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Spatio-Temporal Hierarchical Causal Models

Xintong Li, Haoran Zhang, Xiao Zhou

TL;DR

ST-HCMs address causal inference in nested spatio-temporal data with unobserved unit-level confounders by extending hierarchical causal modeling to the spatio-temporal domain. The core theoretical contribution is the Spatio-Temporal Collapse Theorem, which proves convergence of ST-HCMs to a Dynamic Collapsed Model as subunit data grows, enabling causal identification via sequential adjustability or dynamic instruments. The paper also provides an estimation algorithm using unit-specific conditional dynamics and G-computation, adaptable to Linear Mixed-Effects Models, GBMs, or Gaussian Processes. Empirical studies on synthetic data and a Chicago urban-traffic dataset demonstrate accurate, robust causal estimates and practical gains from modeling both hierarchy and spatial spillovers.

Abstract

The abundance of fine-grained spatio-temporal data, such as traffic sensor networks, offers vast opportunities for scientific discovery. However, inferring causal relationships from such observational data remains challenging, particularly due to unobserved confounders that are specific to units (e.g., geographical locations) yet influence outcomes over time. Most existing methods for spatio-temporal causal inference assume that all confounders are observed, an assumption that is often violated in practice. In this paper, we introduce Spatio-Temporal Hierarchical Causal Models (ST-HCMs), a novel graphical framework that extends hierarchical causal modeling to the spatio-temporal domain. At the core of our approach is the Spatio-Temporal Collapse Theorem, which shows that a complex ST-HCM converges to a simpler flat causal model as the amount of subunit data increases. This theoretical result enables a general procedure for causal identification, allowing ST-HCMs to recover causal effects even in the presence of unobserved, time-invariant unit-level confounders, a scenario where standard non-hierarchical models fail. We validate the effectiveness of our framework on both synthetic and real-world datasets, demonstrating its potential for robust causal inference in complex dynamic systems.

Spatio-Temporal Hierarchical Causal Models

TL;DR

ST-HCMs address causal inference in nested spatio-temporal data with unobserved unit-level confounders by extending hierarchical causal modeling to the spatio-temporal domain. The core theoretical contribution is the Spatio-Temporal Collapse Theorem, which proves convergence of ST-HCMs to a Dynamic Collapsed Model as subunit data grows, enabling causal identification via sequential adjustability or dynamic instruments. The paper also provides an estimation algorithm using unit-specific conditional dynamics and G-computation, adaptable to Linear Mixed-Effects Models, GBMs, or Gaussian Processes. Empirical studies on synthetic data and a Chicago urban-traffic dataset demonstrate accurate, robust causal estimates and practical gains from modeling both hierarchy and spatial spillovers.

Abstract

The abundance of fine-grained spatio-temporal data, such as traffic sensor networks, offers vast opportunities for scientific discovery. However, inferring causal relationships from such observational data remains challenging, particularly due to unobserved confounders that are specific to units (e.g., geographical locations) yet influence outcomes over time. Most existing methods for spatio-temporal causal inference assume that all confounders are observed, an assumption that is often violated in practice. In this paper, we introduce Spatio-Temporal Hierarchical Causal Models (ST-HCMs), a novel graphical framework that extends hierarchical causal modeling to the spatio-temporal domain. At the core of our approach is the Spatio-Temporal Collapse Theorem, which shows that a complex ST-HCM converges to a simpler flat causal model as the amount of subunit data increases. This theoretical result enables a general procedure for causal identification, allowing ST-HCMs to recover causal effects even in the presence of unobserved, time-invariant unit-level confounders, a scenario where standard non-hierarchical models fail. We validate the effectiveness of our framework on both synthetic and real-world datasets, demonstrating its potential for robust causal inference in complex dynamic systems.

Paper Structure

This paper contains 63 sections, 7 theorems, 28 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Causal Graphical Models and HCMs
  5. Spatio-Temporal Hierarchical Causal Models
  6. A Temporal Extension: T-HCMs
  7. The Framework: ST-HCMs
  8. Identification and Estimation
  9. The Identifiability Problem in Dynamic HCMs
  10. Identifiability via Sequential Adjustability
  11. Identifiability via Dynamic Instruments
  12. ATE with Time-Invariant Spatial Confounding
  13. Problem Formulation
  14. Identification Strategy
  15. Estimation Algorithm
  16. ...and 48 more sections

Key Result

Theorem 1

Under Assumptions assump:mech_conv_t and assump:regularity_t, as the number of subunits $m \to \infty$, the joint distribution of the macro-state history from a T-HCGM, marginalized over subunit variables ($P_{m,T}^{marg}$), converges in Kullback-Leibler divergence to the distribution of its corresp

Figures (6)

  • Figure 1: An example of hierarchical causality in traffic.
  • Figure 2: Hierarchy and Collapsing in HCMs.
  • Figure 3: The challenge of identifiability.
  • Figure 4: Validation of ST-HCM estimator.
  • Figure 5: Robustness analysis of estimators.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Definition 1: Temporal Hierarchical Structural Causal Model (T-HSCM)
  • Definition 2: Temporal Hierarchical Causal Graphical Model (T-HCGM)
  • Theorem 1: Convergence of T-HCMs
  • Theorem 1: Convergence of T-HCMs
  • proof : Proof Sketch
  • Definition 3: Spatio-Temporal Hierarchical Causal Graphical Model (ST-HCGM)
  • Theorem 2: Convergence of ST-HCMs
  • proof : Proof Sketch
  • Proposition 1: Identifiability in T-HCMs via Adjustability
  • proof : Proof Sketch
  • ...and 12 more