Clément De

Clément De Chaisemartin
email: clement.dechaisemartin@sciencespo.fr

Trading-off Bias and Variance in Stratified Experiments and in Staggered Adoption Designs, Under a Boundedness Condition on the Magnitude of the Treatment Effect

Keywords: bias-variance trade-off, average treatment effect, mean-squared error, minimaxlinear estimator, bounded normal mean model, stratified randomized experiments, differencesin-differences, staggered adoption designs, shrinkage. JEL Codes: C21, C23

Introduction

I consider the estimation of the average treatment effect (ATE), in a population that can be divided into G groups. I assume that one has unbiased estimators of the conditional average treatment effect (CATE) in each group, with heterogeneous levels of statistical precision. This situation often arises in stratified randomized controlled trials (RCTs), where the treatment probability may vary across strata, thus leading to more (resp. less) precisely estimated CATEs in strata where the treatment probability is close to (resp. far from) 1/2. In such instances, applied researchers typically regress the outcome on the treatment and strata fixed effects. This estimator downweights the least balanced strata, and it often has a lower variance than the unbiased propensity score estimator. But it is also biased if the CATEs vary across strata. Despite its ubiquitousness, it is unclear whether the fixed effects estimator still dominates the unbiased one when bias is accounted for. Another instance where one has unbiased estimators of CATEs with heterogeneous levels of statistical precision are staggered adoption designs, where units adopt a treatment at heterogeneous dates. Then, the long-run treatment effects arising at the end of the panel are often less precisely estimated than the short-run ones, because few units are still untreated and can be used as controls at the end of the panel. Then again, one may want to downweight those long-run effects to reduce variance, but doing so may create bias if short-and long-run effects differ.

To study this bias-variance trade-off, I derive the linear combination of CATEs' estimators with the lowest worst-case mean-squared error, the minimax-linear estimator.

To do so, I first assume that the estimators of the CATEs are uncorrelated, 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. Under those assumptions, I show that the minimax-linear estimator is a weighted sum of the CATEs' estimators, with positive weights that sum to less than 1. The most precisely estimated CATEs receive a weight equal to the share of the population their group accounts for. The least precisely estimated CATEs receive a weight proportional to one over the estimator's variance, and shrunk towards zero. Given B, the optimal estimator is feasible: it only depends on known quantities.

The assumptions outlined in the previous paragraph may be applicable to stratified RCTs.

There, CATE estimators are unbiased and uncorrelated by design. Moreover, normalizing by the outcome's standard deviation is a common practice, and applied researchers often have a good sense, based on the literature, of the effect size that a given intervention may produce. In such instances, they may be able to come up with plausible values of B. When the literature is silent as to which values of B are plausible upper bounds for the CATEs, researchers can conduct a sensitivity analysis by varying B. Finally, my approach also assumes homoscedasticity, but I show that with heteroscedasticity, the minimax-linear estimator still has a lower worst-case MSE than the unbiased one whenever the treated outcome's variance is larger than that of the untreated outcome. When the opposite holds, I show that the minimax-linear estimator still has a lower worst-case MSE, provided the ratio of the treated and untreated outcomes variances is not below a bound that can be readily computed from the data. In my application in Section 5, this bound is equal to 0.06, so the treated outcome's variance would have to be more than 94% smaller than the untreated outcome's for the 2 Electronic copy available at: https://ssrn.com/abstract=3846618 minimax-linear estimator to have a higher worst-case MSE than the unbiased one.

In stratified RCTs, my result implies that the minimax-linear estimator is actually "in between" the unbiased and fixed effects estimators. The unbiased estimator assigns to each CATE a weight equal to group g's share in the population, like the minimax-linear estimator does to the most precisely estimated CATEs, while the fixed effects estimator assigns to each CATE a weight proportional to one over its variance, like the minimax-linear estimator does to the least precisely estimated CATEs.

Then, I replace the assumption that the estimators are uncorrelated by the assumption that they are positively correlated, with covariances known up to the outcome's variance. Under this weaker assumption, I show that the minimax-linear estimator, across all linear combinations of the estimators with positive weights, is the solution of an easy-to-numerically-solve minimization problem, and is still feasible.

This second set-up may be applicable to the differences-in-differences (DID) estimators for staggered adoption designs proposed by [START_REF] Sun | Estimating dynamic treatment effects in event studies with heterogeneous treatment effects[END_REF] I set B = 0.75, meaning that CATEs should all be lower than 75% of the outcome's standard deviation. I find that the minimax-linear estimator downweights some CATEs at period 5. Those are the least precisely estimated CATEs, because there are few units that are still untreated and can be used as controls at that period. If outcomes are indeed iid, the minimaxlinear estimator has substantially lower variance and worst-case MSE than the unbiased one.

The minimax-linear estimator still has substantially lower variance and worst-case MSE if each unit's outcome follows an AR(1), provided the AR(1) coefficient is not too close to 1.

Finally, I consider a number of extensions. I show that assuming that CATEs are positive and bounded, rather than just assuming that they are bounded, does not change the minimaxlinear estimator. I also characterize the minimax-linear estimator with heteroscedasticity.

Its formula remains the same as under homoscedasticity, but it is not feasible anymore: it depends on the estimators' variances, that are typically unknown. A feasible estimator can easily be computed, by replacing those variances by their estimators. [START_REF] Behaghel | Ready for boarding? the effects of a boarding school for disadvantaged students[END_REF], who use a stratified RCT to measure the effect of a boarding school for disadvantaged students in France on students' math test scores.

I use my results to revisit

Their RCT has 363 students and 14 strata. The treatment probability varies substantially across strata, but no stratum has a treatment probability, say, lower than 0.1 or higher than 0.9. Based on a review of several papers that have estimated CATEs of similar interventions (see [START_REF] Curto | The potential of urban boarding schools for the poor: Evidence from seed[END_REF][START_REF] Dobbie | Are high-quality schools enough to increase achievement among the poor? evidence from the harlem children's zone[END_REF][START_REF] Angrist | Inputs and impacts in charter schools: Kipp lynn[END_REF][START_REF] Abdulkadiroğlu | Accountability and flexibility in public schools: Evidence from boston's charters and pilots[END_REF], I argue in Section 5 that 50% of a standard deviation of students' test scores is a plausible upper bound for CATEs in this application. With that value of B, the robust standard error of the minimax-linear estimator is 9% smaller than that of the unbiased estimator, and its worst-case MSE under homoscedasticity is 6% smaller. In the subsample of male students, its robust standard error and worst-case MSE are 15% and 12% smaller.

This paper is related to the pioneering work of [START_REF] Crump | Dealing with limited overlap in estimation of average treatment effects[END_REF], who consider matching estimators when the treatment probability can be close to zero or one conditional on some values of the covariates. My estimator can also be used in such instances, provided the covariates take a finite number of values. [START_REF] Crump | Dealing with limited overlap in estimation of average treatment effects[END_REF] propose to redefine the target parameter as the ATE in the subpopulation whose ATE can be estimated most precisely. In other words, [START_REF] Crump | Dealing with limited overlap in estimation of average treatment effects[END_REF] minimize variance, but do not control bias with respect to the original target parameter (the ATE in the full population). Instead, I propose to use the minimax-linear estimator. This avoids changing the target parameter. But this requires taking a stand on how large CATEs may be, by specifying an upper bound B for them.

Below, I show that when B → +∞, meaning that one does not restrict the magnitude of the CATEs, the minimax-linear estimator converges towards the unbiased one, and the trade-off between bias and variance becomes trivial. Finally, note that the main result in [START_REF] Crump | Dealing with limited overlap in estimation of average treatment effects[END_REF] is derived under homoscedasticity, like the main results in this paper.

This paper is also related to a vast literature in statistics and econometrics, that has studied linear-and affine-minimax estimators. The setting I consider can be cast as a bounded normal mean model, where realizations of G normally distributed random variables are used to estimate a linear combination of their means, which are assumed to be bounded. 1 Donoho (1994), who considers a more general setup than the bounded normal mean model, characterizes the risk of the affine-minimax estimator, and shows that it cannot be more than 25% larger than that of the minimax estimator. [START_REF] Armstrong | Optimal inference in a class of regression models[END_REF] consider a similar set-up as [START_REF] Donoho | Statistical estimation and optimal recovery[END_REF], and show how to construct optimal confidence intervals. My paper makes the following contributions. First, it is the first to apply the bounded normal mean model to derive the minimax-linear estimator of the ATE, under boundedness conditions on the CATEs. Thus, it complements a growing econometrics literature that has applied the setup in [START_REF] Donoho | Statistical estimation and optimal recovery[END_REF] to other estimation problems. 2 Second, the closed-form expression of the minimax-linear estimator that I derive when CATEs are uncorrelated is, to my knowledge, 1 I do not assume that CATEs' estimators are normally distributed, but as noted by Armstrong & Kolesár (2021a), this distributional assumption is not of essence to derive the minimax-linear estimator.

2 Examples include: ATE estimation under uncounfoundedness when the mean outcome conditional on the covariates is a Lipschitz function with a bounded Lipschitz constant (see Armstrong & Kolesár, 2021a); sensitivity analysis in locally misspecified GMM models (see [START_REF] Armstrong | Sensitivity analysis using approximate moment condition models[END_REF]; DID estimation with new. In their Section 4.2., [START_REF] Ibragimov | On nonparametric estimation of the value of a linear functional in gaussian white noise[END_REF] Armstrong & Kolesár, 2021a). Fourth, the realization that with correlated CATE estimators, the minimax-linear estimator with positive weights can be approximated via a simple numerical method is also, to my knowledge, new.

The remainder of the paper is organized as follows. Section 2 presents the paper's main results. Section 3 presents some extensions. Section 4 presents some numerical examples.

Section 5 presents an empirical application.

Main results: feasible minimax-linear estimators

Throughout the paper, I consider the following set-up.

Definition 2.1 (Set-up) One is interested in estimating an unknown parameter τ , equal to a weighted average of unknown parameters (τ g ) 1≤g≤G , with weights (p g ) 1≤g≤G that are known, positive, and sum to 1:

τ = G g=1 p g τ g .
(2.1)

One observes random variables ( τ g ) 1≤g≤G such that E ( τ g ) = τ g for all g.

In the applications I consider, τ is an ATE in a population that can be divided into G groups, τ g is the CATE in group g, p g is the share of the population group g accounts for, and τ g is an unbiased estimator of τ g . The more abstract set-up in Definition 2.1 is useful to connect this paper with the bounded normal mean model (see, e.g., [START_REF] Donoho | Statistical estimation and optimal recovery[END_REF].

bounded departures from parallel trends (see [START_REF] Rambachan | An honest approach to parallel trends[END_REF]; estimation in regression discontinuity designs with bounded second derivatives of the mean of the potential outcomes conditional on the running variable (see [START_REF] Armstrong | Optimal inference in a class of regression models[END_REF][START_REF] Imbens | Optimized regression discontinuity designs[END_REF][START_REF] Noack | Bias-aware inference in fuzzy regression discontinuity designs[END_REF].

3 Relatedly, [START_REF] Berry | Minimax estimation of a bounded normal mean vector[END_REF] studies the minimax estimator in a multivariate bounded normal mean model.

However, the normality assumption is of essence to derive the minimax estimator, so his closed-form expression differs from mine. [START_REF] Vidakovic | On the efficiency of affine minimax rules in estimating a bounded multivariate normal mean: On the efficiency of affine minimax rules[END_REF] studies the minimax-affine estimator in a multivariate bounded normal mean model, assuming that the ℓ2 norm of the vector of means are bounded. Instead, I assume that its ℓ∞ norm is bounded, arguably a more natural assumption for the applications I consider.

5

Electronic copy available at: https://ssrn.com/abstract=3846618

Minimax-linear estimator with uncorrelated and homoscedastic CATE estimators

In this section, I make the following assumption.

Assumption 1 For all g ∈ {1, ..., G}:

1. For all g ′ ̸ = g, cov τ g , τ g ′ = 0.

2. There is a strictly positive unknown real number σ and positive known real numbers

(v g ) 1≤g≤G such that V ( τ g ) ≤ σ 2 v g .

The upper bound in the previous inequality is sharp.

3. There is a strictly positive known real number B such that |τ g | ≤ Bσ.

Assumption 1 requires that the estimators τ g be unbiased, uncorrelated across g, and that their variances can be bounded by the product of an unknown real number σ 2 and known real numbers v g . It also requires that the CATEs be bounded in absolute value by Bσ.

This assumption may for instance be applicable to stratified completely randomized controlled trials (RCTs), see Section 9.3.2 in [START_REF] Imbens | Causal inference in statistics, social, and biomedical sciences[END_REF]. There, groups are equal to the experimental strata. τ g is the CATE in stratum g, and τ g is just the difference between the average outcome of treated and control units in that stratum. Under the assignment mechanism in Section 9.3.2 in [START_REF] Imbens | Causal inference in statistics, social, and biomedical sciences[END_REF], τ g is unbiased for τ g , and cov τ g , τ g ′ = 0 for all g ′ ̸ = g. If one further assumes that the outcome is homoscedastic,

V ( τ g ) ≤ σ 2 1 n 0,g + 1 n 1,g ,
where n 0,g and n 1,g respectively denote the number of treated and control units in stratum g. The upper bound is sharp: it is reached if the treatment effect is homogeneous in stratum g, or if the units in stratum g are randomly drawn from a super population. In stratified RCTs, Point 3 assumes that the CATEs are all bounded in absolute value by B standard deviations of the outcome. Normalizing by the outcome's standard deviation is a common practice in applied research. Based on the literature, researchers often have a pretty good sense of the effect sizes, in percent of the outcome's standard deviation, that the intervention they consider may realistically produce. In such instances, they may be able to come up with a plausible value for B. When the literature is silent as to which value of B may be a plausible upper bound for the CATEs, researchers can conduct a sensitivity analysis by varying B.

Beyond stratified RCTs, there are other instances where this setup is applicable, including for instance treatment effect estimation under uncounfoundedness, when treatment is independent of potential outcomes conditional on covariates taking a finite number of values.

For any 1 × G deterministic vector w = (w 1 , ..., w G ), let

τ (w) = G g=1 w g τ g . (2.2) 6
Electronic copy available at: https://ssrn.com/abstract=3846618 τ (w) is a linear combination of the estimators τ g . Lemma 2.1 gives its worst-case MSE.

Lemma 2.1 (Worst-case MSE of τ (w))

If Assumption 1 holds, then for any 1 × G deterministic vector w = (w 1 , ..., w G )

E ( τ (w) -τ ) 2 ≤ M SE(w) ≡ σ 2    G g=1 w 2 g v g + B 2   G g=1 |w g -p g |   2    .
The upper bound in the previous display is sharp: it is attained if

τ g = σB (1{w g ≥ p g } -1{w g < p g }) and V ( τ g ) = σ 2 v g .
Without loss of generality, assume that

p 1 v 1 ≤ p 2 v 2 ≤ ... ≤ p G v G .
Let g = min{g ∈ {1, ..., G} :

1 1 B 2 + G g ′ =g 1 v g ′ G g ′ =g p g ′ ≤ p g v g }. g is well defined, because 1 1 B 2 + 1 v G p G ≤ p G v G .
For any h ∈ {1, ..., G}, let w h be such that

w g,h = p g for all g < h w g,h = 1 v g 1 1 B 2 + G g ′ =h 1 v g ′ G g ′ =h p g ′ for all g ≥ h. (2.3)
Finally, let

w * = argmin w∈R G M SE(w).
It follows from Lemma 2.1 that τ (w * ) is the minimax-linear estimator of τ .

Theorem 2.2 (Minimax-linear estimator of τ , with bounded CATEs)

If Assumption 1 holds, then τ (w * ) = τ (w h * ), where h * = argmin h∈{g,...,G}

M SE(w h ).

Theorem 2.2 shows that under Assumption 1, the minimax-linear estimator is a weighted sum of the τ g s, with positive weights, that sum to less than 1. For a precisely estimated τ g (one with a low value of p g v g ), the optimal weight is just p g . On the other hand, for an imprecisely estimated τ g (one with a high value of p g v g ), the optimal weight is proportional to one over v g , the non-constant part of its variance. For an imprecisely estimated τ g , the optimal weight is also shrunk towards zero, where the shrinkage depends on how close to zero B is.

Importantly, the minimax-linear estimator in Theorem 2.2 is feasible: given B, the optimal weights w * only depend on the known quantities (v g ) 1≤g≤G and (p g ) 1≤g≤G . For instance, in a stratified RCT, v g = 1/n 0,g + 1/n 1,g and p g = n g /n, where n g = n 0,g + n 1,g and n = G g=1 n g . Accordingly, the weights w * are not stochastic, and

V ( τ (w * )) ≤ σ 2 G g=1 (w * g ) 2 σ 2 0,g /n 0,g + σ 2 1,g /n 1,g , (2.4)
where σ 2 0,g and σ 2 1,g respectively denote the variances of the untreated and treated outcomes in stratum g. The right hand side in the previous display can easily be estimated, thus giving rise to a conservative variance estimator V ( τ (w * )). If the sample size is large enough for τ (w * )/V ( τ (w * )) to be approximately normally distributed,4 one may use τ (w * ) and V ( τ (w * )) to construct conservative confidence intervals for G g=1 w * g τ g . Deriving conservative confidence intervals for τ would require accounting for the estimator's bias. I refer the reader to [START_REF] Armstrong | Optimal inference in a class of regression models[END_REF] for confidence intervals trading-off bias and variance optimally.

One always has w *

G < p G , so τ (w * ) never coincides with the unbiased estimator τ (p). In stratified RCTs, τ (w * ) is somewhere "in between" the unbiased and strata fixed effects estimators. Let p = (p 1 , ..., p G ). The unbiased estimator is equal to τ (p). Let β f e be the coefficient of D ig in the regression of Y ig on a constant, D ig and strata fixed effects. Let

w f e =    1 n 0,1 + 1 n 1,1 -1 G g=1 1 n 0,g + 1 n 1,g -1 , ..., 1 n 0,G + 1 n 1,G -1 G g=1 1 n 0,g + 1 n 1,g -1    .
It follows from, e.g., (3.3.7) in [START_REF] Angrist | Mostly harmless econometrics: An empiricist's companion[END_REF], that τ (w f e ) = β f e . τ (w * ) assigns to precisely estimated τ g s the same weights as the unbiased estimator, but it assigns to imprecisely estimated τ g s weights proportional to those used by the strata fixed effects estimator, shrunk towards zero. Note that in a stratified RCT,

p g v g = 1/(np d,g (1 -p d,g )),
where p d,g = n 1,g /n g is the proportion of treated units in strata g. Accordingly, precisely (resp. imprecisely) estimated τ g s are those for which p d,g is close to (resp. far from) 1/2. Corollary 2.3 below shows that when B → +∞, meaning that one does not restrict the magnitude of the CATEs, τ (w * ) converges towards τ (p), the unbiased estimator.

Corollary 2.3 (Minimax-linear estimator of τ when

B → +∞) lim B→+∞ w * = (p g ) g∈{1,...,G} .
When the bias that can arise from downweighting one of the τ g s goes to infinity, bias dominates variance and the unbiased estimator becomes optimal.

Operationally, to find the minimax-linear estimator, one just needs to compute g, and then evaluate M SE(w) at w h for h ∈ {g, ..., G}. The following lemma shows that to compute g, one just needs to evaluate the inequalities

1 G g ′ =g 1 v g ′ G g ′ =g p g ′ ≤ p g v g for g = G-1, g = G-2,
etc., until one finds a first value where the inequality fails. g is equal to that value plus one.

Lemma 2.2 1 G g ′ =g 1 v g ′ G g ′ =g p g ′ ≤ p g v g ⇒ 1 G g ′ =g+1 1 v g ′ G g ′ =g+1 p g ′ ≤ p g+1 v g+1 .
Finally, I give sufficient conditions under which the minimax-linear estimator still has lower worst-case MSE than the unbiased one with heteroscedasticity. As this is not of essence, below I omit that CATEs' variances can sometimes only be conservatively estimated.

Assumption 2

1. For all g ∈ {1, ..., G}, there is a strictly positive unknown real number σ, unknown real numbers (h g ) 1≤g≤G such that h g ≥ 1 for all g, and positive known real

numbers (v 0,g , v 1,g ) 1≤g≤G , such that V ( τ g ) = σ 2 (v 0,g + h g v 1,g ).
2. For all g ∈ {1, ..., G}, there is a strictly positive unknown real number σ, an unknown real number h such that 0 ≤ h ≤ 1, and positive known real numbers

(v 0,g , v 1,g ) 1≤g≤G , such that V ( τ g ) = σ 2 (v 0,g + hv 1,g ).
In a stratified RCT, Point 1 of Assumption 2 holds if the untreated outcome's variance does not vary across strata, as is the case when in each stratum researchers standardize their outcome by its standard deviation among the stratum's control group, and if in each stratum the variance of the treated outcome is larger than that of the untreated one. Then, σ 2 is the untreated outcome's variance, while h g is the ratio of the treated and untreated outcomes' variances in strata g. Point 2 instead holds if the variance of the treated outcome is lower than that of the untreated one, and if heteroscedasticity is constant across strata.

Corollary 2.4 (Ordering of τ (w * )'s and τ (p)'s worst-case MSE with heteroskedasticity)

1. If Points 1 and 3 of Assumption 1 and Point 1 of Assumption 2 hold, the worst-case MSE of τ (w * ) is lower than that of τ (p).

If Points 1 and 3 of Assumption 1 and Point 2 of Assumption 2 hold, and if

h ≥ B 2 G g=1 |w * g -p g | 2 -G g=1 ((p g ) 2 -(w * g ) 2 )v 0,g G g=1 ((p g ) 2 -(w * g ) 2 )v 1,g
, the worst-case MSE of τ (w * ) is lower than that of τ (p).

In stratified RCTs, Point 1 of Corollary 2.4 implies that the worst-case MSE of τ (w * ) is lower than that of τ (p), if the variance of the outcome with treatment is larger than that of the outcome without treatment. If heteroscedasticity goes in the other direction and does not vary across strata, Point 2 shows that the worst-case MSE of τ (w * ) is still lower than that of τ (p) if the ratio of the treated and untreated outcomes' variances is greater than a lower bound which only depends on the design and can be readily computed. In the application in Section 5, this lower bound is equal to 0.06 so the worst-case MSE of τ (w * ) can only be higher than that of of τ (p) under implausible amounts of heteroscedasticity. Overall, it seems that τ (w * )'s worst-case MSE is still lower than τ (p)'s under realistic heteroscedasticity.5

Minimax-linear estimator with correlated homoscedastic CATE estimators

In this section, I replace Point 1 of Assumption 1 by the following assumption.

Assumption 3 There are known, positive real numbers (c g,g ′ ) 1≤g̸ =g ′ ≤G such that for all g ′ ̸ = g, c g,g ′ = c g ′ ,g and cov τ g , τ g ′ = σ 2 c g,g ′ .

Assumption 3 allows for covariances between the τ g s, but requires that their covariances be equal to the product of the outcome's variance σ 2 , and known real numbers c g,g ′ .

This framework may for instance be applicable to differences-in-differences (DID) estimators in staggered adoption designs. Assume one has a panel of N units observed over T periods.

The design is staggered, meaning that every unit's treatment D i,t is weakly increasing over time: D i,t ≥ D i,t-1 . Let t i be the first date at which unit i is treated. For every k ∈ {1, ..., T }, let N k = N i=1 1{t i = k} be the number of units that start receiving the treatment at period k. For every t ≥ k, let τ k,t denote the average effect of the treatment at period t among units that started receiving it at period k. τ , the average treatment effect on the treated (ATT), can be decomposed as follows:

τ = k:N k >0 T t=k N k N 1 τ k,t , (2.5)
where N 1 = i,t D i,t is the number of treated units. Therefore, (2.1) holds, with a group g being a pair (k, t).

Callaway & Sant'Anna (2020), [START_REF] Sun | Estimating dynamic treatment effects in event studies with heterogeneous treatment effects[END_REF], and de Chaisemartin & D'Haultfoeuille (2020) propose estimators of τ k,t that can all be written as

τ k,t = 1 N k i:t i =k (Y i,t -Y i,k-1 ) - 1 N Ct j∈Ct (Y j,t -Y j,k-1 ),
where C t is a set of control units at t and N Ct is the number of units in that set. To simplify, I assume below that C T ̸ = ∅: there are never-treated units. Then, C t are the never treated units Assume that the potential outcomes without being ever treated Y i,t (0) are independent across (i, t) and homoscedastic with variance σ 2 , that the treatment effects are not stochastic, and that the treatments are non-stochastic. Those are essentially the assumptions of the Gauss-Markov Theorem [START_REF] Borusyak | Revisiting event study designs: Robust and efficient estimation[END_REF]. Then, for any 1

≤ k ≤ t ≤ T , V ( τ k,t ) = 2σ 2 1 N k + 1 N Ct , (2.6)
and for any 1

≤ k ≤ t ≤ T , 1 ≤ k ′ ≤ t ′ ≤ T , k < k ′ , and t < t ′ , cov( τ k,t , τ k,t ′ ) = σ 2 1 N k + 1 N Ct cov( τ k,t , τ k ′ ,t ) = σ 2 1 N Ct cov( τ k,t , τ k ′ ,t ′ ) = 0, ( 2
E ( τ (w) -τ ) 2 ≤ M SE 2 (w) ≡ σ 2    G g=1   w 2 g v g + G g ′ ̸ =g w g w g ′ c g,g ′   + B 2   G g=1 |w g -p g |   2    .
The upper bound in the previous display is sharp: it is attained if

τ g = B (1{w g ≥ p g } -1{w g < p g }) and V ( τ g ) = σ 2 v g . Minimizing M SE 2 (w) across all w such that w g ≥ 0 for all g is equivalent to minimizing G g=1   w 2 g v g + G g ′ ̸ =g w g w g ′ c g,g ′   + B 2   G g=1 (p g -w g )   2 , (2.8) subject to 0 ≤ w g ≤ p g .
While it does not have a closed-form solution, the minimization problem in (2.8) is easy to solve numerically. Given B, this problem is feasible, as it only depends on known quantities.

Accordingly, the optimal weights are non-stochastic, and the variance of the minimax-linear estimator can easily be estimated if one has an estimator of the variance-covariance matrix of ( τ g ) g∈{1,...,G} . In the DID staggered design example, the numbers (v g ) g∈{1,...,G} and (c g,g ′ ) 1≤g̸ =g ′ ≤G depend on the design, so the optimal weights are non-stochastic conditional on the design. Note that the estimator in Theorem 2.5 is minimax across all linear combinations of the τ g s with positive weights. Extending that result to allow for negative weights may be achieved by developing an algorithm similar to the LASSO-LAR algorithm (see [START_REF] Efron | Least angle regression[END_REF][START_REF] Rosset | Piecewise linear regularized solution paths[END_REF][START_REF] Rosset | Piecewise linear regularized solution paths[END_REF] to minimize M SE 2 (w), as Armstrong & Kolesár (2021b) do in the overidentified GMM setup they consider. This extension is left for future work.

Extensions

Feasible minimax-linear estimators with positive CATEs

In this section, I consider again the setup of Section 2.1, but I add the following assumption:

11

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Assumption 4 For all g ∈ {1, ..., G}, 0 ≤ τ g .

Assumption 4 is applicable when CATEs are known to all be positive, on top of being bounded.

It is plausible when the treatment can ex-ante be assumed to not be detrimental. Theorem 3.1 below still holds if one instead assumes that CATEs are all negative. Let

M SE + (w) = σ 2    G g=1 w 2 g v g + B 2 max      G g=1 (w g -p g )1{w g > p g }   2 ,   G g=1 (w g -p g )1{w g < p g }   2       .
Theorem 3.1 (Minimax-linear estimator of τ , with bounded and positive CATEs) If Assumptions 1 and 4 hold, then for any 1 × G deterministic vector w = (w 1 , ..., w G )

E ( τ (w) -τ ) 2 ≤ M SE + (w).
The upper bound in the previous display is sharp. τ (w h * ) = argmin

w∈R G M SE + (w).
Perhaps surprisingly, Theorem 3.1 shows that the minimax-linear estimator is the same when one assumes positive and bounded CATEs and when one only assumes bounded CATEs.

Assuming τ g ≥ 0 can only change the worst-case squared bias of τ (w), not its variance.

Accordingly, the conclusion of Theorem 3.1 still holds with correlated CATE estimators: there as well, assuming positive CATEs does not change the minimax-linear estimator.

Rather than boundedness and sign restrictions, there are alternative restrictions on CATEs one may wish to consider. For instance, one may wish to assume that each CATE is no more than Bσ away from the ATE: |τ g -τ | ≤ Bσ. Deriving the minimax-linear estimator under such restrictions is more complicated than under the restrictions I consider in this paper: as τ is a weighted average of the τ g s, deriving a closed form of a sharp upper bound of the squared bias of τ (w) is not straightforward. This extension is left for future work.

Infeasible minimax-linear estimators without homoscedasticity

A result similar to Theorem 2.2 still holds without the homoscedasticity assumption, and under a modified version of Point 3 in Assumption 1:

Assumption 5 For all g ∈ {1, ..., G}: there is a strictly positive known real number B such

that |τ g | ≤ B.
Without loss of generality, assume that

p 1 V ( τ 1 ) ≤ p 2 V ( τ 2 ) ≤ ... ≤ p G V ( τ G ).
Let g 2 = min{g ∈ {1, ..., G} :

1 1 B 2 + G g ′ =g 1 V ( τ g ′ ) G g ′ =g p g ′ ≤ p g V ( τ g )}.
For any h ∈ {1, ..., G}, let w h,2 be such that

w g,h,2 = p g for all g < h w g,h,2 = 1 V ( τ g ) 1 1 B 2 + G g ′ =h 1 V ( τg) G g ′ =h p g ′ for all g ≥ h.
(3.1)

Finally, let

M SE 3 (w) = G g=1 w 2 g V ( τ g ) + B 2   G g=1 |w g -p g |   2 and w * 2 = argmin w∈R G M SE 3 (w).

Theorem 3.2 (Minimax-linear estimator of τ , with bounded CATEs and heteroskedasticity)

If Point 1 of Assumption 1 and Assumption 5 hold, then for any 1 × G deterministic vector

w = (w 1 , ..., w G ), E ( τ (w) -τ ) 2 ≤ M SE 3 (w).
The upper bound in the previous display is sharp: it is attained if

τ g = B (1{w g ≥ p g } -1{w g < p g }) . τ (w * 2 ) = τ (w h * 2 ), where h * 2 = argmin h∈{g 2 ,...,G} M SE 3 (w h,2 ).
Theorem 3.2 shows that without the homoscedasticity assumption, the minimax-linear estimator is still a weighted sum of the τ g , with positive weights, that sum to less than 1, and where the most precisely estimated CATEs receive a weight equal to the share of the population their group accounts for, while the least precisely estimated CATEs receive a weight proportional to one over the estimator's variance.

While the minimax-linear estimator in Theorem 2.2 is feasible, that in Theorem 3.2 is infeasible, as it depends on the variances of the τ g s, that are unknown. In most instances, it is possible to estimate those variances,6 to then form a feasible estimator τ w * 2 proxying for τ (w * 2 ). Studying the properties of τ w * 2 is left for future work.

Finally, one can also relax the assumption that the τ g s are uncorrelated, or that their correlations has the specific expression in Assumption 3.

Theorem 3.3 (Minimax-linear estimator of τ , with bounded CATEs, heteroskedasticity, and correlations)

If Assumption 5 holds, then for any 1 × G deterministic vector w = (w 1 , ..., w G )

E ( τ (w) -τ ) 2 ≤ M SE 4 (w) ≡ G g=1   w 2 g V ( τ g ) + G g ′ ̸ =g w g w g ′ cov( τ g , τ g ′ )   +B 2   G g=1 |w g -p g |  
The upper bound in the previous display is sharp: it is attained if τ g = B (1{w g ≥ p g } -1{w g < p g }) .

If cov( τ g , τ g ′ ) ≥ 0 for all (g, g ′ ), then minimizing M SE 4 (w) across all w such that w g ≥ 0 for all g is equivalent to minimizing

G g=1   w 2 g V ( τ g ) + G g ′ ̸ =g w g w g ′ cov( τ g , τ g ′ )   + B 2   G g=1 (p g -w g )   2 , (3.2) subject to 0 ≤ w g ≤ p g .
The minimization problem in (3.2) is easy to solve numerically. This problem is not feasible, as it depends on the variance-covariance matrix of ( τ g ) 1≤g≤G , which is typically unknown.

But a feasible estimator can be computed, by replacing those quantities by estimators.

Numerical examples

Stratified RCTs

In this section, I compute w * in stratified RCTs, and compare the performance of τ (w * ) to that of the unbiased and fixed effects estimators τ (p) and τ (w f e ). I first consider an RCT with two strata of 100 units (N = 200). The first strata is perfectly balanced: half of units are treated. The second strata may be imbalanced, and I vary its treatment probability p imb from 0.5 to 0.99. In each design, I compute w * assuming B = 0.75 and B = 0.5. Then, I repeat the same exercise but considering an RCT with two strata of 50 units (N = 100). This gives rise to four sets of estimators, for which results are shown on the four panels of Figure 1 below. On each panel, the blue and brown lines respectively show w * 1 and w * 2 , the weights assigned to the balanced and imbalanced strata by the minimax-linear estimator τ (w * ), as a function of p imb . The green and orange lines respectively show the ratio of the standard error and worst-case MSE of τ (w * ) and τ (p), under the homoscedasticity condition in Point 2 of Assumption 1. Finally, the black and red lines respectively show the ratio of the standard error and worst-case MSE of τ (w * ) and τ (w f e ), under that same assumption. In all panels, w * 1 starts below 0.5, and increases with p imb , until it reaches 0.5. w * 2 on the other hand is decreasing with p imb : the more imbalanced the second stratum is, the more the minimax-linear estimator downweights it. Comparing the four panels, one can see that

w * 2 is increasing in B.
The larger B is, the less the minimax-linear estimator downweights the imbalanced stratum, as downweighting it may result in more bias. w * 2 is also increasing in N . The larger N is, the lower the variances of the τ g estimators, and the less variance matters in the bias-variance trade-off. By construction, τ (w * ) always has lower worst-case MSE and standard error than τ (p). The magnitude of the difference depends on p imb , B, and N . For instance, if B = 1 and N = 200, se( τ (w * ))/se( τ (p)) ≤ 0.9 if p imb is greater than 0.89. If B = 0.5 and N = 200 (resp. B = 1 and N = 100, B = 0.5 and N = 100) 14

Electronic copy available at: https://ssrn.com/abstract=3846618 se( τ (w * ))/se( τ (p)) ≤ 0.9 if p imb ≥ 0.77 (resp. p imb ≥ 0.80, p imb ≥ 0.5). τ (w * ) also always has lower worst-case MSE than τ (w f e ), and this difference is large for values of p imb greater than 0.75. Perhaps surprisingly, τ (w * ) also almost always has lower standard error than τ (w f e ). This is due to the shrinkage embedded in τ (w * ). 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 .5

.6 .7 .8

.9 1

Treatment probability in imbalanced stratum N=200, B=0.75 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

.5 .6 .7 .8

.9 1

Treatment probability in imbalanced stratum N=200, B=0.5 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

.5 .6 .7 .8

.9 1

Treatment probability in imbalanced stratum N=100, B=0.75 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1

.5 .6 .7 .8

.9 1

Treatment probability in imbalanced stratum N=100, B=0.5

Weight balanced stratum Weight imbalanced stratum se(optimal)/se(unbiased) mse(optimal)/mse(unbiased) se(optimal)/se(ols) mse(optimal)/mse(ols)

Difference-in-difference estimators with a staggered adoption design

In this section, I consider another numerical example: a staggered adoption design with 50 units and five time periods, where 10 units get treated in periods 2, 3, 4, and 5, while 10 units remain never treated. For (k, t) ∈ {2, ..., 5} × {k, ..., 5}, let τ k,t = 1 10

i:t i =k (Y i,t -Y i,k-1 ) - 1 10(6-t) i:t i >t (Y i,t -Y i,k-1
) be an estimator of the average treatment effect at period t in the cohort that started receiving the treatment at period k. This estimator is unbiased under a parallel trends assumption (see [START_REF] Callaway | Difference-in-differences with multiple time periods[END_REF].

If one assumes that units' potential outcomes without treatment Y i,t (0) are independent across (i, t) and homoscedastic with variance σ 2 , that the treatment effects are not stochastic, and that the treatments are non-stochastic, one can use (2.7) to derive the variance-covariance matrix of ( τ k,t ) (k,t)∈{2,...,5}×{k,...,5} . Then, for B = 0.75, the solution to the minimization problem in (2.8) is w * = (0.1, 0.1, 0.1, 0.0148, 0.1, 0.1, 0.0565, 0.1, 0.1, 0.1).

The minimax-linear estimator downweights τ 2,5 and τ 3,5 , two effects estimated at the last period of the panel, when few units can be used as controls. Doing so leads to a substantial gain under Assumption 3: se( τ (w * ))/se( τ (p)) = 0.83 and mse( τ (w * ))/mse( τ (p)) = 0.82. It may often be implausible that the outcomes of the same unit are uncorrelated over time.

To assess if results are sensitive to that assumption, I assume instead that V (Y i,t (0)) = 1 for all (i, t) and cov(Y i,t (0), Y i,t ′ (0)) = ρ t ′ -t for all i and 1 ≤ t ≤ t ′ ≤ T . Those conditions for instance hold if Y i,t follows a stationnary AR(1) model with parameter ρ: Y i,1 (0) = ε i,1 , and Y i,t (0) = ρY i,t-1 (0) + 1 -ρ 2 ε i,t , with ε i,t i.i.d. with mean 0 and variance 1. Under those assumptions,

V ( τ k,t ) = 2 1 -ρ t-k+1 1 N k + 1 N Ct ,
and for any 1

≤ k ≤ t ≤ T , 1 ≤ k ′ ≤ t ′ ≤ T , k < k ′ , and t < t ′ , cov( τ k,t , τ k,t ′ ) = ρ t ′ -t -ρ t-k+1 -ρ t ′ -k+1 + 1 1 N k + 1 N Ct cov( τ k,t , τ k ′ ,t ) = 1 -ρ t-k+1 -ρ t-k ′ +1 + ρ k ′ -k 1 N Ct cov( τ k,t , τ k ′ ,t ′ ) = 1{k ′ ≤ t} ρ t ′ -t -ρ t-k ′ +1 -ρ t ′ -k+1 + ρ k ′ -k 1 N Ct .
With ρ = 0.50, se( τ (w * ))/se( τ (p)) = 0.80 and mse( τ (w * ))/mse( τ (p)) = 0.78. Results are similar with ρ = 0.25 and ρ = 0.75. With ρ = 0.9, se( τ (w * ))/se( τ (p)) = 0.76 and mse( τ (w * ))/mse( τ (p)) = 1.05. The weights w * are computed assuming independent outcomes, but in this numerical example the precision gains attached to using τ (w * ) are even larger with positively correlated outcomes. On the other hand, the worst-case MSE gains attached to using τ (w * ) persist with moderately positively correlated outcomes, but disappear with highly positively correlated outcomes.

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Overall, τ (w * ) may lead to substantial precision and MSE gains in staggered adoption designs with a number of units of the same order as the number of US states.

Application

In this section, I use the data from [START_REF] Behaghel | Ready for boarding? the effects of a boarding school for disadvantaged students[END_REF] to illustrate the results in the paper. The authors conducted a stratified RCT to estimate the effect of a boarding school for disadvantaged students in France. The boarding school's pedagogy is similar to that of "No

Excuse" charter schools in the US. It has capacity constraints at the gender × grade level, and in 2009 and 2010, the school had more applicants than seats in 14 gender × grade strata.

In each stratum, seats were randomly offered to some applicants. Thereafter, treatment is defined as receiving an offer to enter the school. This is not the same thing as entering the school, an issue I shall return to. Two years after the randomization, 363 applicants out of the 395 that participated in a lottery took a standardized maths test. The number of treatment and control applicants per strata is shown in Table 1 below. The treatment probability varies across strata, but there is no strata where it is higher than 0.9 or lower than 0.1. Notes: Number of treated and control applicants in the 14 strata in [START_REF] Behaghel | Ready for boarding? the effects of a boarding school for disadvantaged students[END_REF].

I use the data from [START_REF] Behaghel | Ready for boarding? the effects of a boarding school for disadvantaged students[END_REF] to compute the minimax-linear estimator τ (w * ).

Here, τ g is the intention-to-treat effect of receiving an offer on students' test scores two years after the lottery. At that point, the first-stage effect of receiving an offer on the number of than that of the untreated outcome for τ (w * )'s worst-case MSE to be higher than τ (p)'s.

Those precision and MSE gains are achieved by downweighting the least balanced strata. The weights assigned by τ (w * ) to 8th-grade-females-2009 and 10th-grade-males-2009 applicants are roughly half as large as the proportions these strata account for in the sample. The weights assigned by τ (w * ) to all the other strata are equal to their shares in the population.

τ (w * )'s robust standard error is also 3% smaller than that of τ (w f e ), while its worst-case MSE is 33% smaller. When one looks at female and male students separately, the precision and MSE gains attached to using τ (w * ) rather than τ (p) tend to be larger. For instance, among boys τ (w * )'s robust standard error and worst-case MSE are respectively 15 and 12% smaller than that of τ (p). Finally, results are similar with B = 0.6. Then, in the full sample τ (w * )'s robust standard error and worst-case MSE are respectively 7 and 4% smaller than that of τ (p), and 2 and 41% smaller than that of τ (w f e ).

E ( τ (w) -τ ) 2 =V ( τ (w)) + (E ( τ (w)) -τ ) 2 = G g=1 w 2 g V ( τ g ) +   G g=1 (w g -p g )τ g   2 ≤σ 2 G g=1 w 2 g v g +   G g=1 (w g -p g )τ g   2 ≤σ 2 G g=1 w 2 g v g +   G g=1 |w g -p g ||τ g |   2 ≤σ 2    G g=1 w 2 g v g + B 2   G g=1 |w g -p g |   2    .
The first equality follows from the fact that an estimator's MSE is the sum of its variance and squared bias. The second equality follows from the fact w is deterministic, from Equations The sharpness of the upper bound follows from plugging τ g = σB (1{w g ≥ p g } -1{w g < p g })

and V ( τ g ) = σ 2 v g into the second equality in the previous display.

Proof of Theorem 2.2

First, assume that w * has at least one coordinate that is strictly larger than the corresponding coordinate of (p 1 , ..., p G ). Without loss of generality, assume that w * 1 > p 1 . One has M SE(w * ) > M SE(p 1 , w * 2 , ..., w * G ), a contradiction. Therefore, each coordinate of w * is at most as large as the corresponding coordinate of (p 1 , ..., p G ). Accordingly, finding the minimax-linear estimator is equivalent to minimizing M SE(w) with respect to w, across all w = (w 1 , ..., w G ) such that w g ≤ p g for all g ∈ {1, ..., G}.

If w g ≤ p g for all g ∈ {1, ..., G}, M SE(w) = σ 2    G g=1 w 2 g v g + B 2   G g=1 (p g -w g )   2    .
23

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Therefore, w * is the minimizer of

G g=1 w 2 g v g + B 2   G g=1 (p g -w g )   2 , subject to w g -p g ≤ 0 for all g.
The objective function is convex, and the inequality constraints are continuously differentiable and concave. Therefore, the necessary conditions for optimality are also sufficient.

The Lagrangian of this problem is

L(w, µ) = G g=1 w 2 g v g + B 2   G g=1 (p g -w g )   2 + G g=1 2µ g (w g -p g ).
The Karush-Kuhn-Tucker necessary conditions for optimality are

w * g v g -B 2   1 - G g=1 w * g   + µ g = 0 w * g ≤ p g µ g ≥ 0 µ g (w * g -p g ) = 0. (6.1)
Those conditions are equivalent to

w * g = min   1 v g B 2   1 - G g=1 w * g   , p g   (6.2) µ g = max   0, B 2   1 - G g=1 w * g   -p g v g   .
One has that

1 v g B 2   1 - G g=1 w * g   < p g ⇔B 2   1 - G g=1 w * g   < p g v g .
Together with (6.2), the previous display implies that w * g < p g ⇒ w * g+1 < p g+1 . (6.3)

Let g * = min{g ∈ {1, ..., G} : w * g < p g }, with the convention that g * = G + 1 if the set is empty. It follows from Equations (6.2) and (6.3) that

w * g = p g for all g < g * w * g = 1 v g B 2   1 - G g=1 w * g   for all g ≥ g * . (6.4) 24
Electronic copy available at: https://ssrn.com/abstract=3846618 (6.4) implies that

G g=g * w * g = B 2 G g=g * 1 vg 1 + B 2 G g=g * 1 vg G g=g * p g .
Plugging this equation into (6.4) yields

w * g = p g for all g < g * w * g = 1 v g 1 1 B 2 + G g ′ =g * 1 v g ′ G g ′ =g *
p g ′ for all g ≥ g * . (6.5)

Finally, assume that g * < g. Then, w * g * > p g * , a contradiction. Some algebra shows that

M SE(p) -M SE p 1 , ..., p G-1 , 1 v G 1 B 2 + 1 v G p G =p 2 G v G -   p 2 G 1 v G 1 B 2 + 1 v G 2 v G + B 2 p 2 G 1 B 2 1 v G + 1 B 2 2   = p 2 G 1 B 2 + 1 v G 2 v G B 4 + 1 B 2 > 0.
Therefore, g * ∈ {g, ..., G} (6.6)

The result follows from Equations (6.5) and (6.6).

Proof of Corollary 2.3

w * belongs to (w h ) h∈{1,...,G} . For every h, for all g ≥ h,

lim B→+∞ w g,h = 1 v g 1 G g ′ =h 1 v g ′ G g ′ =h p g ′ .
Therefore, for every h,

lim B→+∞ G g=1 w g,h = 1, which then implies that lim B→+∞ G g=1 w * g = 1.
As w * g ≤ p g and G g=1 p g = 1, this implies that for every g,

lim B→+∞ w * g = p g .

Proof of Lemma 2.2

Assume that 1

G g ′ =g 1 v g ′ G g ′ =g p g ′ ≤ p g v g .
Then,

p g+1 v g+1 G g ′ =g+1 1 v g ′ =p g+1 v g+1 G g ′ =g 1 v g ′ -p g+1 v g+1 v g =p g v g G g ′ =g 1 v g ′ + (p g+1 v g+1 -p g v g ) G g ′ =g 1 v g ′ -p g+1 v g+1 v g ≥p g v g G g ′ =g 1 v g ′ + p g+1 v g+1 v g -p g -p g+1 v g+1 v g ≥ G g ′ =g+1 p g ′ .

Proof of Corollary 2.4

Proof of Point 1

Assume that Points 1 and 3 of Assumption 1 hold. If Point 1 of Assumption 2 were to hold with h g = 1 for all g, then Point 2 of Assumption 1 would hold with v g = v 0,g + v 1,g , and τ (w * ) would be minimax-linear. Accordingly, its worst-case MSE under that DGP has to be lower than that of τ (p), which implies that

σ 2 B 2   G g=1 |w * g -p g |   2 ≤ σ 2 G g=1
((p g ) 2 -(w * g ) 2 )(v 0,g + v 1,g ). (6.7)

As for all g, v 1,g ≥ 0, (p g ) 2 -(w * g ) 2 ≥ 0, and h g ≥ 1 under Point 1 of Assumption 2,

σ 2 G g=1 ((p g ) 2 -(w * g ) 2 )(v 0,g + v 1,g ) ≤ σ 2 G g=1
((p g ) 2 -(w * g ) 2 )(v 0,g + h g v 1,g ). (6.8)

Combining (6.7) and (6.8) and rearranging proves the result.

Proof of Point 2

If Points 1 and 3 of Assumption (p g ) 2 (v 0,g + hv 1,g ).

Taking the difference between the two preceding displays, setting that difference lower than 0 and rearranging yields the result.

Proof of Theorem 2.5

That E ( τ (w) -τ ) 2 ≤ M SE 2 (w) follows from the same steps as the proof of Lemma 2.1.

As c g,g ′ ≥ 0 for all (g, g ′ ), if w g ≥ 0 for all g, then using a reasoning similar to that in the proof of Theorem 2.2, one can show that the minimizer of M SE 2 (w) must be such that each of its coordinates are lower than the corresponding coordinate of p. Therefore, this minimization problem is equivalent to that in (2.8).

Proof of Theorem 3.1 0 ≤ τ g ≤ Bσ implies that Bσ(w g -p g )1{w g < p g } ≤ (w g -p g )τ g ≤ Bσ(w g -p g )1{w g > p g }, which in turn implies that (w g -p g )1{w g < p g } , the sharpness of the upper bound follows from plugging τ g = σB1{w g ≥ p g } into (6.9). When (w g -p g )1{w g < p g } , the sharpness of the upper bound follows from from plugging τ g = σB1{w g < p g } into (6.9).

Finally, assume that w * , the argmin of M SE + (w), has at least one coordinate that is strictly larger than the corresponding coordinate of (p 1 , . ), a contradiction. Therefore, each coordinate of w * is at most as large as the corresponding coordinate of (p 1 , ..., p G ).

Accordingly, finding the minimax-linear estimator is equivalent to minimizing M SE + (w)

with respect to w, across all w = (w 1 , ..., w G ) such that w g ≤ p g for all g ∈ {1, ..., G}. On that set,

M SE + (w) =σ 2    G g=1 w 2 g v g + B 2   G g=1 (w g -p g )1{w g < p g }   2    =σ 2    G g=1 w 2 g v g + B 2   G g=1 (w g -p g )1{w g ≤ p g }   2    =σ 2    G g=1 w 2 g v g + B 2   G g=1 (w g -p g )   2   

=M SE(w).

This completes the proof.

Proof of Theorem 3.2

That E ( τ (w) -τ ) 2 ≤ M SE 3 (w) follows from the same steps as the proof of Lemma 2.1.

That M SE 3 (w) is minimized at w h * 2 follows from the same steps as the proof of Theorem 2.2.

Proof of Theorem 3.3

That E ( τ (w) -τ ) 2 ≤ M SE 4 (w) follows from the same steps as the proof of Lemma 2.1.

If cov( τ g , τ g ′ ) ≥ 0 for all (g, g ′ ) and w g ≥ 0 for all g, then using a reasoning similar to that in the proof of Theorem 2.2, one can show that the minimizer of M SE 4 (w) must be such that each of its coordinates are lower than the corresponding coordinate of p. Therefore, this minimization problem is equivalent to that in (3.2).

  , Callaway & Sant'Anna (2020), and de Chaisemartin & D'Haultfoeuille (2020). There, the target parameter is the average treatment effect on the treated (ATT), and the CATEs are the average treatment effect at period t among units that started receiving the treatment at period k. If potential outcomes without treatment are iid, across units and over time, and if the treatment effects and treatments are non-stochastic, the unbiased estimators of those CATEs proposed by Sun & Abraham (2020), Callaway & Sant'Anna (2020), and de Chaisemartin & D'Haultfoeuille (2020) are positively correlated, with covariances known up to the outcome's variance. I consider a numerical example with 50 units (e.g.: the 50 US states) and five periods, where 10 units respectively become treated at periods 2, 3, 4, and 5, and 10 units remain untreated.

in

  [START_REF] Sun | Estimating dynamic treatment effects in event studies with heterogeneous treatment effects[END_REF], the not-yet treated units in de Chaisemartin & D'Haultfoeuille (2020), while Callaway & Sant'Anna (2020) consider both the never-and not-yet treated units. In all cases, C t+1 ⊆ C t . The estimators τ k,t are unbiased for τ k,t under a parallel trends assumption (see[START_REF] Callaway | Difference-in-differences with multiple time periods[END_REF].

  Minimax-linear estimator of τ , with bounded CATEs, correlations, and homoscedasticity) If Points 2 and 3 of Assumption 1 and Assumption 3 hold, then for any 1 × G deterministic vector w = (w 1 , ..., w G )

Figure 1 :

 1 Figure 1: τ (w * ), τ (p), and τ (w f e ) in stratified RCTs Notes: This figure compares the performance of the minimax-linear estimator τ (w * ) to that of the unbiased and fixed effects estimators τ (p) and τ (w f e ) in stratified RCTs with two strata. On each panel, the blue and brown lines respectively show w * 1 and w * 2 , the weights assigned to the balanced and imbalanced strata by τ (w * ), as a function of the treatment probability in the imbalanced strata. The green and orange lines respectively show the ratio of the standard error and worst-case MSE of τ (w * ) and τ (p), under the homoscedasticity condition in Point 2 of Assumption 1. Finally, the black and red lines respectively show the ratio of the standard error and worst-case MSE of τ (w * ) and τ (w f e ), under that same assumption. In the top left panel, each strata has 100 units (N = 200) and B = 0.75. In the top right panel, each strata has 100 units (N = 200) and B = 0.5. In the bottom left panel, each strata has 50 units (N = 100) and B = 0.75. In the bottom right panel, each strata has 50 units (N = 100) and B = 0.5.

  (2.2) and (2.1), and from Definition 2.1 and Point 1 of Assumption 1. The first inequality follows from Point 2 of Assumption 1. The second inequality follows from the fact that for any real number a, a 2 = |a| 2 , from the triangle inequality, and from the fact that x → x 2 is increasing on R + . The third inequality follows from Point 3 of Assumption 1.

  1 and Point 2 of Assumption 2 hold, the worst-case MSEs of τ (w * ) and τ (p) are respectively equal to g ) 2 (v 0,g + hv 1,g ) + B 2

  -p g )1{w g < p g } ≤ G g=1 (w g -p g )τ g ≤ Bσ G g=1 (w g -p g )1{w g > p g }.(6.9)Then, reasoning as in the proof of Lemma 2.1 yieldsE ( τ (w) -τ ) 2 ≤M SE + (w). When G g=1 (w g -p g )1{w g > p g } ≥ G g=1

  -p g )1{w g > p g } < G g=1

Table 1 :

 1 Number of treatment and control applicants in each strata

	Strata	Control applicants Treated applicants
	8th grade, males, 2009	11	15
	8th grade, females, 2009	15	3
	9th grade, females, 2009	8	22
	10th grade, males, 2009	5	22
	10th grade, females, 2009	36	27
	6th grade, males, 2010	6	9
	7th grade, males, 2010	9	8
	7th grade, females, 2010	5	10
	8th grade, males, 2010	5	19
	9th grade, males, 2010	8	6
	9th grade, females, 2010	3	13
	10th grade, males, 2010	12	16
	10th grade, females, 2010	39	24
	11th grade, females, 2010	3	4

Table 2 :

 2 τ (w * ), τ (p), and τ (w f e ) in[START_REF] Behaghel | Ready for boarding? the effects of a boarding school for disadvantaged students[END_REF] Notes: This table shows τ (w * ), τ (p), and τ (w f e ) in[START_REF] Behaghel | Ready for boarding? the effects of a boarding school for disadvantaged students[END_REF]. τ (w * ) is computed with B = 0.5.The treatment is defined as being offered a seat in the boarding school. The outcome is students' standardized maths test scores two years after the lottery. The table shows the point estimates, their robust standard errors, and their worst-case MSE computed assuming homoscedasticity and expressed in percentage points of the outcome's variance. In the first panel, results are shown for the full sample. In the second (resp. third) panel, results are shown for female (resp. male) students.This paper considers situations where one wants to estimate the ATE in a population that can be divided into G groups, and one has unbiased estimators of the CATE in each group 19 Electronic copy available at: https://ssrn.com/abstract=3846618 with heterogeneous levels of statistical precision. To trade-off bias and variance, I derive the minimax estimator of the ATE, across all linear combinations of CATEs' estimators, assuming that CATEs are bounded by B standard deviations of the outcome and homoscedasticity.

	τ (w * ) τ (p) τ (w f e )

  .., p G ). Without loss of generality, assume that w * 1 > p 1 . One has so the squared bias of τ (w * ) is at least as large as that of τ (p 1 , w * 2 , ..., w * G ), while its variance is strictly larger. Then, M SE + (w * ) > M SE + (p 1 , w * 2 , ..., w * G

	and		G		2			G		2
			(w * g -p g )1{w * g < p g } 		=		(w * g -p g )1{w * g < p g } 	,
		g=1					g=2
			G		 2			G
			(w * g -p g )1{w * g > p g }		>		(w *
		g=1					g=2

g -p g )1{w * g > p g }   2 27

Electronic copy available at: https://ssrn.com/abstract=3846618

Electronic copy available at: https://ssrn.com/abstract=3846618

See Li & Ding (2017) for a central limit theorem for completely randomized experiments.

In stratified RCTs, Assumption 2 and Point 3 of Assumption 1 bound the CATEs by B% of the untreated outcome's variance. If one uses instead the treated outcome's variance as the numeraire, the conclusions of Corollary 2.4 revert: τ (w * )'s worst-case MSE is always lower than τ (p)'s if the untreated outcome's variance is larger than that of the treated outcome, while τ (w * )'s worst-case MSE is lower than τ (p)'s under a bound on heteroscedasticity if the opposite holds. Assumption 2 follows the common practice in applied work of standardizing the outcome by its variance in the control group.

In stratified RCTs with non-stochastic potential outcomes, it is only possible to estimate upper bounds of the variances, but that does not affect the result in Theorem 3.2, as those upper bounds are sharp.

years spent in the school is equal to 1.01 (standard error=0.16), 7 so the τ g s can be interpreted as effects of having spent one year in the boarding school. The first-stage effects may vary across strata, which would complicate this interpretation. However, I cannot reject that the first-stages are equal for males and females (t-stat=-1.24), for 2009 and 2010 applicants (t-stat=-1.06), and for middle-and high-school applicants (t-stat=0.53), so the first-stage effect indeed seems to be constant across strata. 8

Based on the literature, 0.5σ is a plausible upper bound for the CATEs. The paper studying the closest intervention is [START_REF] Curto | The potential of urban boarding schools for the poor: Evidence from seed[END_REF], who study a "No Excuse" charter boarding school in Washington DC. In their full sample, they find that one year spent in the school increases students' math test scores by 0.23σ. They also estimate CATEs in eight subgroups of students: males/females, students benefiting/not benefiting from the free lunch program, students in/not in special education, and students above/below the median at baseline. The estimated effects in those 8 subgroups are included between 0.04 and 0.36σ, and in 7 of the 8 subgroups one can reject at the 90% level that the effect is greater than 0.5σ, the only exception being the special education group that only has 30 students. Results from Angrist et al. ( 2010), [START_REF] Dobbie | Are high-quality schools enough to increase achievement among the poor? evidence from the harlem children's zone[END_REF][START_REF] Dobbie | Are high-quality schools enough to increase achievement among the poor? evidence from the harlem children's zone[END_REF][START_REF] Abdulkadiroğlu | Accountability and flexibility in public schools: Evidence from boston's charters and pilots[END_REF], three papers studying successful non-boarding "No Excuse" charter schools in New-York and Boston, also suggest that 0.5σ is a plausible upper bound. Together, these papers estimate 14 CATEs of spending one year in those schools on students' math test scores. All estimates are included between 0.18 and 0.36σ, and one can reject at the 90% level that 13 of the 14 CATEs are greater than 0.5σ. One may argue that the intervention in [START_REF] Behaghel | Ready for boarding? the effects of a boarding school for disadvantaged students[END_REF] takes place in France, a different setting from that in those papers. However, effects of 0.5σ are very rarely found for educational interventions in high-income countries (see [START_REF] Fryer | The production of human capital in developed countries: Evidence from 196 randomized field experiments[END_REF]. As a robustness check, I will show that results do not change much with B = 0.6.

Results are shown in Table 2. In the full sample (Panel A), τ (w * ) is 13% smaller than τ (p).

Its robust standard error, estimated following (2.4), is 9% smaller. Assuming homoskedascity, we can compute the worst-case MSE of both estimators, expressed in percentage points of the outcome's variance. That of τ (w * ) is 6% smaller. As shown in Point 1 of Corollary 2.4, with heteroscedasticity τ (w * )'s worst-case MSE is still lower than τ (p)'s if the treated outcome's variance is higher than that of the untreated outcome. This may be a plausible assumption in this context, as quantile treatment effects of this intervention are higher at the top than at the bottom of the distribution of test scores (see [START_REF] Behaghel | Ready for boarding? the effects of a boarding school for disadvantaged students[END_REF]. I still compute the lower bound in Point 2 of Corollary 2.4, and find that it is equal to 0.06: under Point 2 of Assumption 2, the variance of the treated outcome would have to be more than 94% lower 7 Numbers differ from those in [START_REF] Behaghel | Ready for boarding? the effects of a boarding school for disadvantaged students[END_REF], because they use the doubly-reweighted-ever-offer (DREO) estimators proposed by [START_REF] De Chaisemartin | Estimating the effect of treatments allocated by randomized waiting lists[END_REF]. As the DREO estimators of the treatment effect in each strata do not satisfy Point 2 of Assumption 1, I use instead the initial offer estimators. Moreover, [START_REF] Behaghel | Ready for boarding? the effects of a boarding school for disadvantaged students[END_REF] have control variables in their specification, while I do not have controls.

8 Strata are too small to directly test if the first-stage is the same in all of them.