Quantile Partial Effect (QPE)

• QPE is a statistical measure defined as the derivative of the conditional quantile function with respect to covariates, capturing heterogeneous impacts across the outcome distribution.
• Estimation methods for QPE include conditional quantile regression with kernel, series, or deep learning approaches, which test finite-span assumptions for causal identifiability.
• QPE provides a novel framework for causal discovery by leveraging observable distributional features and Fisher Information, demonstrating strong empirical performance in both synthetic and real-world datasets.

Quantile Partial Effect (QPE) is a statistical functional arising from conditional quantile regression that characterizes how the quantiles of the conditional distribution of an outcome variable respond to changes in covariates. By explicitly focusing on the heterogeneity of impact across the outcome distribution, QPE generalizes traditional mean-based effect notions and serves as a central tool in modern distributional analysis, causal inference, and, more recently, causal discovery using observational data.

1. Formal Definition and Interpretation

QPE is defined as the derivative of the conditional quantile function with respect to a covariate. Let $Y$ be an outcome, $X$ a vector of covariates, and $Q_{Y|X}(x, \tau)$ the $\tau$-th conditional quantile of $Y$ given $X = x$. The Quantile Partial Effect at quantile level $\tau$ and covariate configuration $x$ is

$\text{QPE}(\tau, x) = \nabla_x Q_{Y|X}(x, \tau).$

Alternatively, since the conditional quantile function is the inverse of the conditional cumulative distribution function $F_{Y|X}(y|x)$, the QPE can be equivalently expressed for a given $(y, x)$ pair as

$$\psi_{Y|X}(y|x) = - \frac{\nabla_x F_{Y|X}(y | x)}{p_{Y|X}(y | x)},$$

where $p_{Y|X}$ is the conditional density of $Y$ given $X$ (Chen et al., 16 Sep 2025). This formulation exposes QPE as a measure of the local sensitivity of the cumulative probability mass at $y$ to changes in $x$, "normalized" by the density at that point.

QPE is fundamentally distributional: it describes effects on quantiles—not just means—thereby capturing heterogeneity, including impacts on the tails of the outcome. This makes QPE especially valuable in domains such as risk management, labor economics, health, and complex systems where effects may not be uniform across percentiles (Chao et al., 2016, Houndetoungan, 15 Jun 2025).

2. Theoretical Properties and Identifiability

A key theoretical advance is the demonstration that, under certain parametric restrictions, the QPE can be used to distinguish cause from effect directly from the observational joint distribution. The main parametric assumption is that $\psi_{Y|X}(y|x)$ lies in the finite linear span of known basis functions $\{\phi_j(y)\}_{j=1}^k$:

$\psi_{Y|X}(y|x) = \sum_{j=1}^k c_j(x) \phi_j(y).$

If this assumption holds for one causal direction but not the other, identifiability of causal direction is achieved without explicit modeling of noise or mechanistic structure (Chen et al., 16 Sep 2025). This generalizes earlier results for additive noise models, post-nonlinear models, and others, by leveraging the asymmetry of the observable distribution's shape features as encoded in the QPE.

Crucially, the identifiability argument operates entirely at the level of the observable distribution, eschewing untestable Markov or independence assumptions. A technical criterion, formulated via vanishing Wronskian determinants of QPE-related quantities, provides a sharp (and under regularity conditions, sufficient) characterization of identifiability.

3. Estimation and Basis Function Testing

Estimation of QPE proceeds via conditional quantile regression, using either kernel, series, or deep learning (flow-based) approaches (Chen et al., 16 Sep 2025). To empirically test whether the QPE satisfies the required finite-span assumption for identifiability, the estimated QPE $\hat\psi_{Y|X}(y|x)$ is regressed (in $y$) on the proposed basis functions $\{\phi_j(y)\}$, and the residual "lack of fit" is compared for both candidate causal directions. The direction with the lower residual is adjudged as the cause.

In bivariate settings, kernel-based methods (QPE–k) and neural flow approaches (QPE–f) have demonstrated strong empirical performance. In multivariate scenarios, the relationship between QPE and the score function ($\nabla_x \log p_{X,Y}$) is exploited; specifically, second moment comparisons of scores reveal causal orderings via Fisher Information Causal Ordering (FICO).

4. Mathematical Characterizations

Quantitative characterizations of QPE—core to both estimation and theory—are as follows:

• Direct derivative representation: $\psi_{Y|X}(y|x) = \nabla_x Q_{Y|X}(x, \tau)$ (with $\tau$ such that $y = Q_{Y|X}(x, \tau)$).
• Representation via the CDF:

$$\psi_{Y|X}(y|x) = - \frac{\nabla_x F_{Y|X}(y|x)}{p_{Y|X}(y|x)}.$$

• In the presence of the finite-span condition, for a vector of basis functions $\Phi(y) = (\phi_1(y), ..., \phi_k(y))$:

$$\psi_{Y|X}(y|x) \in \text{span} \{ \Phi(y) \} \implies \text{identifiability of } X \to Y.$$

• When assessing basis inclusion, a Wronskian determinant constructed from QPE and the basis functions is required to vanish identically if the span condition holds.

In the multivariate setting, by leveraging Stein's identity, the paper relates the second moments of score functions and QPE,

$$\mathbb{E}[ (s_{X_i})^2 ] - \mathbb{E}[ (s_Y)^2 ] = \mathbb{E}[ (r_{X_i})^2 ] + \mathbb{E}[ ((\psi_{Y|X})^2 - 1)(s_Y)^2 ] - \mathbb{E}[ (\partial_y \psi_{Y|X})^2 + 2\psi_{Y|X} \partial_y^2 \psi_{Y|X} ].$$

Fisher information is thus sufficient for ordering under assumptions on the QPE's second moment.

5. Empirical Performance and Applications

In benchmark experiments—both synthetic data from functional causal models and real datasets such as the Tübingen cause–effect pairs—QPE-based causal discovery methods (QPE–f in particular) outperform or are competitive with state-of-the-art approaches. This includes settings with additive, heteroscedastic, multiplicative, and post-nonlinear noise, among others (Chen et al., 16 Sep 2025).

For multivariate causal discovery, the FICO algorithm applies the aforementioned Fisher information criterion systematically to infer causal orderings. Empirical results confirm the viability and efficiency of this method in complex settings.

Potential application areas include:

• Causal discovery in systems with complex heterogeneity where functional noise assumptions are inappropriate or unverifiable.
• Conditional impact analysis in economics, epidemiology, and social sciences—where heterogeneity and tail effects are paramount.
• Financial and risk modeling scenarios, particularly where the distributional impacts of covariate shifts are more informative than average effects.

6. Significance and Methodological Implications

The QPE-centered framework moves causal identification and discovery into a new regime by relying on directly observable properties of the data—namely, changes in conditional quantiles—without recourse to structural or noise modeling. This approach allows for novel identifiability results, extends the scope of causal inference, and provides practical, well-defined algorithms for both bivariate and multivariate discovery that are robust to violations of classical assumptions.

Moreover, the QPE yields rich, distributional insight: it enables detection of heterogeneous impacts, tail sensitivity, and nonlinear effects, and under finite-span conditions, provides a rigorous statistical handle for inferring directionality in cause–effect relations (Chen et al., 16 Sep 2025).

7. Limitations and Extensions

While the QPE approach generalizes prior methods, it depends critically on the validity of the finite-span (parametric shape) assumption for identifiability. This is a structural condition on the functional form of the QPE, not directly verifiable in practice, although empirical basis function testing provides a pragmatic diagnostic. The methodology is otherwise free of strong distributional or independence assumptions, increasing its flexibility but also rendering it sensitive to model specification via the choice of basis functions.

A plausible implication is that further research may explore adaptive or data-driven basis selection, theoretical characterization of necessary and sufficient conditions for identifiability in broader classes of structural models, and integration of QPE-based criteria with other distributional or independence-based discovery tools.

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In summary, Quantile Partial Effect provides a versatile, distributionally informed, and observationally anchored framework for both effect sizing and causal discovery, extending the reach and interpretive power of quantile regression into foundational questions of directionality and mechanism in complex systems (Chen et al., 16 Sep 2025).