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Conditional Mean Independence 301

assumption does not. For example, consider an experiment to study the effect on grades in
econometrics of mandatory versus optional homework. Among economics majors (X2i = 1),
75% are assigned to the treatment group (mandatory homework: X1i = 1), while among non-
economics majors (X2i = 0), only 25% are assigned to the treatment group. Because treatment
is randomly assigned within majors and within nonmajors, ui is independent of X1i, given X2i,
so in particular, E(ui ͉ X1i, X2i) = E(ui ͉ X2i). If choice of major is correlated with other charac-
teristics (like prior math) that determine performance in an econometrics course, then
E(ui͉ X2i) 0, and the regression of the final exam grade (Yi) on X1i alone will be subject to
omitted variable bias (X1i is correlated with major and thus with other omitted determinants
of grade). Including major (X2i) in the regression eliminates this omitted variable bias (treat-
ment is randomly assigned, given major), making the OLS estimator of the coefficient on X1i
an unbiased estimator of the causal effect on econometrics grades of requiring homework.
However, the OLS estimator of the coefficient on major is not unbiased for the causal effect of
switching into economics because major is not randomly assigned and is correlated with other
omitted factors that would not change (like prior math) were a student to switch majors.