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Econ-120 · economics · Economics

Mediation inference with a mismeasured mediator

LLM status Open unverified

Statement

CI-44 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Observe independent $(X,W,\widetilde M,Y)$, with $X\in[0,1]^d$, binary $W$, bounded $Y$, and a latent circular mediator $M\in[0,1)$ measured as $\widetilde M=(M+\epsilon)\bmod1$. The known error law is independent of $(X,W,M,Y)$. Its Fourier coefficients $\lambda_k=\E[e^{2\pi\mathrm i k\epsilon}]$ are nonzero and satisfy $c(1+|k|)^{-a}\leq|\lambda_k|\leq C(1+|k|)^{-a}$ for a fixed $a>0$. Define mediator interventions by a jointly measurable response $Y(w,m)=F(w,m,X,U)$ with $U\perp(W,M)\mid X$. Assume consistency, no interference, and treatment propensity in $\mathcal H_d^\alpha(L)$ with fixed overlap. The latent mediator densities $f_w(m\mid x)$ are bounded below and periodic in $m$, and belong to $\mathcal H_{d+1}^{s_f}(L)$ with derivatives periodic in the mediator coordinate, $s_f>0$; controlled causal means $\mu_w$ satisfy the analogous restriction with index $s_\mu>0$. Covariate densities are bounded above and below.

Formal question

For an independent conditional reference draw from $f_{w'}$, the target is \[ \psi(w,w')=\int\!\int_0^1 \mu_w(x,m)f_{w'}(m\mid x)\,dm\,dP_X(x). \] Determine identification from the observed convolution law, then matching minimax squared-error bounds and honest confidence intervals as functions of outcome, mediator-density, and covariate smoothness and error exponent $a$. Construct rate-attaining estimators of direct and indirect contrasts. Extend the problem to an unknown error law learned from $N$ independent validation pairs $(M,\widetilde M)$, quantifying the additional calibration cost and restrictions needed to prevent loss of injectivity.

Scope and status

The circular ordinary-smooth error model is an editorial tractable instance of mediator measurement error, not an assumption in the source. Unknown unrestricted measurement kernels can destroy identification. Deconvolution and causal adjustment are distinct requirements; solving the measurement problem does not justify exchangeability. These targets are stochastic interventional effects, and a noisy-mediator regression substituted into a mediation formula generally estimates a different quantity.

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 [stat.ME], 23 August 2025, §3.4.3, "Conducting mediation analysis in practice," including mediator measurement error. https://arxiv.org/html/2508.17099v1#S3.SS4.SSS3.

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-120-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-120

Identification and a bilinear deconvolution bound for mediation

Catalogue problem: C14-117. Status: Unresolved. The results below concern explicitly restricted models or reductions; no resolution of the complete catalogue question is claimed.

Abstract

Known nonvanishing circular error coefficients identify both the latent mediator density and its outcome-weighted density, hence the stochastic mediation functional under the stated causal assumptions. In a no-covariate submodel with a uniform treated mediator, a split-sample Fourier estimator admits an explicit bias and variance decomposition. It gives a sufficient parametric region, and a conservative calibration bound when the error law is validated. The general unknown-density ratio functional and matching nonparametric lower bounds remain open here.

Target and identification

For $X=x$ and treatment $w$, write $f_w(m\mid x)$ for the latent mediator density and $\mu_w(x,m)={\mathbb{E}}[Y(w,m)\mid X=x]$. The outcome mechanism and exchangeability conditions in the catalogue give ${\mathbb{E}}[Y\mid X=x,W=w,M=m]=\mu_w(x,m)$. The target is \[ \psi(w,w')=\int\int_0^1\mu_w(x,m)f_{w'}(m\mid x)\,dm\,dP_X(x). \] The observed mediator is $\widetilde M=(M+\varepsilon)\bmod1$, with known error independent of all latent and observed causal variables. Use Fourier conventions \[ g_k=\int_0^1g(m)e^{-2\pi\mathrm i km}\,dm, \qquad \kappa_k={\mathbb{E}} e^{-2\pi\mathrm i k\varepsilon}. \] Assume $c(1+|k|)^{-a}\le|\kappa_k|\le C(1+|k|)^{-a}$, $a>0$, and $f_w\ge c_f>0$. This circular measurement model is a restricted formalization of the source's mediator-error agenda [CF].
Proposition (Identification from the full convolution law): For $P_X$-almost every $x$, the observed conditional law identifies $f_w(\cdot\mid x)$ and $h_w(\cdot\mid x)=\mu_w(x,\cdot)f_w(\cdot\mid x)$ as integrable functions up to null sets. Therefore it identifies $\psi(w,w')$. This conclusion uses both error injectivity and the stated causal response assumption.
Proof: Independence of the error gives the observable Fourier moments \[ {\mathbb{E}}[e^{-2\pi\mathrm i k\widetilde M}\mid x,w]=\kappa_k(f_w)_k, \qquad {\mathbb{E}}[Ye^{-2\pi\mathrm i k\widetilde M}\mid x,w]=\kappa_k(h_w)_k. \] Treatment overlap makes both conditional laws available almost everywhere. Divide by the nonzero $\kappa_k$. Fourier coefficients uniquely determine an integrable periodic function: its Fejér means are convolutions with the nonnegative approximate-identity kernels $F_K(u)=K^{-1}|\sum_{j=0}^{K-1}e^{2\pi\mathrm i ju}|^2$ and converge in $L^1$. To see convergence, use continuity of translations in $L^1$, integral one of $F_K$, and the bound $F_K(u)\le C_\delta/K$ away from zero. Thus zero coefficients imply the function is zero. Finally $h_w/f_w=\mu_w$ almost everywhere because $f_w\ge c_f$, and the target integral is determined. ∎

Injectivity does not make inversion statistically stable: dividing by $\kappa_k$ amplifies high-frequency estimation noise. It also does not by itself justify replacing observational means by controlled causal means.

A solvable bilinear submodel

Now omit $X$. Observe independent samples of size $n$ from treatment group 1 and treatment group 0, with known sample allocation. In group 1 let $f_1=1$ be known and let $0\le Y\le1$ with controlled mean $\mu(m)$. In group 0 the mediator has density $f(m)$, bounded above by $C_f$ and below by $c_f$. The target is $\psi=\int\mu f$. Assume the periodic Sobolev bounds \[ \sum_k(1+|k|)^{2\beta}|\mu_k|^2\le L_\mu^2, \qquad \sum_k(1+|k|)^{2\gamma}|f_k|^2\le L_f^2, \quad \beta,\gamma>0. \tag{2} \] Assume $L_\mu,L_f>0$. These are explicit extra restrictions for this rate calculation, not substitutions for every Hölder class in the catalogue.

Define unbiased coefficient estimates \[ \hat\mu_k=\frac1{n\kappa_k}\sum_{i=1}^nY_i^{(1)}e^{-2\pi\mathrm i k\widetilde M_i^{(1)}}, \quad \hat f_k=\frac1{n\kappa_k}\sum_{i=1}^ne^{-2\pi\mathrm i k\widetilde M_i^{(0)}}, \] and $\hat\psi_K=\sum_{|k|\le K}\hat\mu_k\overline{\hat f_k}$. Conjugate symmetry makes this sum real.

Theorem (Complete risk decomposition): For every integer $K\ge1$, under (2), \begin{align*} {\mathbb{E}}(\hat\psi_K-\psi)^2\le C\bigg[&K^{-2(\beta+\gamma)} +\frac{1+K^{(2a-2\beta)_+}+K^{(2a-2\gamma)_+}}n +\frac{K^{4a+1}}{n^2}\bigg], \end{align*} where $x_+=\max(x,0)$ and $C$ depends only on the declared bounds and exponents. In particular if \[ \beta\ge a,\qquad \gamma\ge a,\qquad \beta+\gamma\ge2a+\tfrac12, \] there is a choice of $K$ giving squared risk $O(n^{-1})$.
Proof: By independence of the two samples, ${\mathbb{E}}\hat\psi_K=\psi_K=\sum_{|k|\le K}\mu_k\overline f_k$. Cauchy--Schwarz and (2) give \[ |\psi-\psi_K|\le (1+K)^{-(\beta+\gamma)}L_\mu L_f. \] Let $\Delta\mu_k=\hat\mu_k-\mu_k$ and $\Delta f_k=\hat f_k-f_k$. The centered error is the sum of \[ A=\sum\Delta\mu_k\overline f_k,\quad B=\sum\mu_k\overline{\Delta f_k},\quad C_0=\sum\Delta\mu_k\overline{\Delta f_k}, \] where all sums run over $|k|\le K$. These are pairwise orthogonal in complex $L^2$: for a cross term involving $C_0$, condition on one sample and use centering of the other; for $A,B$, use independence.

For any coefficients $v_k$, Parseval's identity gives \[ {\mathbb{E}}\left|\sum v_k\Delta\mu_k\right|^2 \le\frac1n\sum\frac{|v_k|^2}{|\kappa_k|^2}. \tag{3} \] Indeed the noisy treated mediator remains uniform, $Y^2\le1$, and subtracting a mean only reduces a second moment. The corresponding bound for $\Delta f$ has a factor $C_f$, because convolution with a probability law preserves the bound $f\le C_f$. Applying (3) to $A,B$ bounds their variances by \[ \frac Cn\sum_{|k|\le K}(1+|k|)^{2a} (|f_k|^2+|\mu_k|^2). \] The Sobolev bounds make this at most the displayed first-order variance term. For the last term, condition on the group-0 sample and apply (3), then use ${\mathbb{E}}|\Delta f_k|^2\le C/[n|\kappa_k|^2]$. Thus \[ {\mathbb{E}}|C_0|^2\le\frac C{n^2}\sum_{|k|\le K}|\kappa_k|^{-4} \le Cn^{-2}K^{4a+1}. \] Combining variance and squared bias proves the inequality. In the parametric region choose an integer $K$ between constant multiples of $n^{1/[2(\beta+\gamma)]}$ and $n^{1/(4a+1)}$; these bounds are compatible precisely under the stated sum condition, and each risk term is $O(n^{-1})$. ∎

A constant-$\mu$ Bernoulli submodel with $f=1$ gives a matching $c/n$ lower bound in this sufficient region. Explicitly, fix $\theta_0\in(0,\min\{1,L_\mu\})$ and take means $\theta_0\pm h/\sqrt n$, keeping both in that interval for all sufficiently large $n$. Their $n$-sample KL divergence is bounded by $Ch^2$, with $C$ allowed to depend on $\theta_0$; for small fixed $h$ a nearest-target test has total error bounded away from zero. Wrong selection implies squared error at least $h^2/n$. This gives the claimed lower bound. It does not establish sharp rates when the sufficient smoothness conditions fail.

A conservative validation guarantee

Suppose independently observed validation errors $\varepsilon_1,\ldots,\varepsilon_N$ give empirical coefficients $\hat\kappa_k$. They are computable from circular validation pairs. Let $\eta_N=C\sqrt{\log(4K/\delta)/N}$, increasing $C$ if necessary. Applying the bounded real-variable exponential inequality separately to real and imaginary parts and taking a union bound gives \[ {\mathbb{P}}\left(\max_{|k|\le K}|\hat\kappa_k-\kappa_k|>\eta_N\right)\le\delta. \] Define each estimated deconvolved coefficient to be zero whenever $|\hat\kappa_k|<(c/2)(1+|k|)^{-a}$, and otherwise use $\hat\kappa_k$ as its denominator. This specifies the estimator even on bad validation samples. On the displayed good event, if $2\eta_N\le c(1+K)^{-a}$, all retained coefficients satisfy $|\hat\kappa_k|\ge|\kappa_k|/2$.
Proposition (Conditional calibration cost): Use $\hat\kappa$ in both coefficient estimates and condition on a validation sample satisfying the preceding good event. The risk bound of the theorem then holds with an additional term $C\eta_N^2K^{2a}$ and enlarged constants. Clipping the final estimate to $[0,1]$ makes the unconditional additional bad-event contribution at most $\delta$.
Proof: Conditional on validation, the expected truncated estimator is \[ \sum_{|k|\le K}\frac{|\kappa_k|^2}{|\hat\kappa_k|^2} \mu_k\overline f_k. \] The difference of each multiplicative factor from one is at most $C\eta_N/|\kappa_k|\le C\eta_NK^a$. Since $\sum|\mu_kf_k|\le L_\mu L_f$, the extra bias is at most $C\eta_NK^a$. Squaring the sum of biases costs only an absolute constant. Every inverse denominator in the variance proof increases by at most a factor two, giving the same variance orders. The clipped estimate and target both lie in $[0,1]$, so the loss on the bad event is at most one. ∎

For instance $N\gg nK^{2a}\log(Kn)$ and $\delta\le n^{-1}$ suffice for this crude calibration cost to be negligible at the parametric scale. This is a sufficient condition, not a sharp validation-sample requirement.

Remaining gap

When $f_1$ is unknown, the target is $\int h_1f_0/f_1$; its denominator must be estimated and controlled together with deconvolution. Covariate smoothing and propensity estimation introduce additional interactions. The theorem does not supply matching lower bounds in those models or confidence intervals valid uniformly over them. Even in the bilinear submodel the displayed upper bound alone is not a nonparametric minimax theorem. Identification, attainable upper bounds, and sharp inference are deliberately distinguished.

References

  • [CF] 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 3.4.3, "Conducting mediation analysis in practice." https://arxiv.org/html/2508.17099v1#S3.SS4.SSS3.

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