Technical Summary: Identification and Bounding of Central Moments of Causal Effects Using Marginal Moments Information

Problem Statement

In causal inference, evaluating the heterogeneity of treatment effects is a critical objective beyond estimating the Average Causal Effect (ACE). While the distribution of Individual Causal Effects (ICE), defined as $Y_1 - Y_0$, characterizes this heterogeneity, existing methods for identifying or bounding the distribution of ICE (e.g., Fan and Park [2010], Post and Van Den Heuvel [2025]) and its central moments (Kawakami and Tian [2025]) typically require access to the full marginal distributions of the potential outcomes (POs), $P(Y_1 < y)$ and $P(Y_0 < y)$.

In many empirical settings, however, full individual-level data or complete distribution functions are unavailable due to privacy restrictions, confidentiality, or the nature of historically accumulated studies that report only aggregated summary statistics. In such cases, only marginal central moments (e.g., mean, variance, skewness, kurtosis) of the treatment and control groups are accessible. This paper addresses the problem of identifying and bounding the central moments of the ICE, denoted as $\mu^{(m)}$, using only the marginal central moments of the potential outcomes, $\sigma^{(k)}_1$ and $\sigma^{(k)}_0$, without assuming knowledge of the full marginal distributions.

Methodology

The authors employ the framework of Structural Causal Models (SCMs) and potential outcomes. The central moments of the ICE are defined as $\mu^{(m)} = E[(Y_1 - Y_0 - E[Y_1 - Y_0])^m]$. The marginal central moments of the POs are defined as $\sigma^{(k)}_x = E[(Y_x - E[Y_x])^k]$ for $x \in \{0, 1\}$.

The methodology is divided into two primary approaches:

1. Identification under the Independent Effect Deviation (IED) Assumption:
   The paper investigates identification under the IED condition, where the ICE is independent of the baseline outcome: $(Y_1 - Y_0) \perp\!\!\!\perp Y_0$. Under this assumption, the authors derive recursive identification formulas. Specifically, Theorem 1 shows that if marginal moments up to order $m$ (excluding specific intermediate orders) are available, the $m$-th central moment of the ICE can be recursively identified. Notably, under IED, the variance ($\mu^{(2)}$) is identified simply as the difference in variances of the POs ($\sigma^{(2)}_1 - \sigma^{(2)}_0$), and higher-order moments like skewness and kurtosis can also be identified if the corresponding marginal moments are available.

2. Bounding without IED:
   Recognizing that the IED assumption is strong and often untestable, the paper derives sharp bounds for $\mu^{(m)}$ using only the available marginal moments $(\sigma^{(k)}_1, \sigma^{(k)}_0)$ without assuming independence.

   • Variance ($\mu^{(2)}$): The authors analyze the correlation $\lambda$ between the ICE and the baseline $Y_0$. They derive sharp bounds for $\mu^{(2)}$ based on $\sigma^{(2)}_1$ and $\sigma^{(2)}_0$, showing that the variance is bounded by $(\sqrt{\sigma^{(2)}_1} \pm \sqrt{\sigma^{(2)}_0})^2$. They further demonstrate that knowing higher-order moments (like skewness or kurtosis) does not necessarily tighten the bound for variance unless specific conditions are met.
   • Higher-Order Moments: The paper provides bounds for the third ($\mu^{(3)}$) and fourth ($\mu^{(4)}$) central moments. For odd moments, if only lower-order even moments are known, the bounds are often trivial (i.e., $(-\infty, \infty)$). However, if the $m$-th moment of the POs is known, non-trivial bounds can be derived for even $m$.
   • Sharpness: The authors rigorously define "sharpness" as the property that no strictly tighter closed interval exists given the moment information. They provide counterexamples (in the Supplementary Material) to demonstrate that while some bounds are sharp, combining multiple moment constraints (e.g., knowing both variance and kurtosis) does not always yield sharp bounds for the ICE moments, particularly for higher-order moments.

Key Contributions and Results

1. Identification Results under IED:

   • The paper establishes that under the IED condition, the central moments of the ICE are identifiable from the marginal central moments of the POs.
   • Corollary 1 & 2: Specifically, the skewness and kurtosis of the ICE are identifiable if the corresponding marginal moments (3rd and 4th) are available, alongside the variance.

2. Sharp Bounds for Central Moments:

   • Variance: Theorem 3 provides sharp bounds for $\mu^{(2)}$ using only $\sigma^{(2)}_1$ and $\sigma^{(2)}_0$: $(\sqrt{\sigma^{(2)}_1} - \sqrt{\sigma^{(2)}_0})^2 \leq \mu^{(2)} \leq (\sqrt{\sigma^{(2)}_1} + \sqrt{\sigma^{(2)}_0})^2$.
   • Higher Moments: Theorems 16–18 generalize these results for arbitrary $m$. For even $m$, if $\sigma^{(m)}_1$ and $\sigma^{(m)}_0$ are available, sharp bounds are derived. For odd $m$, the paper shows that knowing a single lower-order moment $k \leq m$ generally does not provide non-trivial bounds for $\mu^{(m)}$ (except in extreme cases where lower moments are zero).
   • Intersection of Bounds: The paper investigates whether intersecting bounds from different available moments (e.g., using both $\sigma^{(2)}$ and $\sigma^{(4)}$) yields tighter results. The authors find that while this often narrows the range, the resulting bounds are generally not sharp (Theorem 7, 11, 15), except in specific cases involving variance and higher-order moments.

3. Empirical Case Studies:
   The authors apply their methods to two real-world studies where only summary statistics were reported:

   • Moseley et al. [2002] (Knee Arthroscopy): Using reported means and standard deviations, the authors bound the variance of the ICE. They find substantial heterogeneity, with the variance potentially ranging from 19.36 to 806.56. Under the assumption of positive correlation between baseline and ICE, the upper bound tightens significantly.
   • Špirtovi´c et al. [2023] (Aerobics Program): Using reported mean, SD, skewness, and kurtosis, the authors bound the variance, skewness, and kurtosis of the ICE. The resulting bounds are wide, reflecting the fundamental non-identifiability given limited information, but they still allow for the characterization of the distribution's shape (e.g., positive skewness under IED).

4. Numerical Experiments:
   Simulations comparing the proposed moment-based bounds with a Monte Carlo integration (MCI) estimator (which assumes full marginal distributions) show that:

   • In some settings, the moment-based bounds are comparable to or even tighter than MCI bounds.
   • The proposed method is robust in finite-sample settings, though confidence intervals widen with smaller sample sizes.

Significance and Claims

The paper claims that its identification and bounding results are novel, as the problem of evaluating ICE central moments using only marginal central moments has not been systematically studied before.

• Practical Relevance: The primary significance lies in enabling heterogeneity analysis for studies that report only summary statistics (means, variances, skewness, kurtosis) rather than full datasets. This allows researchers to re-analyze historically accumulated studies or those with privacy-restricted data to infer the extent and structure of treatment effect heterogeneity.
• Information Limits: The authors modestly claim that their work explicitly characterizes "what can and cannot be learned" under moment-only information. They demonstrate that while variance can often be bounded meaningfully, higher-order features (like skewness and kurtosis) are inherently underdetermined without stronger assumptions (like IED) or full distributional knowledge.
• Sharpness: The paper emphasizes that the derived bounds are sharp where claimed, providing the tightest possible intervals given the specific moment information. However, it also acknowledges that for many combinations of available moments, the bounds are not sharp, highlighting the fundamental limits of inference from aggregated statistics.

The authors conclude that while the IED assumption allows for point identification, the bounds derived without it provide a necessary characterization of the uncertainty inherent in causal effect heterogeneity when only marginal moments are available. They note that future work could explore tightening these bounds using additional structural assumptions, such as causal graphs or monotonicity.