We study linear regressions with period and group fixed effects , with several treatment variables. We show that under a parallel trends assumption, the coefficient of each treatment identifies the sum of two terms. The first term is a weighted sum of the average effect of that treatment in each group and period, with weights that may be negative  . The second term is a weighted sum of the average effect of the other treatments in each group and period , with weights that may again be negative . Accordingly, the treatment coefficients in those regressions are not robust to heterogeneous effects across groups and over time , and may also be contaminated by the effect of other treatments. When a single treatment is interacted with mutually exclusive binary variables to investigate treatment effect heterogeneity, our result implies that the coefficient for a given subgroup may actually be contaminated by treatment effects in other subgroups. We propose an alternative estimator that does not suffer from those issues .
We study regressions with period and group fixed effects and several treatment variables. Under a parallel trends assumption, the coefficient on each treatment identifies the sum of two terms. The first term is a weighted sum of the effect of that treatment in each group and period, with weights that may be negative and sum to one . The second term is a sum of the effects of the other treatments , with weights summing to zero . Accordingly, coefficients in those regressions are not robust to heterogeneous effects , and may be contaminated by the effect of other treatments. We propose alternative estimators .
D
linear
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, with
and
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We show that under
Under
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of
on
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average
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and sum to one
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weighted
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average effect
effects
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in each group and period
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that may again be negative
summing to zero
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the treatment
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across groups and over time
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also
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When a single treatment is interacted with mutually exclusive binary variables to investigate treatment effect heterogeneity, our result implies that the coefficient for a given subgroup may actually be contaminated by treatment effects in other subgroups. We propose an alternative estimator that does not suffer from those issues
We propose alternative estimators