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Econ-195 · economics · Economics

Conditional distributional treatment heterogeneity

LLM status Open unverified

Statement

CI-16 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Use the binary causal baseline with independent observations, consistency, no interference, conditional exchangeability, and overlap $e(x)\in[\varepsilon,1-\varepsilon]$. Let $X$ be uniform on $[0,1]^d$ and $Y(w)\in[-B,B]$. Define $F_w(y\mid x)=\Pr(Y(w)\leq y\mid X=x)$ and $Q_w(u\mid x)=\inf\{y:F_w(y\mid x)\geq u\}$. Distinct distributional contrasts include \[ v(x)=\operatorname{Var}(Y(1)\mid X=x)- \operatorname{Var}(Y(0)\mid X=x),\qquad q_u(x)=Q_1(u\mid x)-Q_0(u\mid x), \] and $a_c(x)=\Pr(Y(1)>c\mid X=x)-\Pr(Y(0)>c\mid X=x)$.

Declare $e\in\mathcal H_d^\alpha(L)$ and $F_w(y\mid\cdot)\in\mathcal H_d^\beta(L)$ uniformly in $y$. For quantile targets, restrict $u$ to $[\eta,1-\eta]$. On fixed-radius neighborhoods of all relevant quantiles, bound conditional densities between fixed constants $0<c<C<\infty$, uniformly over models, treatments, covariates, and $u$. Optionally impose a separate smoothness $\gamma$ on the selected contrast.

Formal question

For each contrast separately, determine its sharp minimax integrated squared-error risk and honest confidence-set size as functions of $n,d,\alpha,\beta,\gamma$. Identify when contrast structure permits faster estimation than estimating both conditional distributions individually, and determine necessary nuisance-regularity thresholds. Establish matching lower bounds, including classes where quantile density conditions fail.

Scope and status

These contrasts involve marginal counterfactual distributions and are identified under the stated baseline assumptions. They are not the conditional variance or quantiles of the individual effect $Y(1)-Y(0)$, which generally require unidentified cross-world dependence. Basic conditional distribution estimation is established; the attached agenda motivates the stated sharp-rate and structure questions without certifying each subclass as currently unsolved.

References for the source of the problem

Carlos Cinelli, Avi Feller, Guido Imbens, Edward Kennedy, Sara Magliacane, and Jose Zubizarreta, Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference, arXiv:2508.17099v1, 23 August 2025, Section 3.3.3, paragraph Beyond expectations. https://arxiv.org/html/2508.17099v1.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-195-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-195

Distributional treatment contrasts Sharp baseline rates and a quantile nonregularity obstruction

Problem ID: C21-191. Status: Unresolved; rigorous restricted results.

Restricted result and normalization

The target asks for separate minimax theories for exceedance, variance and quantile contrasts, including gains from an unusually smooth contrast. We establish the baseline integrated-risk rate when $0<\beta\leq1$ and no extra contrast smoothness is imposed, under arbitrary unknown propensity satisfying overlap. We also construct honest $L^2$ balls and prove that removing quantile density conditions can destroy uniform consistency altogether. Rates exploiting a separate contrast index $\gamma$ and optimal confidence-set diameters remain unresolved.

An affine rescaling reduces bounded outcomes to $Y(w)\in[0,1]$. This changes only fixed constants: variance contrasts rescale quadratically, quantile contrasts linearly, and the exceedance threshold transforms with the outcome. For an informative exceedance target fix $c\in(0,1)$; at thresholds outside the outcome support the contrast is identically zero. Fix $u\in[\eta,1-\eta]$ for a supplied $\eta>0$.

Use independent observations under the catalogue's binary causal assumptions, with $X$ uniform on $[0,1]^d$, $\varepsilon\leq e\leq1-\varepsilon$, and \[ |F_w(y\mid x)-F_w(y\mid x')|\leq L\|x-x'\|^\beta \quad\text{for all }y,x,x',w. \] For quantiles, assume the stipulated fixed-radius neighborhood and density bounds $0<c_*\leq f_w\leq C_*$, uniformly over $x,w$ and relevant $u$. Source motivation is [challenge, Section 3.3.3]. The results below concern marginal counterfactual contrasts, not the unidentified distribution of individual treatment effects.

An armwise histogram works without learning the propensity

Partition the cube into $J=q^d$ cells of side $h=1/q$. Within every nonempty cell-arm, use the empirical distribution of its outcomes. Let $\widehat m_w,\widehat r_w$ be its first and second moments, $\widehat F_w(c)$ its CDF at $c$, and $\widehat Q_w(u)$ its empirical $u$-quantile. These estimates are constant over the cell. On an empty cell-arm use any fixed probability distribution on $[0,1]$, for example the point mass at zero. Define \[ \widehat a_c=\widehat F_0(c)-\widehat F_1(c),\quad \widehat v=(\widehat r_1-\widehat m_1^2) -(\widehat r_0-\widehat m_0^2),\quad \widehat q_u=\widehat Q_1(u)-\widehat Q_0(u). \] The sign in $a_c$ uses $\Pr(Y>c)=1-F(c)$, including distributions with atoms.
Lemma (Cell mixtures preserve CDF approximation): Let $G_{jw}$ be the conditional outcome distribution given cell $A_j$ and arm $w$. For every $x\in A_j$, \[ \sup_y|G_{jw}(y)-F_w(y\mid x)|\leq b_h:=Ld^{\beta/2}h^\beta. \] Its first and second moments differ from their conditional counterparts at $x$ by at most $b_h$ each.
Proof: The mixture defining $G_{jw}$ averages $F_w(y\mid x')$ over the actual law of $X$ in the cell-arm. Every such $x'$ is at distance at most $\sqrt d h$ from $x$, proving the uniform CDF inequality. For a variable $V\in[0,1]$, ${\mathbb{E}} V=\int_0^1(1-F_V(y))\,dy$ and ${\mathbb{E}} V^2=\int_0^1 2y(1-F_V(y))\,dy$ by Tonelli's theorem applied to the pointwise integral identities. Both weighting functions integrate to one, proving the moment bounds. No regularity of $e$ inside the cell was used. ∎
Lemma (Quantile risk from two Bernoulli tail bounds): Let $F$ have a density at least $c_*>0$ on the fixed-radius $r_*>0$ neighborhood of its $u$-quantile $Q$, with $F(Q)=u$. If a distribution $G$ obeys $\|G-F\|_\infty\leq b\leq c_*r_*/4$, then the empirical $u$-quantile from $N\geq1$ independent $G$ draws satisfies \[ {\mathbb{E}}|\widehat Q-Q|^2\leq C(b^2+N^{-1}). \] The constant depends only on $c_*,r_*$; outcomes lie in $[0,1]$.
Proof: For $0<t\leq r_*$, density positivity implies $F(Q+t)\geq u+c_*t$ and $F(Q-t)\leq u-c_*t$. Thus when $c_*t>b$, Hoeffding applied to the empirical proportions below these two fixed points gives \[ \Pr\{\widehat Q>Q+t\}\leq e^{-2N(c_*t-b)^2},\qquad \Pr\{\widehat Q<Q-t\}\leq e^{-2N(c_*t-b)^2}. \] For example, the first event forces the empirical CDF at $Q+t$ below $u$, while its expectation is at least $u+c_*t-b$. The second is analogous. Boundary equalities can be handled by limits in $t$ and do not alter the tail integral.

For $t\geq2b/c_*$ up to $r_*$, the combined bound is at most $2\exp(-Nc_*^2t^2/2)$. For larger $t$ use the bound at $r_*$. Since $|\widehat Q-Q|\leq1$, integrate ${\mathbb{E}}|\widehat Q-Q|^2=\int_0^1 2t\Pr(|\widehat Q-Q|>t)\,dt$. The portion below $2b/c_*$ is at most $4b^2/c_*^2$; the Gaussian integral is at most $C/N$; the remaining bounded interval contributes at most $2e^{-Nc_*^2r_*^2/2}\leq C'/N$. This proves the claim. ∎

Theorem (Upper rates for each contrast): For all sufficiently small $h$, each of the three estimators satisfies \[ \sup_P{\mathbb{E}}\|\widehat\theta-\theta\|_2^2 \leq C\{h^{2\beta}+J/n\}, \qquad \theta\in\{a_c,v,q_u\}. \] The quantile assertion uses its density assumptions; the other two do not. With $h$ of order $n^{-1/(2\beta+d)}$, the rate is $n^{-2\beta/(2\beta+d)}$. It holds for every propensity smoothness $\alpha$ compatible with overlap, since the estimator never learns $e$.
Proof: Conditional on a cell-arm count $N>0$, its empirical indicator and moment averages have variance at most $1/N$. Lemma Lemma (Cell mixtures preserve CDF approximation) bounds their squared bias by $b_h^2$. For variance, the inequality \[ |(\widehat r-\widehat m^2)-(r-m^2)| \leq|\widehat r-r|+2|\widehat m-m| \] holds because all means are in $[0,1]$. Squaring gives the same risk order. Lemmas Lemma (Cell mixtures preserve CDF approximation) and Lemma (Quantile risk from two Bernoulli tail bounds) give it for quantiles once $b_h\leq c_*r_*/4$. Both armwise target and estimates are bounded, so an empty group contributes at most a constant times its probability.

The cell-arm probability is $p_{jw}\geq\varepsilon h^d$. With $N\sim\operatorname{Bin}(n,p_{jw})$, \[ {\mathbb{E}}\frac{{\mathbf 1}\{N>0\}}N\leq\frac2{(n+1)p_{jw}},\qquad \Pr(N=0)\leq\frac1{(n+1)p_{jw}}. \] The first follows from $1/N\leq2/(N+1)$ and the binomial integral identity for ${\mathbb{E}}[1/(N+1)]$; the second follows by maximizing $(n+1)p(1-p)^n$. Integrating over a cell of volume $h^d$ and summing $J$ cells gives variance and empty-group contribution at most $CJ/n$. The bias integrates to $Cb_h^2$. Finally the squared error of the difference of two armwise estimates is at most twice their summed squared errors. This proves all three claims. ∎

Matching lower bounds for the baseline classes

Theorem (Sharp integrated-risk exponent): For each contrast separately, with no extra smoothness restriction $\gamma$ on that contrast and fixed $0<\beta\leq1$, compatible nondegenerate radius and density constants give \[ \inf_{\widehat\theta}\sup_P{\mathbb{E}}\|\widehat\theta-\theta\|_2^2 \geq c n^{-2\beta/(2\beta+d)}. \] This applies to every interior threshold $c\in(0,1)$ for $a_c$ and every fixed $u\in[\eta,1-\eta]$ for $q_u$.
Proof: Take a fixed nonnegative Lipschitz bump $\varphi$, supported in the central half of $(0,1)^d$, with maximum one and positive squared integral. On $J=q^d$ grid cells define \[ t_\omega(x)=a\sum_{j=1}^J\omega_j\varphi((x-x_j)/h), \quad a=\kappa h^\beta,\quad \omega_j\in\{-1,1\}. \] Separation of supports and vanishing at their boundaries make these uniformly $\beta$-Hölder after choosing $\kappa$ small enough. Always use uniform covariates and randomized treatment $e=1/2$.

For exceedance and variance, let control outcomes be Bernoulli with success probability $p_0=1/4$, and treated outcomes Bernoulli with success probability $p_0+t_\omega(x)$. Any interior threshold separates zero from one, so $a_c(x)=t_\omega(x)$. The variance contrast is $(p_0+t_\omega)(1-p_0-t_\omega)-p_0(1-p_0)$. On $|t_\omega|\leq1/8$, this scalar map has derivative $1-2(p_0+t_\omega)\geq1/4$, so differences of contrasts dominate a fixed constant times differences of $t_\omega$. The conditional CDF is affine in $t_\omega$, meeting the uniform $\beta$-Hölder condition for every $y$.

For quantiles use uniform control outcomes and treated density \[ f_t(y)=1+t(2y-1),\quad 0\leq y\leq1,\quad |t|\leq1/4. \] Its CDF is $F_t(y)=y+t(y^2-y)$, so it has the required covariate regularity and density between $3/4$ and $5/4$. Its quantile $Q_t(u)$ remains uniformly inside $(0,1)$ for $u\in[\eta,1-\eta]$, providing a common neighborhood radius. Implicit differentiation gives \[ \frac{dQ_t(u)}{dt} =\frac{Q_t(u)(1-Q_t(u))}{1+t(2Q_t(u)-1)}\geq c_\eta>0. \] Thus quantile contrasts $Q_{t_\omega}(u)-u$ also separate in $L^2$ by a fixed multiple of the underlying bump separation. The density conditions are genuine here, rather than borrowed from the binary construction.

In each family, neighboring sign vectors differ on one cube. For Bernoulli outcomes, \[{\operatorname{KL}}(\operatorname{Ber}(p),\operatorname{Ber}(q))\leq(p-q)^2/[q(1-q)]\] gives neighbor divergence at most $Cna^2h^d$. For the density family, $\log z\leq z-1$ bounds KL by $\int(f_t-f_{t'})^2/f_{t'}$, at most $C(t-t')^2$; integration in $X$ gives the same neighbor bound. Choose $q$ of order $n^{1/(2\beta+d)}$ and $\kappa$ small enough that every neighbor total variation is at most $1/2$.

On each cube the two possible local contrast functions have squared distance at least $c a^2h^d$. Decode an arbitrary estimator by the nearer local candidate. The triangle inequality makes a decoding error cost at least one quarter of this distance. Pairing neighboring sign alternatives gives average bit-error probability at least $(1-{\operatorname{TV}})/2\geq1/4$. Summing $J$ cubes under a uniform sign prior proves average risk at least $c'Ja^2h^d=c'a^2$. Its maximum risk is no smaller. All these observable laws admit potential outcomes independent of randomized treatment, so the statistical lower bounds are inside the stipulated causal experiment. ∎

Inference and failure without quantile regularity

For each estimator above take a supplied constant $C$ from Theorem Theorem (Upper rates for each contrast) and define the $L^2$ ball with radius $\sqrt{C\{h^{2\beta}+J/n\}/\zeta}$, $0<\zeta<1$. Markov's inequality gives finite-sample coverage at least $1-\zeta$, uniformly over the declared class. Its diameter has order $n^{-\beta/(2\beta+d)}$ for fixed $\zeta$ at the chosen mesh. This is an upper construction, not a proof that the diameter is optimal or adaptive.
Proposition (Without a density condition, median contrast can be discontinuous): Remove the quantile density restriction but retain bounded outcomes, overlap and arbitrarily high covariate smoothness of both CDFs. The median treatment contrast has minimax integrated squared risk bounded away from zero. For confidence level $1-\zeta$ with $\zeta<1/2$, any uniformly honest $L^2$ confidence set must also have nonvanishing worst-case expected diameter.
Proof: Let all distributions be constant in $x$, take randomized treatment, and let control outcomes be uniform on $[0,1]$, with median $1/2$. In two treatment laws set $\Pr(Y(1)=0)=1/2+\epsilon$ or $1/2-\epsilon$, with remaining mass at one. Under the infimum definition of quantiles their medians are respectively zero and one, so the median contrasts are the constant functions $-1/2$ and $1/2$, at $L^2$ distance one. All CDFs are constant in $x$, so every covariate smoothness restriction is satisfied.

Couple one observation under the two laws using the same covariate, treatment, and uniform random seed for the binary treated outcome. Their mismatch probability is at most $2\epsilon$, and the full sample total variation is at most $2n\epsilon$ by a union bound. With $\epsilon=n^{-2}$ this tends to zero. Nearest-target testing from any estimator makes squared risk at least one quarter on a testing error. The sum of the two testing errors is at least $1-{\operatorname{TV}}$, so the larger risk is at least $(1-2/n)/8$.

For a uniformly honest confidence set, under the first law coverage of its own target is at least $1-\zeta-o(1)$. Coverage of the second target under the second law is also at least that amount, hence at least $1-\zeta-o(1)$ under the first law by total variation. With probability at least $1-2\zeta-o(1)$ it contains both targets and has diameter at least one. This proves the diameter obstruction. It concerns a class without density regularity and does not contradict the earlier regular-quantile theorem. ∎

Remaining contrast-structure problem

For $0<\beta\leq1$ and no extra $\gamma$, the baseline rate is sharp and propensity regularity imposes no extra cost: armwise distributions are already sufficiently learnable. If a contrast is much simpler than either conditional distribution, this construction may waste that structure. Deriving faster contrast-specific estimators, the corresponding nuisance thresholds, and sharp honest set sizes is the unresolved part. The three targets must remain separate: the quantile lower bound requires a density family, while exceedance and variance lower bounds can use atoms. No claim about cross-world dependence or the variance of $Y(1)-Y(0)$ follows from these marginal results.

References

  • [challenge] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane and J. Zubizarreta (2025), Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference, arXiv:2508.17099v1, Section 3.3.3, "Beyond expectations," inspected 9 October 2026. https://arxiv.org/html/2508.17099v1#S3.SS3.SSS3.

Completion Estimate

45%

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