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Granger Causality in Extremes

Juraj Bodik, Olivier C. Pasche

Abstract

We introduce a rigorous mathematical framework for Granger causality in extremes, designed to identify causal links from extreme events in time series. Granger causality plays a pivotal role in uncovering directional relationships among time-varying variables. While this notion gains heightened importance during extreme and highly volatile periods, state-of-the-art methods primarily focus on causality within the body of the distribution, often overlooking causal mechanisms that manifest only during extreme events. Our framework is designed to infer causality mainly from extreme events by leveraging the causal tail coefficient. We establish equivalences between causality in extremes and other causal concepts, including (classical) Granger causality, Sims causality, and structural causality. We prove other key properties of Granger causality in extremes and show that the framework is especially helpful under the presence of hidden confounders. We also propose a novel inference method for detecting the presence of Granger causality in extremes from data. Our method is model-free, can handle non-linear and high-dimensional time series, outperforms current state-of-the-art methods in all considered setups, both in performance and speed, and was found to uncover coherent effects when applied to financial and extreme weather observations.

Granger Causality in Extremes

Abstract

We introduce a rigorous mathematical framework for Granger causality in extremes, designed to identify causal links from extreme events in time series. Granger causality plays a pivotal role in uncovering directional relationships among time-varying variables. While this notion gains heightened importance during extreme and highly volatile periods, state-of-the-art methods primarily focus on causality within the body of the distribution, often overlooking causal mechanisms that manifest only during extreme events. Our framework is designed to infer causality mainly from extreme events by leveraging the causal tail coefficient. We establish equivalences between causality in extremes and other causal concepts, including (classical) Granger causality, Sims causality, and structural causality. We prove other key properties of Granger causality in extremes and show that the framework is especially helpful under the presence of hidden confounders. We also propose a novel inference method for detecting the presence of Granger causality in extremes from data. Our method is model-free, can handle non-linear and high-dimensional time series, outperforms current state-of-the-art methods in all considered setups, both in performance and speed, and was found to uncover coherent effects when applied to financial and extreme weather observations.
Paper Structure (55 sections, 37 theorems, 157 equations, 8 figures, 3 algorithms)

This paper contains 55 sections, 37 theorems, 157 equations, 8 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. From Granger causality via structural causality to causality in extremes
  3. Granger and structural causality
  4. Causality in extremes
  5. Connections between the definitions
  6. Robustness of causality in extremes to hidden confounders under regular variation
  7. Preliminaries for regular variation and SRE
  8. Causality in extremes under regular variation
  9. Estimation and causal discovery
  10. Conditioning on confounders being non-extreme
  11. Discovery of causality in extremes
  12. Testing tail causality
  13. Multivariate extension and hardness of testing
  14. Causal graph estimation
  15. Hardness of testing
  16. ...and 40 more sections

Key Result

Proposition 1

If $\textbf{X}\overset{{\rm ext}}{\longrightarrow}\textbf{Y}$, then $\textbf{X}\overset{{\rm tail}}{\longrightarrow}\textbf{Y}$. Under Assumptions AssumptionA1 and (IndependenceOfNoise), if $\textbf{X}\overset{{\rm tail}}{\longrightarrow}\textbf{Y}$, then $\textbf{X}\overset{{\rm ext}}{\longrightarr

Figures (8)

  • Figure 1: Comparison of the average model errors between our approach and the competitors for different numbers of variables (x-axis), data processes (columns) and sample sizes (rows). The average error is computed as the average distance between the true graph and the estimated graph, standardized between 0 and 1. The "random algorithm" generates a random graph with each edge present with probability $\frac{1}{2}$. Due to time complexity constraints, PCMCI with GPDC independence test is estimated only for $n=500, m\leq 7$.
  • Figure 2: Topographic map showing all 68 gauging stations in Switzerland Pasche. The purple dot 'M1' represents the meteorological station.
  • Figure 3: Estimated summary causal graph indicating Granger causality in extremes among the log returns of cryptocurrencies. The graph is obtained using Algorithm \ref{['Algorithm2']} incorporating the testing procedure outlined in Section \ref{['Section_testing']}. The width of each edge represents the magnitude of the p-value; a value close to $0$ results in a wider edge.
  • Figure S.1: Performance of Algorithm \ref{['Algorithm1']} for a range of causal strengths $\alpha_X$, for different choices of $q_F$ in $\hat{F}_{}^{truc(q_F)}$, and for all four considered data models (Models \ref{['VAR_time_series_definition']} and \ref{['GARCH_time_series_definition']} with heavy- and non-heavy-tailed noise distributions).
  • Figure S.2: Aggregated performance of Algorithm \ref{['Algorithm1']} with $k_n = \left \lfloor{n^\nu}\right \rfloor$ as a function of $\nu$ over all data models, when the confounder is accounted for in the estimation (left) or ignored to simulate hidden confounding (right).
  • ...and 3 more figures

Theorems & Definitions (77)

  • Definition 1: Granger causality GRANGER1980329
  • Definition 2: Structural causality
  • Definition 3: Causality in extremes
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • proof
  • Theorem 1
  • Definition 4
  • Definition 5
  • ...and 67 more