[Y|D = 0]$
Clearly, you can have both problems, and they might interact. For example, $D$ might be randomized, but if $D$ and $U$ influence $S$, and $U$ is unobserved and also influences $Y$, the relationship between $D$ and $Y$ in the sample (S = 1) will be biased for the causal effect of $D$ on $Y$. 
However, you can have sample selection problems even if you are only interested in descriptive inferences on $E[Y]$ or $E[Y|D]$, while the treatment selection problem is about causal inference.
The Heckman correction is intended to solve sample selection problems, which is very clear if you look in the original Heckman 1979 paper (p. 154, last equation). The correction often (but not necessarily) relies on "instruments" that affect sample selection $S$, but not the outcome $Y$. 
To quote from the comments, in the Heckman model "the exclusion restriction helps explain the sample selection, but this does not determine causality of Y. The intention of the instruments (to correct for self-selection) used in the Heckman correction is different from the intention of the instruments (to show causal inference) used in instrumental variables."
Finally, see Paul Hünermund's blog ("Sample Selection vs. Selection into Treatment") for a longer discussion of this issue.
A: There is a "treatment effects" version of the Heckman model, in which individuals select into (or out of) treatment in addition to the original "wage equation" version. The treatment effects version with an exclusion provides a close analogue to IV.
A paper with an excellent (and accessible by economics standards) discussion is 
Blundell, Richard, Lorraine Dearden and Barbara Sianesi. 2003. “Evaluating the Impact of Education on Earnings in the UK: Models, Methods and Results from the NCDS.” Journal of the Royal Statistical Society, Series A 168(3): 473-512.
A: From Heckman, Urzua and Vytlacil (2006):
Two main approaches have been adopted to solve the selection bias problem: (a) selection models and (b) instrumental variable models.
The selection approach models levels of conditional means. The IV approach models the slopes of the conditional means. IV does not identify the constants estimated in selection models.
The IV approach does not condition on D (the treatment). The selection (control function) estimator identifies the conditional means using control functions.
When using control functions with curvature assumptions, one does not require an exclusion restriction (does not require Z≠X) in the selection model. By assuming a functional form for the distribution of the error terms, one rules out the possibility that the conditional mean of the outcome equation equals the conditional control function, and thus you can correct for selection without exclusion restrictions. See also Heckman and Navarro (2004).
