Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
124,980 characters · 18 sections · 77 citation commands
Sharp Hybrid Confidence Bands for Partially Identified Treatment Effects under Tail Uncertainty with an Application to Workforce Gender Diversity and Firm Performance
\noindentKeywords: concATE; partial identification; interval-identified parameters; average treatment effect; finite-sample inference; confidence bands; threshold effects; workforce gender diversity
\noindentJEL Classification: C21; C14; M14; L25; J16
Estimating the causal effects of a treatment in the presence of partially observed counterfactuals, where for each unit only one of the two potential outcomes is observed, i.e., the one corresponding to the treatment status actually realised - is a central problem within the causal inference literature. When only one of two potential outcomes is observed, the average treatment effect (ATE hereafter) is not point-identified without imposing additional assumptions. A large literature therefore studies partial identification, characterising causal effects estimands as sets rather than points honore2017advances. In the canonical case of a scalar ATE, this yields an interval-identified parameter, whose lower and upper endpoints depend on the observed outcomes with minimal restrictions on the outcome support manski1990nonparametric,manski2003partial. A critical but largely unexamined restriction in this literature is that the outcome support - the range of values the potential outcomes can take - is treated as known or fixed. In many empirical settings, however, credible global support bounds are unavailable, and outcomes may be heavy-tailed, so that common implementations relying on ad-hoc truncation or observed sample extrema can compromise finite-sample coverage d3fc5a28-c5bb-3e85-a388-e8132ffff151. In this paper, we propose concATE, a hybrid confidence band for interval-identified ATEs that treats the latent outcome support as an inferential object rather than an identification assumption. The method combines finite-sample, distribution-free control of tail uncertainty via the Dvoretzky-Kiefer-Wolfowitz inequality with asymptotic delta-method inference for the smooth components of the Manski bound endpoints, and we show that the resulting tail-bracketing procedure achieves the minimax optimal rate among all distribution-free confidence procedures over the class of distributions with finite mean.
The problem of inference for set-identified causal effects has been studied extensively. imbens2004confidence and stoye2009more develop confidence intervals for partially identified scalar parameters by exploiting the asymptotic normality of the estimated bound endpoints. While imbens2004confidence provide sharp inference for partially identified parameters, they assume the identification region is consistently estimable. More generally, a broad literature develops inference for identified sets and functions thereof, including approaches based on moment inequalities, set estimation, and resampling methods chernozhukov2007estimation,rosen2008confidence,romano2008inference,andrews2010inference,andrews2009validity,bugni2017inference. Related work studies intersection and projection bounds, providing asymptotic inference for parameters defined as extrema over index sets chernozhukov2013intersection,kaido2019confidence. Surveys of these developments emphasize the richness of available asymptotic tools for partially identified models that guarantee asymptotic validity, but not finite-sample guarantees honore2017advances,MOLINARI2020355.
While these studies provide tools for large-sample inference on partially identified estimators, they typically assume a fixed class of data-generating processes in which the outcome support adheres to regularity conditions that ensure the asymptotic validity of estimated bound endpoints. The moment inequality framework, for instance, takes population moment conditions as given: for Manski bounds on the ATE with known support, the moment inequalities are well-defined, and the inference problem reduces to accounting for sampling uncertainty in the estimated moments. Support uncertainty is categorically different. It concerns whether the bounding constants entering Manski's identified region, which depend on the outcome support, are themselves correctly specified. When the support is unknown or unbounded, the standard moment inequalities may be misspecified, and no amount of asymptotic refinement addresses this. In practice, applied researchers confronting this problem often substitute observed sample extrema or ad-hoc quantile truncations for the unknown support bounds. As our Monte Carlo results in Section (ref) demonstrate, while plug-in methods using sample extrema achieve valid coverage, they do so at the cost of substantially wider intervals, often 2--4 times the width of the concATE band, reflecting the systematic overshoot of sample extrema relative to the true population support.
The recent literature has also aimed at addressing the issue of exact finite-sample inference in partially identified models. li2022finite consider the general class of incomplete structural models, though their paradigm requires the parametric specification of latent variable distribution and that the support restriction is known a priori. Furthermore, their approach relies on simulation-based test inversion with known latent distributions. song2024categorical construct finite-sample confidence intervals for partially identified ATE under categorical IV models, using the Kullback-Leibler divergence concentration bounds, but this is restricted to categorical variables. cox2023simple provide exact finite-sample size control, but only in DGPs arising from normal models. In the point-identified setting, armstrong2021finite derive finite-sample minimax optimal confidence intervals for the ATE under unconfoundedness with known outcome support, though their framework requires point-identification and known bounds on the outcome.
This paper studies finite-sample and hybrid inference procedures for interval-identified ATEs when the outcome support and its corresponding data-generating process are unknown. We propose concATE, a hybrid concentration-driven confidence band that explicitly accounts for tail uncertainty without imposing parametric assumptions on the distribution of the unobserved counterfactuals. Our approach separates uncertainty arising from smooth functional components\footnote{In the Hadamard differentiability sense.} of the Manski bound endpoints, such as treated and untreated means, from uncertainty about unobserved outcome tails. For the latter, we exploit the Dvoretzky-Kiefer-Wolfowitz inequality dvoretzky1956asymptotic to construct a finite-sample, distribution-free envelope for the latent outcome distribution. This envelope induces, with controlled probability, a confidence set for the Manski identified interval. For the smooth components, we employ standard delta-method asymptotic approximations, and we combine the two sources of uncertainty using Bonferroni's inequality to obtain a simultaneous confidence band that guarantees joint coverage of the lower and upper ATE bounds. While many existing confidence regions for partially identified parameters are only pointwise asymptotically valid and may fail in finite samples, we construct confidence bands with explicit finite-sample coverage guarantees for the tail components and asymptotic guarantees for the smooth components, yielding a hybrid band that remains valid under weak distributional assumptions.
We further extend concATE and its finite-sample counterpart to an optimal stopping setting, in which a group-sequential procedure is employed to identify the first threshold at which the ATE becomes non-zero while controlling the family-wise error rate. This extension is motivated by empirical contexts featuring threshold or “tipping point” effects and provides a principled approach to post-selection inference under partial identification.
To demonstrate the utility of our approach, we apply it to the question: Does greater senior-level gender workforce diversity causally improve firm performance? This question has taken on renewed importance as many firms have invested heavily in Diversity, Equity, and Inclusion (DEI) initiatives, yet establishing causality is difficult due to non-random adoption of diversity practices. A rich body of literature in management and economics has examined the links between top management team composition and organisational outcomes. The foundational “upper echelons” theory of hambrick1984upper posits that a firm’s strategies and performance reflect the backgrounds of its senior executives. Consistent with this view, numerous studies document associations between executive attributes and firm outcomes such as innovation and financial performance. For example, prior research finds that gender-diverse boards tend to exhibit improved internal governance (e.g., better oversight and attendance) although the average impact on firm profitability or market value is mixed adams2009women. A comprehensive meta-analysis by post2015women reports that female board representation is positively related to accounting returns, especially in societies with greater gender parity, but the correlation with market-based performance metrics is weaker. More recent work has begun to address endogeneity in this relationship: safiullah2022gender, analyzing Spain’s Gender Equality Act, use GMM techniques and find that while gender-diverse boards outperform on accounting measures, they can underperform on market valuation measures, suggesting investors may respond differently than internal metrics. Similarly, a study of Russian firms by kuzmina2021gender finds that gender-diverse boards are associated with higher profitability and market value, with the benefits particularly pronounced during economic downturns. Beyond gender, other aspects of diversity have been linked to innovation outcomes: ostergaard2011does show that employee gender and educational diversity positively predict firm innovation, and in a study of London firms, nathan2013cultural find cultural diversity in management boosts product innovation and entrepreneurship. Field experiments also echo these benefits. hoogendoorn2013impact conducted a randomized experiment with startup teams and found that gender-balanced teams outperformed male-dominated teams in terms of sales and profits.
An intriguing hypothesis in this literature is the existence of non-linear effects, or “critical mass” thresholds, in the diversity-performance relationship. Sociologist Rosabeth Kanter’s classic work on tokenism kanter1977some,kanter1987men theorised that women in extreme minority (a “token” few) face marginalisation, whereas once a minority group reaches a substantial share of the team, dynamics shift and their influence grows disproportionately. Kanter’s typology categorises group gender composition into skewed (up to $\sim$15% women), tilted ($\sim$20-35% women), and balanced ($\sim$40-50% women) categories, proposing that performance benefits might emerge when moving from skewed to tilted or balanced distributions. Subsequent studies have sought empirical evidence of such tipping points. For instance, torchia2011women find that having at least three women directors (roughly a critical mass on many boards) is associated with a jump in innovation outputs, consistent with moving beyond token representation. ali2011gender report an inverted U-shaped relationship between female representation and firm performance in certain contexts, suggesting that the strongest returns may occur at intermediate diversity levels before tapering off. These studies, while suggestive, largely report correlations or rely on linear/quadratic models that may not capture the true causal threshold. Our study contributes to this literature by using a robust, partially identified approach to formally test for causal tipping points. By refraining from imposing a specific functional form, we let the data reveal whether and where increasing diversity has a statistically reliable positive effect on firm value.
Our empirical analysis uses a panel of 901 publicly listed firms observed quarterly from 2015 Q2 to 2022 Q1. We focus on Tobin’s $Q$ (the ratio of market value to the replacement cost of assets) as the outcome of interest, which is a standard proxy for a firm’s growth opportunities and innovative performance. Originally introduced by tobin1969general and later expounded in Tobin’s subsequent work tobin1978monetary, the $Q$-ratio captures market expectations of future returns. A value above 1 indicates that the firm’s market valuation exceeds book value, signalling strong investment incentives brainard1968pitfalls,tobin1976asset. For each quarter, we define the “treatment” as whether the firm’s top management team or board exceeds a given diversity threshold. In separate analyses, we consider thresholds for the percentage of women in senior leadership (e.g., 30%, 40%, 50%, etc.), reflecting the critical mass levels discussed above. We then estimate the nonparametric bounds on the ATE of diversity at each threshold using our concATE procedure. This approach does not assume that firms with different diversity levels are comparable on unobservables; instead, it provides an interval estimate for the possible causal effect, given the observable data, without invoking full identification. In contrast to most prior studies that report point estimates after making identification assumptions, our results will highlight the range of plausible causal impacts of diversity on Tobin’s $Q$, emphasising what can be learned with minimal assumptions.
Our findings yield informative insights. In broad terms, the concATE analysis suggests that senior-level gender diversity has a significantly positive causal effect on Tobin's $Q$, but only after a certain threshold of representation is achieved. In innovation-driven sectors (such as technology and healthcare, where overall growth opportunities are high), we find that once female representation in leadership surpasses roughly 55%, the lower bound of the ATE becomes positive, and the confidence band excludes zero. The estimated effect size grows as diversity increases, with particularly strong gains evident as teams approach and exceed gender balance (around 50% female). This provides causal empirical support for the notion of a “tipping point” around high diversity levels in dynamic industries. One interpretation is that innovation-oriented firms, facing fast-moving and competitive markets, have strong incentives to harness the benefits of workforce diversity. Such firms may actively invest in inclusive cultures and leadership practices that allow diverse perspectives to be heard and integrated, thereby capturing value from diversity once a basic critical mass is present.
By contrast, in more traditional or cyclically oriented industries, the data suggest that a higher critical mass (on the order of 60% female representation) is needed before we detect a reliably positive impact on firm value. Below that level, the confidence bands include zero, indicating we cannot rule out no effect in those sectors (but we can rule out negative effects). This stringent “tipping point” in traditional industries may reflect a lack of inclusion---when women remain a small minority, they may not experience the psychological safety needed to freely voice their insights or challenge prevailing viewpoints. Indeed, diversity alone (without an inclusive environment) can lead to friction or marginalization of minority members, whereas inclusion actively involves and values those members. This aligns with prior research suggesting that diversity must be complemented by inclusion to unlock its benefits: simply adding a few token individuals from underrepresented groups often fails to improve outcomes unless the organizational climate empowers those individuals to participate fully roberson2006disentangling,nishii2013benefits,josten2025makes. Alternatively, it is possible that the intrinsic gains to diversity are lower in these contexts. Importantly, these conclusions are drawn with rigorous uncertainty quantification. The concATE bands allow us to assert, for example, that at 95% confidence a firm in a growth industry with a gender-balanced leadership enjoys an ATE on Tobin's $Q$ that is positive (bounded away from zero), whereas at lower diversity levels the effect cannot be distinguished from zero. Such results illustrate how our methodological innovation can uncover nuanced causal relationships that might be obscured or misstated by conventional point estimation approaches.
The remainder of the paper is organised as follows. Section (ref) formalises the problem and presents the theoretical framework for nonparametric identification, extending Manski’s bounds to our context. Section (ref) discusses the estimation procedure of nonparametric bounds in practice. Section (ref) details the construction of finite-sample concentration-inequality-driven inference and the hybrid concATE confidence band, and extends it to sequential threshold analysis, including theoretical guarantees. Section (ref) derives the minimax lower and upper bounds for the convergence rate of the hybrid concATE approach and compares it with a few distribution-free methods, and Section (ref) reports results from a Monte Carlo simulation that compares the finite-sample performance of concATE to traditional methods under a variety of data-generating processes. Section (ref) then presents the empirical findings from our panel of firms, highlighting the estimated diversity tipping points and their interpretation. Finally, Section (ref) concludes with a discussion of implications for research and policy, and potential extensions of our framework.
In this section, inspired by the noted shortcomings in causal inference methodologies rigorously discussed by angrist2009mostly, we extend the theoretical framework of manski1990nonparametric, manski2003partial to derive nonparametric bounds on the “average” diversity treatment effect.
Let us denote the potential outcomes for firm $i$ in sector $j$ in quarter $t$ by $Y_{ijt}^{(0)}$ and $Y_{ijt}^{(1)}$, corresponding to the scenarios of no diversity efforts (no treatment) and with diversity efforts (treatment), respectively. Regardless of whether firm $i$ actually adopts diversity, $Y_{ijt}^{(0)}$ represents the hypothetical (counterfactual) outcome had the firm not exercised any diversity efforts, and $Y_{ijt}^{(1)}$ represents the outcome if the firm did adopt diversity. In essence, the question we seek to investigate is whether these potential outcomes differ (i.e., whether diversity efforts affect $Y_{ijt}$).
For simplicity of exposition, assume that $Y_{ijt}^{(k)}\in\mathbb{R}$ for $k\in\{0,1\}$ and define the treatment indicator
where $\mathbbm{1}\{\cdot\}$ is the indicator function, $\mathcal{D}$ is a diversity signal, and $\tau$ is a threshold chosen by the investigator. Let $\mathbf{X}_{ijt}=(X_{ijt}^{1},\dots,X_{ijt}^{d})^{\top}\in\mathbb{R}^{d}$ denote a $(d\times1)$ vector of covariates. Our goal is to learn the conditional treatment effect $Y_{ijt}^{(1)}-Y_{ijt}^{(0)}\mid\mathbf{X}_{ijt}$. Following the notation of angrist2009mostly, the observed outcome can be written in terms of potential outcomes as
Because only one potential outcome is ever observed for a given firm--quarter $(i,j,t)$, a naïve comparison of conditional means by treatment status is
Substituting (ref) into (ref) gives
where $\rho(\mathbf{X})$ is the (conditional) treatment effect and $\mathcal{B}(\mathbf{X})$ is the selection bias. As a corollary, the unconditional mean‐comparison parameter $\delta$ of angrist2009mostly is obtained by integrating $\delta(\mathbf{X})$ over the (marginal) distribution of $\mathbf{X}\mid Z(\tau)$, i.e.,
hence
The latter may be non-zero because firms that adopt diversity efforts might do so precisely when they face innovation shortfalls, either to signal responsiveness to investors or to diversify their workforce in search of new ideas; in such cases $\mathcal{B}(\mathbf{X})<0$. Conversely, if a firm scales up diversity after large innovation gains, aiming to sustain that momentum, then $\mathcal{B}(\mathbf{X})>0$.
manski1990nonparametric, manski2003partial formalise the problem differently. For firms characterised by attributes $\mathbf{X}$, define the difference in expected outcomes as
Using the law of total expectation, each conditional mean in (ref) can be decomposed; for example,
and an analogous expression holds for $k=0$.
Equation (ref) makes clear that two latent expectations,
are never observed in the data. Put differently, we do not observe the innovation outcome a firm would have achieved without diversity efforts when it actually implemented them ($Z_{ijt}(\tau)=1$), nor the outcome with diversity efforts when it did not implement them ($Z_{ijt}(\tau)=0$).
In both scenarios, one can conduct a randomized experiment, as noted by angrist2009mostly, which coincides with the mean-independence assumption in manski2003partial. In that case:
Then, $\delta(\mathbf{X}) = \rho(\mathbf{X})$, and the expression in (ref) simplifies to:
Hence, under random assignment, $\delta(\mathbf{X})$ suffers no selection bias, and $\Re(\mathbf{X})$ is point-identified.
The mean-independence assumption, however, is rather strict. For expositional purposes, consider bounding arguments that replace unknown support limits with quantile-indexed truncations of tail mass:
where $ Q_Y(p) = \inf\{ y : F_Y(y) \geq p \} $, with $ F_Y(\cdot) $ the CDF of $Y$ and $p^c$ is the complement of $p$, i.e., $p^c=1-p$. Suppose further that we are interested in the unconditional average treatment effect rather than the effect for each unit. Using the law of iterated expectations:
Now, since some conditional expectations remain latent (as shown in (ref)), one may bound them using either known outcome supports $ [L^{(k)},\; U^{(k)}] $ or their quantile-based versions $ [Q_Y(p), Q_Y(1-p)] $. manski1990nonparametric, manski2003partial propose the following nonparametric bounds for the treatment effect:
These expressions illustrate how the identified set depends on tail behaviour. Replacing $ L^{(k)} $ and $ U^{(k)} $ with quantile-indexed limits, say $ Q_Y(p) $ and $ Q_Y(1-p)$, correspond to a heuristic truncation of tail mass and does not define a new estimand. In essence, using a similar notation to manski2003partial, the region $\mathcal{H}[\Re]$ is the identification region for $\Re$, where $\mathcal{H}[\Re]$ is defined as the bound in (ref). Note that $\mathcal{H}[\Re]$ is only partially identified when $0<\Pr[Z_{ijt}(\tau)=k]<1$ for $k=0,1$, as otherwise, $\mathcal{H}[\Re]$ is simply a singleton. In other words, if, say, $\Pr[Z_{ijt}(\tau)=1]=1$, then both upper and lower bounds coincide with the treated mean, so $\mathcal{H}[\Re]$ collapses.
Throughout, the target parameter remains the original Manski-identified set $\mathcal{H}[\Re]$; quantile-based bounds are introduced only as a device to illustrate sensitivity to tail behaviour and will be formally calibrated using finite-sample concentration inequalities in subsequent sections.
In the following sections, we outline estimation procedures for both the naïve unconditional difference and the nonparametric bounds. We also construct $100(1 - \alpha)\%$ confidence intervals for the bounds using Bonferroni-adjusted intervals as proposed by horowitz1998censoring, and derive standard errors via the delta method [see casella2024statistical].
For a fixed $\tau$, the bounding constants \[ L^{(1)} \le \mathbb{E}\left[Y_{ijt}^{(1)} \mid Z_{ijt}(\tau)=0\right] \le U^{(1)}, \quad L^{(0)} \le \mathbb{E}\left[Y_{ijt}^{(0)} \mid Z_{ijt}(\tau)=1\right] \le U^{(0)}, \] state that the latent (never-observed) mean outcome a “treated’’ firm would have realised had it not been treated cannot be lower than $L^{(1)}$ nor higher than $U^{(1)}$; analogously for an “untreated’’ firm under treatment. Without bounding these counterfactual means, the ATE, $\Re$, is not point-identifiable: any value between $-\infty$ and $\infty$ could be rationalised by suitable (and untestable) choices of $\mathbb{E}[Y_{ijt}^{(1)}\mid Z_{ijt}(\tau)=0]$ and $\mathbb{E}[Y_{ijt}^{(0)}\mid Z_{ijt}(\tau)=1]$. Because our outcome of interest (Tobin’s $Q$) is in theory unbounded from above, we use quantile-based limits here, e.g.\ $L^{(k)} = 0$ and $U^{(k)} = Q_Y^{(k)}(0.90)$, purely for illustration; in Sections (ref)--(ref) these fixed cut-offs are replaced by data-driven, DKW-calibrated endpoints that deliver valid outer confidence sets for the original Manski identified region: they confine the worst-case counterfactual means to the central 90% of the empirical outcome distribution, ruling out only the most extreme tail behaviour while introducing minimal additional assumptions. The 10% tails trimmed was deemed a reasonable balance between realism and conservatism in our firm performance data, which can be heavy-tailed. In Section (ref), when applying Manski bounds to study the causal impact of workforce gender diversity on Tobin’s $Q$, we further experiment with tighter bounds (such as the 10\textsuperscript{th}--90\textsuperscript{th} percentiles) to assess how the results are affected.
Under these mild restrictions, the interval in (ref) remains robust to selection on unobservables, yet is now finite, so if the entire interval lies above (below) 0 we may still conclude a positive (negative) causal effect even when ignorability fails. We therefore describe $\mathcal{H}[\Re]$ as a set of “worst-case bounds’’ for the ATE under no unverifiable assumptions beyond the outcome range.
As it is evident from Eq. (ref)--(ref), the sampling composition of the treated and untreated firms depends on the threshold $\tau$. In theory one could fix $\tau$ and analyse the resulting samples accordingly, though in the context of this work, our goal is different: we seek the tipping point at which the average treatment effect $\Re$ becomes strictly positive (or negative). Thus, $\tau=\tau(\omega)$ must be regarded as a random stopping time with values in $\mathcal{M}$, where $\omega$ is the realization.
Formally, let $(\Omega,\mathcal{F},\mathbb{P})$ be the underlying probability space, with $\{\mathcal{F}_u\}_{u\in\mathcal{M}}$ denoting the filtration generated by the sequential statistics up to look $u$; for instance, $\mathcal{F}_u:=\sigma(\Re_v:v\in\mathcal{M},v\le u)$, where $\sigma(.)$ is the $\sigma$-field spanned by the random variables $\{\Re_v:v\leq u\}$. The random threshold $\tau=\tau(\omega)$, for each $u \in \mathcal{M}$ with \[ \{\omega: \tau(\omega)\leq u\} \in \mathcal{F}_u,\quad \forall u \in \mathcal{M}. \] is selected by the procedure will be a stopping time with respect to $\{\mathcal{F}_u\}_{u\in\mathcal{M}}$.
Let $\mathcal{D}_{ijt}$ denote the diversity signal for firm $i$ in sector $j$ at time $t$ (e.g.\ the percentage of female or non-white executives). A firm is labelled “treated” when $\mathcal{D}_{ijt}\ge\tau$. Rather than prespecify $\tau$, we examine a grid of meaningful cut-offs, \[ \tau_m := m, \qquad m\in\mathcal{M}, \] where in our context $\mathcal{M}=\{5,10,15,\dots,90,95\}$, and $\overline{\mathcal{M}}:=\lvert\mathcal{M}\rvert$. For each $m$ we define \[ Z_{ijt}(\tau_m):=\mathbbm{1}\{\mathcal{D}_{ijt}\ge \tau_m\}. \]
For a chosen significance level $\alpha\in(0,1)$ we test
Following siegmund2013sequential, define the stopping rule \[ \tilde{\tau} := \inf\bigl\{\tau_u:\ u\in\mathcal{M},\ \mathcal{H}_*[\Re_u]>0\ \text{or}\ \mathcal{H}^*[\Re_u]<0\bigr\}, \] with the convention $\inf\varnothing=+\infty$. We then reject $H_0$ if $\tilde{\tau}\le \tau_{m_1}$. Denote the rejection event at look $u$ by \[ S^{\tau_u} := \{\mathcal{H}_*[\Re_u]>0\}\cup\{\mathcal{H}^*[\Re_u]<0\}, \] where $\mathcal{H}_*[\Re_u]:=\inf \mathcal{H}[\Re_u]$ and $\mathcal{H}^*[\Re_u]:=\sup \mathcal{H}[\Re_u]$. By construction, $S^{\tau_u}\in\mathcal{F}_u$, and $\tilde{\tau}$ is a stopping time with respect to $\{\mathcal{F}_u\}_{u\in\mathcal{M}}$. For a fixed threshold $\tau_u$, the test can be sized at level $\alpha_u$, i.e. \[ \Pr\left(S^{\tau_u}\mid H_0\right)\le \alpha_u. \] For the sequential procedure, the family-wise Type I error requirement is \[ \Pr\left( \exists u\in\{m_0,\dots,m_1\}:\ (\tilde{\tau}=\tau_u)\cap S^{\tau_u} \ \Big\vert\ H_0 \right)\le\alpha. \] Since the rejection event is a union over looks, \[ \{\exists u:\tilde{\tau}=\tau_u\text{ and }S^{\tau_u}\} = \bigcup_{u=m_0}^{m_1} \left\{(\tilde{\tau}=\tau_u)\cap S^{\tau_u}\right\}, \] Boole's inequality yields
where (ref) holds, because, $\left\{(\tilde{\tau}=\tau_u)\cap S^{\tau_u}\right\}\subseteq \left\{S^{\tau_u}\right\}$, and \[ \Pr\left((\tilde{\tau}=\tau_u)\cap S^{\tau_u}\mid H_0\right)\leq \Pr\left( S^{\tau_u}\mid H_0\right)\leq \alpha_u. \] Thus, a sufficient condition for family-wise control at level $\alpha$ is \[ \sum_{u=m_0}^{m_1}\alpha_u=\alpha. \] Since all $\overline{\mathcal M}=19$ looks are pre-scheduled, we adopt the equal-spending rule of pocock1977group, \[ \alpha_u=\frac{\alpha}{\overline{\mathcal{M}}},\qquad u=m_0,\dots,m_1. \] The corresponding two-sided critical value is $c_u=\Phi^{-1}(1-\alpha_u/2)\approx 3.007$ for $\alpha=0.05$ and $\overline{\mathcal M}=19$.\footnote{Alternative allocations include O'Brien-Fleming o1979multiple or the Lan-DeMets spending function gordon1983discrete.}
As before, for a fixed $\tau$, let $k\in\{0,1\}$ index the untreated and treated groups respectively. For the indices $i,j,t$, denote the population and empirical distribution functions by \[ F^{(k)}(y)=\Pr\left(Y_{ijt}^{(k)}\le y\right), \quad \hat F^{(k)}_{N_k}(y)=\frac{1}{N_k}\sum_{i,j,t}^{N_k}\mathbbm{1}\{Y_{ijt}^{(k)} \le y\}, \] respectively, and define the uniform deviation $\Delta_k:=\sup_{y\in\mathbb R}\lvert \hat F^{(k)}_{N_k}(y)-F^{(k)}(y)\rvert$. As in Sections (ref)--(ref), the problem consists of finding suitable bounding thresholds for the latent conditional expectations $\mathbb{E}[Y_{ijt}^{(0)}\mid Z_{ijt}(\tau)=1]$ and $\mathbb{E}[Y_{ijt}^{(1)}\mid Z_{ijt}(\tau)=0]$. The problem of controlling the family-wise size $\alpha$ for a given level $\alpha\in (0,1)$ using the Bonferroni approach is discussed extensively in Section (ref). However, in this Section, we discuss how to control for an arbitrary size $\alpha'$ for the uniform divergence of the true CDF $F^{(k)}(y)$ and its empirical counterpart $\hat{F}_{N_k}^{(k)}(y)$.
For group $k$, the Dvoretzky-Kiefer-Wolfowitz inequality (Massart-sharp) dvoretzky1956asymptotic implies that for every $c>0$,
Selecting $c=\varepsilon_{k}$ with
yields the DKW event $\mathcal E_{k}:=\{\Delta_k\le \varepsilon_{k}\}$ with $\Pr(\mathcal E_{k})\ge 1-\alpha'$. Thus,
so the construction is jointly valid at the targeted family-wise level.
Uniform control of the CDFs translates into control of tail quantiles. On the event $\{\Delta_k\le\varepsilon_{k}\}$, for every $p\in[\varepsilon_{k},1-\varepsilon_{k}]$ and noting Lemma (ref) in the Appendix, one has
where the superscript $(k)$ is dropped from the inverse CDFs hereafter for the ease of exposition. Choosing $p=2\varepsilon_{k}$ and $p=1-2\varepsilon_{k}$ (which lie strictly inside the admissible interval) and writing $Y^{(k)}_{(1)}\le\cdots\le Y^{(k)}_{(N_k)}$ for the order statistics, \[ F^{-1}(\varepsilon_{k}) \le \hat F^{-1}_{N_k}(2\varepsilon_{k}) =Y^{(k)}_{(\lceil 2\varepsilon_{k} N_k\rceil)}, \quad \hat F^{-1}_{N_k}(1-2\varepsilon_{k}) =Y^{(k)}_{(\lceil (1-2\varepsilon_{k}) N_k\rceil)} \le F^{-1}(1-\varepsilon_{k}). \] We therefore define the DKW-calibrated tail endpoints
These are “outward" in the sense that up to $\varepsilon_{k}$ probability mass is allowed to lie beyond each observed tail; the widening occurs on the probability axis via (ref), and the resulting endpoints are empirical quantiles (order statistics). When population bounds $L^{(k)},U^{(k)}$ are unknown in Sections (ref) and (ref), we plug in $L^{(k)}\leftarrow L_{k}$ and $U^{(k)}\leftarrow U_{k}$ for the relevant group $k$.
Figure (ref) illustrates the construction for a sample of size $N_k=200$. Taking $\alpha'=0.05$ gives \[ \varepsilon_{k} =\sqrt{\frac{\ln(2/0.05)}{2\cdot 200}} =0.096, \] and the shaded band $F^{(k)}\pm \varepsilon_{k}$ envelopes the empirical curve uniformly. In this case the indices in (ref) are $\lceil 2\varepsilon_{k} N_k\rceil=39$ and $\lceil (1-2\varepsilon_{k})N_k\rceil=162$, which are the data-driven tail endpoints replacing unknown support extremes in the bounds and tests that follow.
In Section (ref) we defined the unconditional mean‐comparison parameter $\delta$ (Eq. (ref)) and Manski’s bounds $\Re$ (Eq. (ref)). We now give their sample estimates and show how to tighten the bounds via quantiles.
For a fixed $\tau$, the estimator of $\delta$ can be written as a single weighted sum: \[ \hat\delta = \sum_{j=1}^{K}\sum_{i=1}^{n^j}\sum_{t=1}^{T} Y_{ijt}\,w_{ijt}, \qquad w_{ijt} = \frac{Z_{ijt}(\tau)}{N_1} -\frac{1 - Z_{ijt}(\tau)}{N_0}, \] where \[ N_k =\sum_{i,j,t}\mathbbm{1}\{Z_{ijt}(\tau)=k\}, \quad k\in\{0,1\}, \quad N=N_0+N_1. \] It then follows by the Central Limit Theorem (CLT hereafter) that
To estimate $\Re$, recall from (ref) that $\Re$ involves the four quantities $\mathbb{E}[Y_{ijt}^{(k)}\mid Z_{ijt}(\tau)=k]$ and $\Pr(Z_{ijt}(\tau)=k)$, $k=0,1$, plus the endpoints $\{L^{(k)},U^{(k)}\}$. We estimate them by \[ \hat\delta_k = \frac{1}{N_k}\sum_{i,j,t}Y_{ijt}\mathbbm{1}\{Z_{ijt}(\tau)=k\}, \qquad \hat p_k = \frac{N_k}{N}, \] and in the case of classical manski1990nonparametric bounds \[ L^{(k)} = \min\{Y_{ijt}:Z_{ijt}(\tau)=k\}, \quad U^{(k)} = \max\{Y_{ijt}:Z_{ijt}(\tau)=k\}, \] noting that $\hat\delta_1-\hat\delta_0=\hat\delta$ and $\hat p_1=1-\hat p_0$.
To tighten the raw‐support bounds, as means of illustrating the sensitivity of the bounds, one may replace $L^{(k)},U^{(k)}$ by the sample $p$- and $(1-p)$-quantiles in each group: \[ \hat F^{(k)}(y) = \frac{1}{N_k} \sum_{i,j,t}\mathbbm{1}\{Y_{ijt}\le y,\;Z_{ijt}(\tau)=k\}, \qquad \hat Q_Y^{(k)}(p) = \inf\{y:\hat F^{(k)}(y)\ge p\}, \] or equivalently $\hat Q_Y^{(k)}(p)=Y^{(k)}_{(\lceil pN_k\rceil)}$ when $Y_{(1)}^{(k)}\le\cdots\le Y_{(N_k)}^{(k)}$ are group‑$k$ order stats. Finally, in (ref) substitute \[ L^{(k)}\mapsto \hat Q_Y^{(k)}(p), \qquad U^{(k)}\mapsto \hat Q_Y^{(k)}(1-p), \] to obtain the quantile‐based bounds.
In Section (ref) below we describe how to construct $(1-\alpha)\%$ confidence bands for $\hat\delta$ and for the nonparametric bounds via the Bonferroni‐adjusted delta‐method.
For $u = m_0,\dots,m_1$, constructing $(1-\alpha_u)$ confidence intervals for the naïve estimator $\hat\delta$ is rather straightforward, because by definition, $\hat\delta$ is simply a difference of sample means. Recall from (ref) that \[ \sqrt{N}(\hat\delta - \delta)\xrightarrow{\mathcal{D}} N(0,\sigma^2). \] where
is the asymptotic variance parameter and $p_k = \Pr(Z_{ijt}(\tau_u)=k)$. Hereafter, for notational simplicity and without loss of generality, we write $N_k$ instead of $N_k(\tau_u)$ for each look $u$, keeping the dependence on $\tau_u$ implicit.
A consistent estimator of $\sigma^2(\hat\delta)$ is the usual difference-in-means estimator:
where \[ \hat\sigma_k^2 = \frac{1}{N_k-1}\sum_{Z_{ijt}(\tau)=k}(Y_{ijt}-\bar Y_k)^2, \qquad \bar Y_k = \frac{1}{N_k}\sum_{Z_{ijt}(\tau)=k} Y_{ijt}, \] and $\hat p_k = N_k/N$.
Since $\text{Var}(\hat\delta)\approx \sigma^2(\hat\delta)/N$, a $(1-\alpha_u)$ Wald-type confidence interval for $\delta$ is therefore
where $\Phi^{-1}(\cdot)$ denotes the standard normal quantile function.
Obtaining confidence intervals for the nonparametric bounds is less straightforward, since the upper and lower bound estimators are nonlinear functions of the data. We therefore rely on a Bonferroni-adjusted delta method, as formalized in the following proposition.
A major shortcoming of the nonparametric bounds proposed by manski1990nonparametric,manski2003partial and introduced in Section (ref) is the strong assumption that the latent conditional expectations in Eq. (ref) lie inside a known bounded interval \[ [L^{(k)},\;U^{(k)}], \qquad k\in\{0,1\}. \] In practice, these expectations may be unbounded. To address this, we reformulate the problem probabilistically and study the probability \[ \Pr\left(\forall u \in \mathcal{M},\;\Re_u \in \mathcal{H}_{\alpha_u}[\Re_u]\right). \]
We first analyse a stylized setting in which the observations $Y_{ijt}$ are independent across all indices $i,j,t$. Although independence may be reasonable for purely cross-sectional snapshots (for instance, a single quarter across many sectors) it is unrealistic in panel settings, where serial correlation is typically present. Accordingly, we later extend our results to allow for temporal dependence within firm, while maintaining cross-sectional independence across firms.
For each threshold value $\tau_u$, we define \[ \delta_k := \mathbb{E}\left[Y^{(k)}_{ijt}\mid Z_{ijt}(\tau_u)=k\right], \qquad p_k := \Pr\left(Z_{ijt}(\tau_u)=k\right), \qquad k\in\{0,1\}, \] with the dependence on $u$ left implicit to lighten notation. Following manski1990nonparametric,manski2003partial, the identified set for the tipping-point functional $\Re_u$ is then
The latent cross-terms $\delta_{10} := \mathbb{E}[Y^{(1)}_{ijt}\mid Z_{ijt}(\tau_u)=0]$ and $\delta_{01} := \mathbb{E}[Y^{(0)}_{ijt}\mid Z_{ijt}(\tau_u)=1]$ are replaced by their support bounds $(L^{(k)},\;U^{(k)})$ to obtain the interval in (ref). In the first setting, we assume that the data exhibit neither cross-sectional nor serial dependence.
Proposition (ref) states that the data-driven set $\mathcal H_{\alpha_u}[\Re_u]$ in (ref) is a $100(1-\alpha_u)\%$-level confidence region for the average treatment effect $\Re_u$ under nothing more than i.i.d. sampling. Because $\mathbb{E}[Y_{ijt}^{(1)}\mid Z_{ijt}(\tau_u)=0]$ and $\mathbb{E}[Y_{ijt}^{(0)}\mid Z_{ijt}(\tau_u)=1]$ are latent, point identification is impossible without additional assumptions; the proposition nevertheless guarantees that the random interval constructed from the empirical means, treatment proportions, and DKW-calibrated empirical quantiles will cover the true $\Re_u$ in at least $100(1-\alpha_u)\%$ of repeated samples. Practically, one computes $\mathcal H_{\alpha_u}[\Re_u]$ by (i) splitting the sample into treated and untreated subsamples, (ii) forming the subsample-specific means $\hat\mu_k$ and proportions $\hat{p}_k$, (iii) computing $\varepsilon_k=\sqrt{\log(12/\alpha_u)/(2N_k)}$ and choosing any $r_k\in(0,\frac12-\varepsilon_k]$, then setting the tail endpoints \[ L_{\alpha_u}^{(k)}=Y^{(k)}_{\left(\lceil (r_k+\varepsilon_k)N_k\rceil\right)}, \quad U_{\alpha_u}^{(k)}=Y^{(k)}_{\left(\lceil (1-r_k-\varepsilon_k)N_k\rceil\right)}, \] (in particular, if $\varepsilon_k\le \frac14$ one may take $r_k=\varepsilon_k$, yielding the convenient indices $\lceil 2\varepsilon_k N_k\rceil$ and $\lceil(1-2\varepsilon_k)N_k\rceil$), and (iv) plugging these objects into (ref). The resulting band can be used exactly like an ordinary confidence interval: the null hypothesis $H_0: 0\in \Re_u$ is rejected at level $\alpha_u$ whenever $0\notin \mathcal H_{\alpha_u}[\Re_u]$.
If substantive knowledge implies that the latent outcomes are truncated on one or both tails, the extreme-value inputs in Manski’s bounds can be replaced by the true population limits. Let $\lambda$ (lower) and $\Lambda$ (upper) denote any such known bounds. When both limits are known one sets $L^{(k)}=\lambda$ and $U^{(k)}=\Lambda$ in the plug-in formulas; the resulting $100(1-\alpha)\%$ simultaneous band coincides with Proposition (ref) and requires no DKW calibration. When only one tail is known, say $Y\ge \lambda$, but the upper support is unknown, we fix the lower extreme at $\lambda$ while retaining the DKW-calibrated endpoint on the upper side. The next Corollary shows that this hybrid construction preserves the nominal family-wise coverage probability even when the first significant threshold is data-selected.
From here on, we weaken the i.i.d.\ assumption and allow the data to exhibit weak dependence by assuming each series is a stationary $\upalpha$-mixing process. For example, any stationary AR($1$) model satisfies this condition.
Similar to Corollary (ref), the extension of Proposition (ref) to the case of one-tail truncated latent conditional expectations simply requires modifying the endpoints. Specifically, for the case of lower-tail truncation, replace \[ L^{(k)} := \lambda, \quad U^{(k)}_{\alpha_u} := \hat F^{-1}_{N_k}\left(1 - r_k - \tilde{\varepsilon}_k\right) = Y^{(k)}_{\left(\lceil (1 - r_k - \tilde{\varepsilon}_k)\,N_k\rceil\right)}, \] with the fixed lower bound and a one-sided DKW-calibrated upper empirical quantile, where \[ \tilde{\varepsilon}_k:=(1+4C_{\upalpha})\sqrt{\frac{2\log\!\left(6/\alpha_u\right)}{N_k}}, \qquad r_k\in\left(0,1-\tilde{\varepsilon}_k\right], \] for $k = 0,1$, and the observed order statistics satisfy $\lambda \leq Y_{(1)}^{(k)} \leq \cdots \leq Y_{\left(\lceil (1-r_k-\tilde{\varepsilon}_k)(N_k)\rceil\right)}^{(k)}$. (When $\tilde{\varepsilon}_k\le \tfrac14$, a convenient choice is $r_k=\tilde{\varepsilon}_k$, yielding the index $\lceil (1-2\tilde{\varepsilon}_k)N_k\rceil$.)
A drawback of the finite‐sample bands in Propositions (ref) and (ref) is that the Bernstein and Hoeffding‐type paddings for sub‐exponential tails depend on multiple nuisance constants (mixing rates, sub‐exponential parameters, etc.), which quickly becomes cumbersome in practice. Moreover, although the sub‐exponential assumption is fairly general, it is still a substantive restriction on the data. In Proposition (ref) we therefore introduce a hybrid confidence band that combines
Although asymptotic normality can be used for smooth mean components, asymptotic quantile inference would require additional tail regularity conditions and delivers only pointwise validity. The DKW inequality instead provides uniform, finite-sample control of tail uncertainty without imposing distributional assumptions, which is essential for constructing joint confidence bands for interval-identified parameters.
Proposition (ref) thus guarantees that our hybrid concATE band achieves the desired confidence level for the ATE bounds even when data are dependent, by using the DKW inequality with appropriate mixing corrections.
Following the same logic as Corollary (ref), the extension of Proposition (ref) to the case of one-tail truncated latent conditional expectations replaces the endpoints by \[ L^{(k)} := \lambda, \qquad U^{(k)}_{\alpha_u} := \hat F^{(k),-1}_{N_k}\left(1 - r_k - \tilde{\varepsilon}_k\right) = Y^{(k)}_{\left(\lceil (1 - r_k - \tilde{\varepsilon}_k) N_k\rceil\right)}, \] where \(\tilde{\varepsilon}_k=(1+4C_{\upalpha})\sqrt{2\log(4/\alpha_u)/N_k}\) and \(r_k\in(0,1-\tilde{\varepsilon}_k]\).
Note that in the case of weakly dependent data, the hybrid concATE approach in Proposition (ref) depends on the nuisance constant $C_{\upalpha}=\sum_{k\geq 1}\upalpha(k)^{1/2}$. For the important special case of a stable Gaussian AR($1$) process, this constant admits a closed-form upper bound.
To further strengthen these results, we formally derive and discuss the optimality of the concATE approach in the minimax sense in Section (ref).
The hybrid concATE approach introduced in Proposition (ref) uses DKW-calibrated order statistics of the treated and untreated outcome variables to derive $\alpha'$-sized lower and upper bands on the set-identified ATE. A natural question is whether a different distribution-free procedure (e.g., bootstrapping) could achieve a tighter worst-case interval while maintaining the same coverage. In what follows, we show that the DKW-calibrated endpoints $L_{\alpha'}^{(k)}$ and $U_{\alpha'}^{(k)}$ of concATE, for $k\in\{0,1\}$, are minimax-rate-optimal tsybakov2008nonparametric, which implies that no other distribution-free procedure can achieve a faster rate of worst-case excess bracket width, while maintaining valid coverage over the class $\mathcal{F}$ of all outcome distribution with finite mean\footnote{Note that the definition of $\mathcal{F}$ is distinct from the $\sigma$-field definition in Section (ref)}. The minimax results concern the tail-bracketing component of the confidence band, since the smooth components, i.e., $\hat{\delta}_k$ and $\hat{p}_k$, rely on the asymptotically valid delta-method, where the CLT applies.
Before introducing the first set of results, let us denote $\mathcal{F}$ as the class of all Borel probability measures on $\mathbb{R}$ with finite mean. The results presented are for each arm $k\in\{0,1\}$, but for ease of exposition, we drop the subscript (and subscripts) $k$. Furthermore, following the notations in Section (ref), $\alpha'$ is some generic $\alpha'\in(0,1)$ and the thresholds $\tau$ are suppressed.
For $F\in\mathcal{F}$ and a fixed tail probability $r\in\left(0,\tfrac{1}{2}\right)$, define the true tail endpoints \[ q_r:=F^{-1}(r),\qquad q_{1-r}:=F^{-1}(1-r), \] where $G^{-1}(u)=\inf\{y:G(y)\geq u\}$ as in Lemma (ref). In what follows, we find the minimax lower and upper bounds for concATE.
The lower and upper bounds of the the bracket width of the DKW-calibrated tail endpoints in Propositions (ref) and (ref) lead to the following Corollary.
The statements from Corollary (ref) can be verified as follows: let us first consider the case of plug-in sample extrema. Given the order statistics $Y_{(1)},\dots,Y_{(N)}$, using $Y_{(N)}$ are the upper bound tail endpoint, fails coverage whenever the hypothesised class of distributions $F_{1}$ concentrates mass above the observed sample range with positive probability (e.g., DGP E in Section (ref). Hence, the sample maximum does not satisfy the distribution-free bracket condition over $\mathcal{F}$. Similarly, boostrap procedures achieve the rate $O\left(N^{-1/2}\right)$ only under additional regularity condition (e.g., bounded support of Donsker conditions van1996weak, and can fail for heavy-tailed distributions $F\in \mathcal{F}$.
To study the finite sample behaviour of the hybrid band in Proposition (ref), we run a Monte Carlo experiment with six data generating processes (DGPs). Each design is replicated $B=2{,}000$ times on a single sector with $n=50$ firms observed for $T\in\{1,2,5\}$ periods, giving sample sizes $N=nT\in\{50,100,250\}$ respectively. A single diversity cut-off $\tau^{\circ}=50\%$ is analyzed; hence no Bonferroni size split is required. The overall two-sided size is fixed at $\alpha=0.05$, giving the critical values \[ c_{M}=\Phi^{-1}\!\left(1-\alpha/2\right)\approx1.96 \quad\text{and}\quad c_{H}=\Phi^{-1}\!\left(1-\alpha/4\right)\approx2.24 \] correspondingly for the manski1990nonparametric and Hybrid (concATE) approaches.
Within each replication and arm $k\in\{0,1\}$ we compute \[ \varepsilon_k:=\sqrt{\frac{\log\left(8/\alpha\right)}{2N_k}}, \quad r_k\in\left(0,\frac12-\varepsilon_k\right], \] and set the outward empirical-quantile endpoints \[ L^{(k)}:=Y^{(k)}_{\left(\lceil (r_k+\varepsilon_k)N_k\rceil\right)}, \quad U^{(k)}:=Y^{(k)}_{\left(\lceil (1-r_k-\varepsilon_k)N_k\rceil\right)}. \] (When $\varepsilon_k\le\frac14$ we take $r_k=\varepsilon_k$, yielding the convenient indices $\lceil 2\varepsilon_kN_k\rceil$ and $\lceil(1-2\varepsilon_k)N_k\rceil$; if $\varepsilon_k>\frac14$, we set $r_k=\frac12-\varepsilon_k-\frac1{N_k}$ to preserve ordering.) The Manski benchmark uses the usual plug-in with known support when available (DGP F), and otherwise the sample extrema. The realized outcome is \[ Y_{it}^{\mathrm{obs}} = Y_{it}^{0} + \Delta D_{it}, \qquad \Delta = 4, \] where $Y_{it}^{0}$ follows the distribution listed below and $D_{it}\sim\mathrm{Bernoulli}(0.3)$.
For each replication, we also compute: (i) the imbens2004confidence confidence interval, which solves for the critical value $c_n$, satisfying $\Phi(c_N+\sqrt{N}\Delta_N)-\Phi(-c_N)=1-\alpha$, where $\Delta_N=(\hat{U}-\hat{L})/\hat{\sigma}_{\max}$; (ii) The stoye2009more interval, which sets $c=\max(c_N,z_{\alpha/2})$ to ensure uniform validity; and (iii) percentile bootstrap with $B_{boot}=999$ replications, using a Sieve bootstrap for DGP C, when $T> 1$ buhlmann1998sieve.
DGP D is specifically constructed so that the finite sample has a tunable probability of not observing the population extremes, the precise scenario for which the hybrid band was developed.
Table (ref) reports the pointwise Monte Carlo coverage of the identified set for the average treatment effect evaluated at the tipping point $\tau^{\circ} = 50\%$. For each replication, we check whether $\Delta$ lies inside the manski1990nonparametric, imbens2004confidence, stoye2009more, Bootstrap and Hybrid (concATE) bounds, at this single threshold.
At first glance, it is evident that all the bounds are valid with a 95% coverage, with the concATE followed by the bootstrap approaches being the closest to the nominal 95% level. In fact, their slightly-below-100% coverage reflects better calibration rather than a deficiency, since a method that always reports 100% coverage is being unnecessarily conservative. As the sample size increases, Hybrid coverage converges to 100% as the asymptotic delta-method component improves, while remaining well above the 95% nominal threshold throughout.
The width of the bounds reveals an entirely different story. For DGPs A--D the Manski, IM, Stoye and Bootstrap approaches all exhibit similar performances. As it is evident, in the case of DGP A, the Manski, IM and Stoye approaches all seem to converge to the same bound width of $\approx 5.6$ as the sample size increases. Similar converging phenomena is observed for DGP C, which are sampled from a normal distribution, but with a weakly dependent structure, which has increased the width. However, in the case of heavy-tailed distributions, while the results of the different techniques are similar for larger samples, the width seems to get extensively larger. This would simply be because a larger sample from the heavy-tailed DGPs implies larger deviations and hence variances.
In all instances, the width of the concATE approach shrinks with larger samples and the method provides sharper bounds on the average treatment effect. Quantitatively, the Hybrid band is 2--4 times narrower than all competing methods across DGPs A--D. For example, at $N=50$ under DGP A, the Hybrid width is 2.0 compared to 5.2 for Manski - a 62% reduction. This efficiency gain arises because the hybrid replaces arm-specific sample extrema, which systematically overshoot the population support, with DKW-calibrated quantile endpoints that trim tail mass with controlled probability, yielding a tighter identification region without sacrificing coverage.
The advantage of the Hybrid is particularly pronounced under DGP D, which is specifically constructed so that finite samples have a tunable probability of missing the population extremes. At $N=50$, the Hybrid width is 7.9 compared to 14.1 for Manski, while both methods maintain coverage above 95%. Even as $N$ increases to 250 and the competing methods' coverage remains at 100%, their widths grow to 16.4 whereas the Hybrid width is only 7.5 - a gap that widens rather than closes, precisely because larger samples from heavy-tailed DGPs produce more extreme order statistics that inflate the plug-in support bounds. The Hybrid, by contrast, is insulated from this effect through its quantile-based construction.
In this section, we ask “Does gender-based board diversity causally affect firm innovation?". We begin by outlining the data and summarizing its key descriptive statistics. We then present the nonparametric bounds approach of manski1990nonparametric, manski2003partial with simultaneous confidence bands in Proposition (ref), the hybrid band proposed in Proposition (ref), and the naïve mean‐comparison framework of angrist2009mostly (reported in Appendix (ref)). The common objective is to test the null hypothesis of a zero average treatment effect of diversity on innovation, against a positive or negative alternative, when the diversity cut-off is selected endogenously (see Eq. (ref)).
The empirical analysis uses a panel of publicly listed firms compiled from FactSet, with quarterly observations from 2015 Q2 through 2022 Q1. The initial sample includes 945 firms, yielding a short panel of 945 cross-sectional units over 28 quarters (totalling 26,460 firm-quarter observations).
In our analysis, we categorize the eleven Global Industry Classification Standard sectors (GICS hereafter) into five broader groups: Cyclicals (Consumer Discretionary, Materials, Industrials, Real Estate), Defensives (Health Care, Consumer Staples, Utilities), Growth & Innovation (Information Technology, Communication Services), Financials, and Energy. This classification reflects the economic sensitivities of these sectors, as identified by Morgan Stanley Capital International (MSCI hereafter). Specifically, MSCI's Cyclical and Defensive Sectors Indexes classify sectors based on their performance correlation with the business cycle, using the OECD Composite Leading Indicator. According to MSCI, sectors like Consumer Discretionary, Materials, Industrials, Real Estate, Information Technology, Communication Services, and Financials are considered cyclical due to their positive correlation with economic expansions. Conversely, sectors such as Health Care, Consumer Staples, Utilities, and Energy are deemed defensive, exhibiting resilience during economic downturns. By adopting this grouping, we aim to capture the nuanced behaviours of these sectors in relation to macroeconomic conditions, facilitating a more informed analysis of sectoral dynamics. This classification can be found in Table (ref).
Following the Corporate Sustainability Reporting Directive definition (CSRD hereon) of a ‘large undertaking’ (Directive 2022/2464/EU, Art. 3 Pt 4) and the 250-employee threshold used in EU and UK gender-pay-gap statutes, we restrict the sample to firms whose time-average workforce is at least 250 employees over the sample horizon to ensure they fall under harmonized disclosure regimes. The restriction yields $n=901$ firms and a total of $N=25,228$ firm-quarter observations.
The key “treatment” variable is the percentage of women in senior leadership positions. To develop measures of senior management diversity, we use data from Revelio Labs. Revelio Labs is a data provider that aggregates workforce data from multiple sources, including online professional profiles (e.g., LinkedIn), job postings, company websites, government records and census data. Data are aggregated by firm and time and weighted to correct for underrepresentation in online professional profiles, such as lower-skilled or lower-paid roles, as well as to address issues such as mapping discrepancies and the presence of fake or duplicate profiles. For each firm, Revelio provides global and local headcounts and demographic characteristics of employees including gender and ethnicity, based on probabilistic estimations. From this data, we focus on the gender and ethnicity diversity of employees located in the US or the U.K., serving as a proxy for headquarters location, since the firms in our sample are listed on the MSCI US and S&P U.K. indices. Our analysis is concentrated on senior management positions, defined as executive and senior executive roles (e.g., Managing Director, Partner, CEO, CFO). We restrict the sample to senior management in the headquartered country as there is a growing literature indicating that executive and senior management characteristics impact management practices and firm performance flabbi2019female. Additionally, these roles are less likely to be underrepresented in the Revelio dataset. These diversity measures are constructed using a supervised machine learning algorithm applied to senior executives’ names, which infers gender from linguistic patterns. If the algorithm cannot assign a gender with high confidence, the individual is labelled as “unknown”. Importantly, the incidence of unknown classifications is very low: on average only about 0.03% (Table (ref)).
The outcome of interest is Tobin’s $Q$, defined as the ratio of the firm’s market value to the replacement cost of its assets, a standard measure of firm performance and growth opportunities tobin1969general,tobin1978monetary. We also utilize several control variables for descriptive analysis, including firm size (log total assets), leverage (debt-to-assets ratio), and total employees. Summary statistics for all main variables are provided in Table (ref). After excluding observations with missing data on key fields, the average percentage of women in senior roles is about 27.5%. The standard deviation (around 12 percentage points for female share) indicates considerable cross-firm variation. Notably, a non-trivial subset of firm-quarters have zero diversity: roughly 4% of observations have no women in senior leadership, at least at some point in the sample. The distribution of the diversity variables is right-skewed. Figure (ref) illustrates kernel density estimates of the percentage of female senior leaders across all firm-quarters. The distribution is skewed to the right with a primary mode around 25--35%, and a secondary mass at 0% corresponding to firms and periods with homogeneous leadership teams.
We next explore the raw association between gender diversity measure and firm performance. In the full sample (pooled across all sectors and time periods), there is a strong positive correlation between senior-team diversity and Tobin’s $Q$. Figure (ref) plots rolling correlations over time, using a moving window of half the sample period ($T/2 \approx 14$ quarters) to track how the relationship evolves. The Pearson correlation between the percentage of women in leadership and Tobin’s $Q$ is in the range of +0.6 to +0.7 for most of the sample, indicating a fairly strong linear association. The Figure also reports Kendall’s $\tau$ rank correlation, which captures monotonic association; this measure corroborates the positive link while being slightly lower in magnitude, suggesting the relationship is broadly monotonic even if not perfectly linear. The association appears to strengthen from 2015 up to about 2019, consistent with increasing awareness and implementation of diversity initiatives, but then shows a noticeable drop around 2019-2020. After 2019, the rolling correlations decline, implying that the previously tight diversity-performance relationship loosened and is increasing again after 2021. One possible interpretation is that external shocks or changing market conditions (for instance, the disruptive impact of the COVID-19 pandemic or the murder of George Floyd) temporarily weakened the correlation between diversity and market valuations.
In light of this, and in addition to the sectoral group analysis, we examine the overall rolling correlations for the period preceding this drop. The corresponding Pearson correlations for this classification are reported in Figure (ref), while the rolling associations are shown in Figure (ref).
Several noteworthy patterns emerge. First, the Growth & Innovation sector consistently exhibits a strong positive correlation between gender diversity measures and Tobin’s $Q$ across all years, and this sector does not experience the 2019 drop in correlation seen in the aggregate data. Second, the Energy sector shows a markedly different pattern: the percentage of women in senior positions in energy firms is actually negatively correlated with Tobin’s $Q$ in most years. These observations may reflect unique dynamics or reverse causality in the energy industry (for example, struggling firms may appoint more women to leadership roles as part of restructuring). Third, in the Financials sector, the correlation with diversity is negative in the earlier part of the sample (implying more homogenous banks were associated with slightly higher $Q$ ratios pre-2019), but this relationship reverses sign around 2019. By the end of the sample period, financial firms with more diverse leadership tend to have higher Tobin’s $Q$, indicating a possible structural change in how markets value diversity in finance or how an increase in inclusion that enabled diversity to be leveraged for business gains.
\FloatBarrier
While the descriptive results suggest a concordance between greater senior-level gender diversity and higher firm performance, correlation alone cannot establish causality. In this section, we formally test whether increases in executive diversity causally impact Tobin’s $Q$, using the methodology developed in Sections (ref)-(ref). Because the “treatment” (crossing a diversity threshold) is not randomly assigned, a naïve estimation of this effect risks bias from selection on unobservables. We therefore implement both a conventional point-estimation approach under strong assumptions and a robust partial-identification approach under minimal assumptions, and compare the findings.
First, we apply an unconditional mean-comparison framework following angrist2009mostly. For each candidate diversity threshold $\tau$ (e.g. 5%, 10%, …, 50%, etc.), firms are split into a treated group (above the threshold) and a control group (below the threshold). We then estimate the difference in mean Tobin’s $Q$ between treated and control firms for that threshold. This difference-in-means is a point estimate of the ATE if one assumes mean independence (i.e. that, conditional on crossing the threshold, potential outcomes are the same for treated and control firms on average). We construct simultaneous 95% confidence bands for these ATE estimates across all thresholds in the set $\mathcal M = {5, 10, 15, …, 90, 95}$, applying a Bonferroni or Šidák correction to account for the multiple comparisons. This yields a series of tests for the null hypothesis of no effect at each diversity level, adjusted so that the overall family-wise error rate is 5%. It is important to note that this point-identified approach treats the threshold “treatment” as if random; in practice, firms that surpass a given diversity level could differ systematically from those that do not (for instance, more progressive or better-governed firms might both adopt diverse leadership and perform well for other reasons). As a result, the point estimates of $\delta$ may capture more than the true causal effect of diversity. We use this method as a benchmark, fully aware that its validity hinges on strong assumptions.
We next relax the strong assumptions by employing a partial identification strategy manski1990nonparametric,manski2003partial. Instead of assuming we can precisely identify the counterfactual outcome for each firm, we derive bounds on the possible ATE. Instead of point identification, we partially identify the region in which the average treatment effect $\Re$ lies, as characterized by Eq.(ref). We denote this set the identification region $\mathcal H_{\alpha_u}[\Re_u]$ for all $u$ in $\mathcal{M}$, where as noted in Section (ref), $\mathcal{M}=\{5,10,15,\cdots,90,95\}$ which represents the random diversity thresholds. As previously noted, estimation of Eq. (ref) involves latent quantities $\mathbb{E}[Y_{ijt}^{(0)} \mid Z_{ijt}(\tau_u)=1]$ and $\mathbb{E}[Y_{ijt}^{(1)} \mid Z_{ijt}(\tau_u)=0]$, which are not observed but can be bounded by quantities $L^{(k)}$ and $U^{(k)}$. On one hand, we may acknowledge that the extrema of the latent outcomes within the finite sample may not capture the true population extrema (and consequently the true treatment effect interval), in which case we rely on the finite sample hybrid approach. On the other hand, one may argue that since using the full range of outcomes ($\min$ and $\max$) can lead to overly conservative bounds, we also construct Manksi bounds using the (5\textsuperscript{th}, 95\textsuperscript{th}) and (10\textsuperscript{th}, 90\textsuperscript{th}) quantiles of $Y_{ijt}^{(k)}$. Finally, we build a simultaneous joint 95% confidence region for the estimated bounds to make causal inference claims.
Before turning to results, we address some practical implementation details. As noted in Section (ref), it is necessary for both the treated and control groups to be non-empty (and sufficiently large) at each threshold to estimate meaningful effects. In our panel, some extreme diversity thresholds (especially very high ones) result in very few firms in one group. We therefore discard threshold levels $\tau$ for which one of the groups contains fewer than 10 observations (approximately, we require at least 10 firm-quarters above and below the threshold). If too many high-$\tau$ values are discarded for a particular subset of the data, that subset is excluded from the threshold analysis due to lack of support. In practice, this means that for some sector-specific analyses we cannot evaluate very high diversity percentages because, for example, no firm in a given sector ever reaches 90% female leadership. Based on this criterion, certain combinations of sector and diversity type are dropped from the causal analysis. In particular, we exclude female leadership in sectors that never approach gender parity (notably the Financials and Energy sectors). These exclusions are a matter of data availability and ensure that the identification regions for ATE do not trivially collapse to a point. All remaining sector clusters and diversity measures satisfy $0 < \Pr(Z(\tau_u)=1) < 1$ at the thresholds of interest, so both treated and control outcomes can be observed in those cases.
We first examine the impact of senior-level gender diversity on firm performance in the full sample, comparing a naive point estimate approach to our partial identification methods. Using the unconditional difference-in-means (Angrist and Pischke's approach), we find that greater female representation is associated with higher Tobin's $Q$, with an apparent “tipping point” at moderate diversity levels. In particular, once women comprise roughly one-third of the top management team, the naive ATE estimate becomes positive and statistically significant. For example, crossing about 30--35% female leadership is associated with a jump in Tobin's $Q$ (see Table (ref), which summarizes the estimated threshold levels at which the treatment effect becomes significant under each method): beyond this threshold the simple treated-control difference excludes zero at the 5% level. This suggests that, under strong assumptions of ignorability, even a moderately gender-diverse leadership team might boost firm market value.
However, when we relax those assumptions, the evidence is less definitive. Manski's nonparametric bounds, which allow for arbitrary selection and unobserved heterogeneity, remain wide in finite samples, and their associated confidence band always includes zero. At very low diversity levels, the lower bound on the ATE is substantially negative (reflecting the worst-case scenario that “token” diversity could coincidentally occur in poorly performing firms). As the female share increases, this lower bound rises toward zero. We observe an inflection around 20--25% female representation, roughly consistent with kanter1977some's notion of moving from tokenism to a more influential minority. Beyond that point, the worst-case impact of diversity is no longer severely negative; by around 50% female leadership, the lower bound is near zero and the upper bound is positive. Nevertheless, without additional information about outcome limits, even at the highest diversity levels observed (e.g.\ 80--90% female), the 95% confidence region for the ATE still straddles zero. In other words, under minimal assumptions the data do not allow us to conclusively rule out no effect (or even a small negative effect) for the overall sample. This highlights how misleading the precise naive estimate can be: what appears as a clearly positive effect with a simple mean comparison becomes statistically ambiguous once we account for uncertainty about counterfactual outcomes.
Imposing mild outcome bounds yields somewhat tighter inference. If we assume, for instance, that Tobin's $Q$ outcomes are effectively constrained within the central 90--95% of the observed sample range (excluding extreme tail realizations), the identified ATE interval narrows. Under these plausible restrictions, the partial identification bounds move inward: the lower bound is higher (less negative) and the upper bound lower (less positive) than the unbounded Manski case. As a result, the concATE confidence band becomes more optimistic at high diversity levels. For example, using the 5\textsuperscript{th} and 95\textsuperscript{th} percentiles of $Q$ as rough bounds, we find that at very high diversity (above about 60% women in leadership) the lower bound on the ATE is nearly zero or slightly positive. With an even tighter 10\textsuperscript{th}--90\textsuperscript{th} percentile restriction, the lower bound actually rises above zero at some thresholds. These results hint that a real positive effect may emerge once diversity is sufficiently high: with female leadership above roughly two-thirds, even the worst-case impact is likely to be zero. However, we emphasize that even under these trimmed-outcome assumptions, the joint 95% confidence band for the ATE barely excludes zero. In the full sample, no diversity threshold produces a completely robust positive effect at the 5% significance level unless one accepts some outcome-range assumptions. Thus, our most cautious conclusion for the overall dataset is that greater gender diversity could improve firm value, but the evidence is not statistically conclusive under minimal assumptions. The contrast between the naive point estimate (significant at $\sim$30% diversity) and the conservative bounds (no significance without assumptions) underscores the importance of conservative, rigorous inference: smaller apparent gains may reflect unobserved biases or heavy-tailed outcomes rather than true causal effects.
\noindentSector-Specific “Tipping Points"
We next investigate whether the diversity--performance relationship exhibits stronger effects in particular types of firms. To explore this, we apply an identical analysis within more homogeneous sector groupings. In each case we report the threshold at which the concATE band indicates a significant effect, and compare it to the naive and classical bounds results. This reveals several interesting findings.
First, in high-growth, innovation-intensive industries, we find clear evidence of a diversity tipping point. Firms in these sectors show relatively high variance in leadership composition, with some approaching gender-balanced teams. The unconditional mean comparison suggests a positive effect of diversity that becomes sizeable at upper diversity levels. However, due to the smaller sample of firms in this category, the naive threshold for significance is somewhat high: only at nearly half female representation does the simple difference in Tobin's $Q$ become significant. Our robust analysis confirms and sharpens this finding. The concATE confidence band for the average treatment effect in Growth & Innovation firms first excludes zero at approximately 55% female representation in senior roles. In other words, once a firm's top team is roughly half women, we can confidently assert a positive causal impact on market valuation in this sector. Below that threshold, the partial-identification interval still includes zero, meaning the effect cannot be distinguished from zero (i.e.\ while we cannot assert a positive effect, we can rule out a negative effect) with high confidence. Notably, the estimated ATE grows larger as diversity increases beyond 55%; for firms that actually achieve gender-balanced or women-majority leadership, even the lower bound of the effect is distinctly above zero. This pattern aligns with the idea that innovative companies reap substantial benefits from diverse perspectives only after achieving a critical mass of diversity. Before that point, female voices may be too diluted to change organizational outcomes, but around parity their influence on decision-making and the innovation climate becomes strong. It is encouraging that both the naive method and the more rigorous concATE method point to a similar threshold in these sectors (around 50--55% female): this convergence suggests the result is not merely an artefact of assumptions. In sum, for Growth & Innovation firms we find a statistically significant positive causal effect of diversity emerging at just over half women in leadership.
Second, for Defensive sectors (Healthcare/Staples/Utilities), traditionally “stable” industries that historically have lower female leadership representation, fewer firms in our sample reach high diversity levels. A naive analysis indicates that even moderate diversity might help performance: the difference-in-means suggests an uptick in Tobin's $Q$ once the female share surpasses roughly 40% in these sectors. Indeed, raw correlations in the Defensive group are positive, hinting that more diverse leadership teams tend to coincide with slightly higher $Q$ ratios. However, our robust inference reveals that the bar for significance is higher in this context. The concATE confidence band does not exclude zero until female representation reaches around 60% or more in Defensive-sector firms. In other words, only when women form a substantial majority of top management do we find a clear positive effect on firm value with 95% confidence. This implies that at the 60% threshold there are true performance gains from diversity. One interpretation is that these traditional industries require a larger critical mass to overcome legacy cultures and realize the advantages of inclusion. When women remain a small minority (say 20--30%), they may lack the influence or psychological safety needed to affect corporate strategy, yielding no measurable gain. By contrast, if a firm reaches 60% female leadership (an uncommon achievement), it likely reflects deep organizational changes that unlock diversity's benefits (e.g.\ improved problem-solving, stakeholder alignment, or innovation even in mature markets). Thus, for Defensive sectors our findings suggest a delayed tipping point: meaningful performance improvements emerge only at a high level of representation, higher than in fast-paced growth industries. This result underscores how the required “critical mass” can vary by context.
The remaining sectors (Cyclical, Financial, Energy) either showed no robust threshold within our data range or could not be rigorously analysed due to limited support. Cyclical sectors (e.g.\ Consumer Discretionary, Industrials) have intermediate diversity levels. The naive analysis in cyclicals suggested a possible positive effect emerging at about 30% female leadership (similar to the overall sample). Yet, using our more cautious approach, we found that the confidence intervals for the ATE in cyclicals still included zero at all feasible thresholds. In short, we cannot confirm a statistically significant benefit even if the point estimates are positive. We can, however, rule out a negative impact. This does not mean diversity has no effect in cyclical firms, but rather that the data do not provide high-confidence evidence of an effect under minimal assumptions. It is possible that unobserved factors or heavy-tailed outcomes obscure the impact in this mixed group of industries.
For Financial firms, we were unable to identify a tipping point because virtually none of the sampled banks or insurers exceeded 45--50% female leadership during the study period. Since testing a threshold requires some treated and control firms on either side, the lack of any instances of very high diversity meant we could not apply our sequential threshold test in Financials. Interestingly, the correlation between diversity and Tobin's $Q$ in finance was negative in earlier years and then became positive toward the end of our sample (as noted in our descriptive analysis), suggesting a shifting dynamic. Our method would need a longer horizon or more variation to pin down where a critical mass effect might occur in finance, if at all.
The Energy sector remains an outlier. Energy companies not only had the lowest levels of female leadership (maxing out around 50% in our data, with most far lower), but they also exhibited a negative raw correlation between diversity and performance. This negative association could reflect reverse causality or industry-specific factors. For instance, struggling energy firms might appoint more women to leadership in response to external pressures, creating a spurious negative link. In any case, our partial identification analysis did not find any significant positive effect of diversity in Energy. Even at the highest observed female share (just above 50%), the ATE bounds encompass zero and even negative values. Thus, we find no evidence of a beneficial tipping point in Energy firms. We caution that this does not prove diversity harms performance in energy---only that, given the data and minimal assumptions, we cannot confirm any uplift. It is a reminder that the advantages of diversity may not be universal and could depend on complementary organizational changes, such as a culture of inclusion.
In summary, the causality analysis using nonparametric bounds and finite-sample confidence bands paints a more nuanced picture than the raw correlations. The data provide qualified evidence of “tipping points”: in certain high-growth or defence sectors, reaching a critical mass of diversity (for example, women comprising about half of senior leadership roles) is associated with a reliable increase in firm value. For the remaining sectors (with the exception of the energy sector), we can conclude that diversity does not change firm value (i.e.\ it does not have any negative effects).
\FloatBarrier
This paper introduces concATE as a general framework for robust causal inference when point identification is not possible or reliable. By marrying Manski's nonparametric bounds with finite-sample concentration inequalities, concATE offers researchers a new tool to obtain ATE confidence bands without assuming away heavy-tailed outcomes or requiring strong parametric models. Our Monte Carlo results demonstrate that concATE achieves confidence bands that are 2--4 times narrower than those of imbens2004confidence, stoye2009more, and the nonparametric bootstrap, while maintaining valid coverage across all data generating processes considered. Moreover, the DKW-calibrated tail endpoints underlying concATE are minimax-rate-optimal among all distribution-free confidence procedures (Corollary (ref)), implying that no alternative distribution-free method can achieve a strictly faster rate of convergence while preserving coverage. The methodology's broader relevance lies in its ability to deliver valid and efficient inference under minimal assumptions (even with weakly dependent data), thereby guarding against the unnecessarily conservative intervals that can arise from plug-in methods when the outcome support is unknown or heavy-tailed.
Our empirical findings on workforce diversity illustrate the importance of such rigorous inference. While na\"ive regressions might suggest that even modest increases in female leadership yield significant gains, the concATE approach paints a more nuanced picture. We find that substantive benefits of gender diversity materialise only once a sufficient representation level is achieved. In practice, this means token diversity---for example, a lone woman or two in senior leadership---is unlikely to drive measurable performance improvement. By contrast, reaching a critical mass of women in leadership (roughly half or more in growth-oriented industries, and a clear majority in others) is associated with a reliably positive impact on firm value. These conclusions align with the critical mass hypothesis: diversity can boost performance, but only after crossing a threshold that moves an organisation beyond tokenism kanter1977some.
By confirming this pattern under stringent inference, our study provides guidance for firms and policymakers. It emphasises that real gains from diversity require either significant numbers of women or, alternatively, substantial inclusion efforts. Indeed, an inclusive organisational culture may allow firms to reap performance gains at lower diversity levels than this critical mass. Evidence suggests that diversity alone is not sufficient and must be accompanied by inclusion to realise its full benefits nishii2013benefits,roberson2006disentangling,almeida2024diversity,josten2025makes. Investigating the interplay between inclusion and diversity outcomes---and in particular whether inclusive cultures shift the tipping point downward---remains an important area for future research.
On the methodological side, several extensions merit investigation. First, combining the DKW-calibrated endpoints of concATE with the adaptive critical value of imbens2004confidence could yield further efficiency gains by sharpening inference on the smooth components while retaining finite-sample tail protection. Second, practical guidance on estimating the mixing parameter $C_\upalpha$ from data, rather than relying on known dependence structures, would broaden the applicability of the method to general panel settings. Finally, our application demonstrates how concATE can be deployed in other domains, such as programme evaluation, environmental policy, or clinical trials, to uncover robust causal insights where traditional methods may produce misleadingly precise conclusions.
This work is supported by the Gnanalingam family through their sponsorship of the G&G Hub at The Inclusion Initiative, The London School of Economics and Political Science, and the Economic and Social Research Council (ESRC) under the “Diversity and Productivity: from Education to Work” (DAPEW) project [Grant Ref: ES/W010224/1].
\FloatBarrier