SSRN Author: Clément de ChaisemartinClément de Chaisemartin SSRN Content
https://www.ssrn.com/author=2953365
https://www.ssrn.com/rss/en-usThu, 25 Nov 2021 01:13:27 GMTeditor@ssrn.com (Editor)Thu, 25 Nov 2021 01:13:27 GMTwebmaster@ssrn.com (WebMaster)SSRN RSS Generator 1.0REVISION: Two-way Fixed Effects Regressions with Several TreatmentsWe 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 that are robust to heterogeneous effects, and that do not suffer from the contamination problem.
https://www.ssrn.com/abstract=3751060
https://www.ssrn.com/2079605.htmlWed, 24 Nov 2021 04:09:38 GMTREVISION: Two-way Fixed Effects Regressions with Several TreatmentsWe 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 that are robust to heterogeneous effects, and that do not suffer from the contamination problem.
https://www.ssrn.com/abstract=3751060
https://www.ssrn.com/2066248.htmlWed, 06 Oct 2021 15:22:35 GMTREVISION: Two-way Fixed Effects Regressions with Several TreatmentsWe 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.
https://www.ssrn.com/abstract=3751060
https://www.ssrn.com/2063884.htmlTue, 28 Sep 2021 13:26:04 GMTREVISION: Difference-in-Differences Estimators of Intertemporal Treatment EffectsWe 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 counterfactual status 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 ...
https://www.ssrn.com/abstract=3731856
https://www.ssrn.com/2060808.htmlFri, 17 Sep 2021 13:18:15 GMTREVISION: Two-way Fixed Effects Regressions with Several TreatmentsWe 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.
https://www.ssrn.com/abstract=3751060
https://www.ssrn.com/2060802.htmlFri, 17 Sep 2021 13:15:14 GMTREVISION: The Minimax Estimator of the Average Treatment Effect, among Linear Combinations of Estimators of Bounded Conditional Average Treatment EffectsI consider estimation of the average treatment effect (ATE), in a population composed of $G$ groups, when one has unbiased and uncorrelated estimators of each group's conditional average treatment effect (CATE). These conditions are met in stratified randomized experiments. I assume that the outcome is homoscedastic, and that each CATE is bounded in absolute value by $B$ standard deviations of the outcome, for some known $B$. I derive, across all linear combinations of the CATEs' estimators, the estimator of the ATE with the lowest worst-case mean-squared error. This minimax-linear estimator assigns a weight equal to group $g$'s share in the population to the most precisely estimated CATEs, and a weight proportional to one over the estimator's variance to the least precisely estimated CATEs. I also derive the minimax-linear estimator when the CATEs' estimators are positively correlated, a condition that may be met by differences-in-differences estimators in staggered adoption ...
https://www.ssrn.com/abstract=3846618
https://www.ssrn.com/2057554.htmlTue, 07 Sep 2021 10:32:15 GMTREVISION: The Minimax Estimator of the Average Treatment Effect, among Linear Combinations of Conditional Average Treatment Effects EstimatorsI consider the estimation of the average treatment effect (ATE), in a population that can be divided into $G$ groups, and such that one has unbiased and uncorrelated estimators of the conditional average treatment effect (CATE) in each group. These conditions are for instance met in stratified randomized experiments. I assume that the outcome is homoscedastic, and that each CATE is bounded in absolute value by $B$ standard deviations of the outcome, for some known constant $B$. I derive, across all linear combinations of the CATEs' estimators, the estimator of the ATE with the lowest worst-case mean-squared error. This estimator assigns a weight equal to group $g$'s share in the population to the most precisely estimated CATEs, and a weight proportional to one over the estimator's variance to the least precisely estimated CATEs. Given $B$, this optimal estimator is feasible: the weights only depend on known quantities. I then allow for positive covariances known up to the outcome's ...
https://www.ssrn.com/abstract=3846618
https://www.ssrn.com/2054210.htmlThu, 26 Aug 2021 14:44:00 GMTREVISION: The Minimax Estimator of the Average Treatment Effect, among Linear Combinations of Conditional Average Treatment Effects EstimatorsI consider the estimation of the average treatment effect (ATE), in a population that can be divided into $G$ groups, and such that one has unbiased and uncorrelated estimators of the conditional average treatment effect (CATE) in each group. These conditions are for instance met in stratified randomized experiments. I assume that the outcome is homoscedastic, and that each CATE is bounded in absolute value by $B$ standard deviations of the outcome, for some known constant $B$. I derive, across all linear combinations of the CATEs' estimators, the estimator of the ATE with the lowest worst-case mean-squared error. This estimator assigns a weight equal to group $g$'s share in the population to the most precisely estimated CATEs, and a weight proportional to one over the estimator's variance to the least precisely estimated CATEs. This optimal estimator is feasible: the weights only depend on known quantities. I then allow for positive covariances known up to the outcome's variance ...
https://www.ssrn.com/abstract=3846618
https://www.ssrn.com/2049810.htmlWed, 11 Aug 2021 09:00:45 GMTREVISION: Two-way Fixed Effects Regressions with Several TreatmentsWe 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.
https://www.ssrn.com/abstract=3751060
https://www.ssrn.com/2043462.htmlTue, 20 Jul 2021 13:37:44 GMTREVISION: Two-way Fixed Effects Regressions with Several TreatmentsWe 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.
https://www.ssrn.com/abstract=3751060
https://www.ssrn.com/2041569.htmlMon, 12 Jul 2021 10:44:25 GMTREVISION: Difference-in-Differences Estimators of Intertemporal Treatment EffectsWe 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 counterfactual status 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 ...
https://www.ssrn.com/abstract=3731856
https://www.ssrn.com/2033216.htmlSat, 12 Jun 2021 10:15:38 GMTREVISION: The Minimax Estimator of the Average Treatment Effect, among Linear Combinations of Conditional Average Treatment Effects EstimatorsI consider the estimation of the average treatment effect (ATE), in a population that can be divided into $G$ groups, and such that one has unbiased and uncorrelated estimators of the conditional average treatment effect (CATE) in each group. These conditions are for instance met in stratified randomized experiments. I assume that the outcome is homoscedastic, and that each CATE is bounded in absolute value by $B$ standard deviations of the outcome, for some known constant $B$. I derive, across all linear combinations of the CATEs' estimators, the estimator of the ATE with the lowest worst-case mean-squared error. This optimal estimator assigns a weight equal to group $g$'s share in the population to the most precisely estimated CATEs, and a weight proportional to one over the CATE's variance to the least precisely estimated CATEs. This optimal estimator is feasible: the weights only depend on known quantities. I then allow for positive covariances known up to the outcome's variance ...
https://www.ssrn.com/abstract=3846618
https://www.ssrn.com/2029096.htmlTue, 01 Jun 2021 10:16:12 GMTREVISION: The Minimax Estimator of the Average Treatment Effect, among Linear Combinations of Conditional Average Treatment Effects EstimatorsI consider the estimation of the average treatment effect (ATE), in a population that can be divided into G groups, and such that one has unbiased and uncorrelated estimators of the conditional average treatment effect (CATE) in each group. These conditions are for instance met in stratified randomized experiments. I first assume that the outcome is homoscedastic, and that each CATE is bounded in absolute value by $B$ standard deviations of the outcome, for some known constant $B$. I derive, across all linear combinations of the CATEs' estimators, the estimator of the ATE with the lowest worst-case mean-squared error. This optimal estimator assigns a weight equal to group $g$'s share in the population to the most precisely estimated CATEs, and a weight proportional to one over the CATE's variance to the least precisely estimated CATEs. This optimal estimator is feasible: the weights only depend on known quantities. I then allow for heteroskedasticity and for positive correlations ...
https://www.ssrn.com/abstract=3846618
https://www.ssrn.com/2025948.htmlTue, 18 May 2021 14:39:15 GMTREVISION: At What Level Should One Cluster Standard Errors in Paired and Small-Strata Experiments?In paired experiments, units are matched into pairs, and one unit of each pair is randomly assigned to treatment. To estimate the treatment effect, researchers often regress their outcome on a treatment indicator and pair fixed effects, clustering standard errors at the unit-of-randomization level. We show that the variance estimator in this regression may be severely downward biased: under constant treatment effect, its expectation equals 1/2 of the true variance. Instead, we show that researchers should cluster their standard errors at the pair level. Using simulations, we show that those results extend to stratified experiments with few units per strata.
https://www.ssrn.com/abstract=3520820
https://www.ssrn.com/2021847.htmlThu, 06 May 2021 08:37:10 GMTREVISION: Two-way Fixed Effects Regressions with Several TreatmentsWe 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 ...
https://www.ssrn.com/abstract=3751060
https://www.ssrn.com/2014948.htmlSat, 17 Apr 2021 09:32:44 GMTREVISION: Difference-in-Differences Estimators of Intertemporal Treatment EffectsWe 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.
https://www.ssrn.com/abstract=3731856
https://www.ssrn.com/1980417.htmlWed, 13 Jan 2021 15:21:15 GMTREVISION: Two-way Fixed Effects Regressions with Several TreatmentsWe 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. We propose an alternative estimator that does not suffer from those issues.
https://www.ssrn.com/abstract=3751060
https://www.ssrn.com/1980415.htmlWed, 13 Jan 2021 15:18:51 GMT