  In studies of discrimination, researchers often seek to estimate a causal
effect of race or gender on outcomes. For example, in the criminal justice
context, one might ask whether arrested individuals would have been
subsequently charged or convicted had they been a different race. It has long
been known that such counterfactual questions face measurement challenges
related to omitted-variable bias, and conceptual challenges related to the
definition of causal estimands for largely immutable characteristics. Another
concern, which has been the subject of recent debates, is post-treatment bias:
many studies of discrimination condition on apparently intermediate outcomes,
like being arrested, that themselves may be the product of discrimination,
potentially corrupting statistical estimates. There is, however, reason to be
optimistic. By carefully defining the estimand -- and by considering the
precise timing of events -- we show that a primary causal quantity of interest
in discrimination studies can be estimated under an ignorability condition that
may hold approximately in some observational settings. We illustrate these
ideas by analyzing both simulated data and the charging decisions of a
prosecutor's office in a large county in the United States.
