We consider the estimation of the effect of a policy or treatment, using panel data where different groups of units are exposed to  the treatment at different times. We focus on parameters aggregating instantaneous and dynamic treatment effects, as a way to evaluate the welfare effects of the policies that occurred over the duration of the  panel. We show that under common trends conditions, these parameters can be unbiasedly estimated by weighted sums of differences-in-differences , provided that at least one group is always untreated, and another group is always treated . Our estimators are valid if the treatment effect is heterogeneous, contrary to the commonly-used event-study regression . We also propose estimators of a dynamic linear model, with group-specific but time-invariant effects of the current and lagged treatments, which may be used to evaluate ex-ante the effect of future policies  .
We consider the estimation of the effect of a treatment, using panel data where groups of units are exposed to different doses of the treatment at different times. We consider two sets of parameters of interest. The first are the average effects of having changed treatment for the first time \ell periods ago. Those parameters generalize the average effect of having started receiving the treatment \ell periods ago that has often been estimated in applications with a binary treatment and staggered adoption. We also consider cost-benefit ratios a planner may use to compare the treatments actually assigned to a counterfactualstatus quo scenario where groups receive their period-one treatment throughout the panel. We show that under common trends conditions, all these parameters are unbiasedly estimated by weighted sums of differences-in-differences . Our estimators are valid if the treatment effect is heterogeneous, contrary to the commonly-used dynamic two-way fixed effects regressions . We also propose estimators of a dynamic linear model, with group-specific but time-invariant effects of the current and lagged treatments, which may be used to evaluate ex-ante the effect of future policies . In an application, we find that our estimators differ substantially from those obtained with dynamic two-way fixed effects regressions .
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policy or
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different
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different doses of
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focus on parameters aggregating instantaneous and dynamic treatment effects, as a way to evaluate the welfare effects of the policies that occurred over the duration of the
consider two sets of parameters of interest. The first are the average effects of having changed treatment for the first time \ell periods ago. Those parameters generalize the average effect of having started receiving the treatment \ell periods ago that has often been estimated in applications with a binary treatment and staggered adoption. We also consider cost-benefit ratios a planner may use to compare the treatments actually assigned to a counterfactual
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status quo
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scenario where groups receive their period-one treatment throughout the
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these parameters can be
all these parameters are
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, provided that at least one group is always untreated, and another group is always treated
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event-study regression
dynamic two-way fixed effects regressions
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. In an application, we find that our estimators differ substantially from those obtained with dynamic two-way fixed effects regressions