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Econ-101 · economics · Economics

Time-varying stochastic mediation effects

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Statement

CI-41 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Observe independent trajectories $(L_1,A_1,M_1,\ldots,L_T,A_T,M_T,Y)$, with binary treatments, finite-valued mediators, and bounded terminal outcome. Let $H_t$ contain all measured variables before $A_t$. Treatment precedes the mediator, which precedes $L_{t+1}$. Assume consistency, no interference between trajectories, sequential exchangeability for interventions on both treatment and mediator, and positivity for every history visited by the specified interventions. Thus measured post-treatment mediator--outcome confounders are included in the appropriate history. For a fixed treatment sequence $a$, define reference kernels $g_t^a(m\mid h)=P(M_t=m\mid H_t=h,A_t=a_t)$, evaluated on the histories generated by the actual intervention.

Formal question

Let $\psi(a,g)=\E[Y^{a,g}]$ be the mean under treatment sequence $a$ and mediator kernels $g$. Define \[ D(a,a')=\psi(a,g^{a'})-\psi(a',g^{a'}),\qquad I(a,a')=\psi(a,g^a)-\psi(a,g^{a'}). \] Their sum equals $\psi(a,g^a)-\psi(a',g^{a'})$, the total sequence contrast under the stipulated observed kernels. Determine identification conditions when intermediate covariates are treatment-affected, then sharp minimax risk and honest inference with transition densities and outcome means in Hölder balls of fixed indices $s_f,s_y>0$ and radius $L$, separately on continuous coordinates within each discrete-history stratum. Quantify horizon, history-dimension, and mediator-support dependence; specify when cumulative positivity forces deterioration even if each stage has overlap.

Scope and status

The reference kernels and rate question are editorial. They do not replace a subject's mediator by that same subject's counterfactual mediator trajectory. Mixed-history kernels must be specified and supported; $g^{a'}$ at a history generated under $a$ need not be a natural trajectory distribution under $a'$. No cross-world assumption is needed for these stochastic interventions, whereas equivalence to natural longitudinal effects requires additional restrictions. The assumptions are sufficient model requirements to investigate, not conclusions of the agenda itself.

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, "Studying time-varying mediation." 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-101-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-101

Time-varying stochastic mediation effects: Mixed-history identification and separate treatment/mediator overlap costs

Stochastic mediation target

Treatments precede finite mediators, which precede the next measured covariates. Let $q_t^a(h)=P(A_t=a_t\mid H_t=h)$ and $r_t^b(m\mid h)=P(M_t=m\mid H_t=h,A_t=b_t)$. For treatment sequence $a$ and reference sequence $b$, write \[ \psi(a,g^b)=\mathbb E[Y^{a,g^b}],\quad D=\psi(a,g^{a'})-\psi(a',g^{a'}),\quad I=\psi(a,g^a)-\psi(a,g^{a'}). \] The longitudinal mediation agenda is in [agenda, Section 3.4.3]. We prove the mixed-history identification formula, an exact second-moment bound with supplied kernels, finite-state honest inference with estimated kernels, and rare-path lower bounds. Full continuous-history Hölder minimax rates are not obtained.

Identification without freezing intermediate covariates

Write $f_{t+1}(dl\mid h,a_t,m)$ for the observed next-covariate kernel and $\mu(h_T,a_T,m_T)$ for the terminal conditional mean. Let $p_1$ be the initial covariate law. Under consistency, sequential exchangeability for treatment and mediator interventions, and support on all generated histories, the identified formula is \[ \psi(a,g^b)=\int \mu(h_T,a_T,m_T)\,p_1(dl_1) \prod_{t=1}^T r_t^b(dm_t\mid h_t) \prod_{t=1}^{T-1}f_{t+1}(dl_{t+1}\mid h_t,a_t,m_t). \tag{1} \] All $h_t$ in this formula contain the actual intervened past. In particular, $r_t^b$ is evaluated on that history, not on a separately simulated natural history under $b$.
Proposition: Under the stated assumptions and absolute continuity of each replaced kernel, define the weight \[ R^{a,b}=\prod_{t=1}^T \frac{\mathbf1\{A_t=a_t\}}{q_t^a(H_t)} \frac{r_t^b(M_t\mid H_t)}{r_t^a(M_t\mid H_t)}. \] Then $\psi(a,g^b)=\mathbb E[R^{a,b}Y]$ and $\mathbb E R^{a,b}=1$.
Proof: The observed density factors into $p_1$, the treatment kernels $q_t$, the mediator kernels $r_t^{A_t}$, covariate transitions $f_{t+1}$, and the terminal outcome law. Multiplication by $R^{a,b}$ restricts every treatment to $a_t$, cancels $q_t^a$ and $r_t^a$, and inserts $r_t^b$. Integrating the outcome gives (1). The same cancellation with terminal integrand one gives expectation one. Sequential exchangeability and consistency justify interpreting this replaced factorization as the causal intervention law; the algebra alone would not establish that interpretation. ∎

A cumulative information bound

Theorem: Assume $|Y|\le1$, all relevant assignment and mediator kernels are supplied, and for each stage \[ c_t=\sup_{h}\frac1{q_t^a(h)} \sum_m\frac{r_t^b(m\mid h)^2}{r_t^a(m\mid h)}<\infty, \tag{2} \] where the supremum is over supported intervened histories. Then the empirical weighted mean has risk at most $C_T/n$, where $C_T=\prod_{t=1}^T c_t$. If $q_t^a\ge\varepsilon$ and $r_t^b/r_t^a\le\Lambda_t$, then $C_T\le\varepsilon^{-T}\prod_t\Lambda_t$. For $b=a$, the mediator factors disappear and one can take $C_T=\varepsilon^{-T}$.
Proof: For the one-stage factor $Z_t$ of $R^{a,b}$, $\mathbb E[Z_t^2\mid H_t]= (q_t^a(H_t))^{-1}\sum_m r_t^b(m\mid H_t)^2/r_t^a(m\mid H_t)$. Starting from the last stage, bound this conditional expectation by $c_T$, then iterate backward through the natural covariate transition kernels. This proves $\mathbb E(R^{a,b})^2\le\prod_t c_t$. Since $Y^2\le1$, variance of its iid empirical mean is at most $C_T/n$. The ratio bound gives $\sum_m r_t^b(m)^2/r_t^a(m)\le\Lambda_t\sum_mr_t^b(m)=\Lambda_t$. For equal mediator kernels the sum is exactly one. ∎
For contrasts, taking the difference of two weighted means on the same sample yields variance at most $2(C_T^{(1)}+C_T^{(2)})/n$. Chebyshev intervals using this bound are finite-sample honest. The cardinality of mediator support alone cannot bound (2): with two mediator states a reference kernel can concentrate on a state whose actual-treatment probability is arbitrarily small. These are oracle-kernel results; estimating the law-dependent reference kernels requires additional analysis, supplied next in a restricted case.

Finite-state plug-in inference

Assume all covariate and mediator states are finite, each conditional kernel has at most $B$ possible outputs, and every history/assignment configuration needed to evaluate (1) under the three required interventions has observed probability at least $\beta>0$. Let $C$ count these configurations, also including initial and terminal configurations. Estimate all probabilities and terminal means by empirical conditional frequencies and averages. The plug-in expression uses these estimates at the mixed histories in (1).
Proposition: If $n\beta\ge8\log(4C/\alpha)$, set \[ t=\sqrt{\frac{4\log(4BC/\alpha)}{n\beta}},\qquad E=4T\min\{1,Bt/2\}+t. \] With probability at least $1-\alpha$, all three plug-in values have error at most $E$, and the plug-in estimates of $D,I$ have error at most $2E$. Thus their respective intervals of radius $2E$ are simultaneously honest.
Proof: A binomial lower-tail union bound makes every required count at least $n\beta/2$ with failure probability at most $\alpha/4$. Conditional Hoeffding bounds then simultaneously bound every category probability error by $t$ and terminal mean error by $t$, with remaining failure probability at most $3\alpha/4$; for $[-1,1]$ outcomes the exponent is at least $n\beta t^2/4$. Each kernel therefore has total variation error at most $\min(1,Bt/2)$. The initial, mediator and intermediate-covariate kernels total $2T$. Couple the true and estimated simulated intervention histories stage by stage. The probability of any first disagreement is at most the sum of these $2T$ total variation bounds. An outcome mean in $[-1,1]$ changes by at most twice that probability, plus its own estimation error $t$. This proves the error $E$ for every intervention simultaneously on the same kernel-estimation event. Taking differences proves the contrast bound. ∎
This procedure includes estimation of $g^b$, unlike the supplied-kernel weighted mean. Its dependence on $C$ and $\beta$ can still be exponential in $T$ because full histories are counted.

Necessary loss from rare treatment trajectories

Theorem: Within smooth continuous-coordinate submodels of the canonical class, for each of the targets $D$ and $I$ there is a sequence of experiments with per-stage treatment overlap $\varepsilon$ such that its minimax squared risk is at least \[ c\min\{1,(n\varepsilon^T)^{-1}\}. \tag{3} \] For $0<\alpha<1/2$, any uniformly honest confidence interval in these subexperiments has worst-case expected length at least $c_\alpha\min\{1,(n\varepsilon^T)^{-1/2}\}$.
Proof: Take independent Bernoulli$(\varepsilon)$ treatments, $a=(1,\ldots,1)$, $a'=(0,\ldots,0)$, and let all continuous coordinates be independent uniforms. For the direct effect, take mediator kernels independent of treatment, and terminal Bernoulli mean $1/2+\theta$ on the all-one path, $1/2$ on every other path. Then $D=\theta$. For the indirect effect, let only the final mediator differ by treatment: its probability of one is $3/4$ for treatment one and $1/4$ for treatment zero; all earlier mediator kernels agree. Give the outcome mean $1/2+\theta M_T$ on the all-one path and $1/2$ elsewhere. Then $I=\theta/2$. Both mediator values are supported under both treatments. Means and transition densities are constant in continuous coordinates, so satisfy every smoothness index when the fixed radii admit these constants. In either construction, the two laws $\theta=\pm\Delta$ differ only on an event of probability at most $\varepsilon^T$ and have sample KL at most $Cn\varepsilon^T\Delta^2$. Choose $\Delta=b\min\{1,(n\varepsilon^T)^{-1/2}\}$. A sufficiently small $b$ makes total variation at most $1/2$, and two-point testing yields (3). For intervals reduce $b$ further so that total variation is at most $(1-2\alpha)/2$. Under either one of the two laws, the probability that the interval contains both target values is at least $1-2\alpha-\operatorname{TV}>0$, by transferring the other law's coverage event. Multiplying this probability by target separation proves the expected-length bound. Uniform asymptotic coverage gives the same result with an $o(1)$ coverage term. ∎

Unresolved smoothness problem

The lower bound uses no roughness and therefore isolates cumulative positivity loss. The finite-state upper bound supplies a complete honest procedure, but does not determine the sharp dependence on continuous history dimension or the indices $s_f,s_y$. Higher-order treatment and mediator corrections, data-dependent reference-kernel estimation under Hölder restrictions, and matching nonparametric lower bounds remain unresolved. None of the calculations equates the stochastic kernels with a subject's counterfactual natural mediator trajectory.

References

  • [agenda] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane and J. Zubizarreta. Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference (2025), arXiv:2508.17099v1. https://arxiv.org/html/2508.17099v1.

Completion Estimate

35%

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