  Instrumental variables (IV) estimation suffers selection bias when the
analysis conditions on the treatment. Judea Pearl's early graphical definition
of instrumental variables explicitly prohibited conditioning on the treatment.
Nonetheless, the practice remains common. In this paper, we derive exact
analytic expressions for IV selection bias across a range of data-generating
models, and for various selection-inducing procedures. We present four sets of
results for linear models. First, IV selection bias depends on the conditioning
procedure (covariate adjustment vs. sample truncation). Second, IV selection
bias due to covariate adjustment is the limiting case of IV selection bias due
to sample truncation. Third, in certain models, the IV and OLS estimators under
selection bound the true causal effect in large samples. Fourth, we
characterize situations where IV remains preferred to OLS despite selection on
the treatment. These results broaden the notion of IV selection bias beyond
sample truncation, replace prior simulation findings with exact analytic
formulas, and enable formal sensitivity analyses.
