A causal interactions indicator between two time series using extreme variations in the first eigenvalue of lagged correlation matrices

ArXiv ID: 2307.04953 “View on arXiv”

Authors: Unknown

Abstract

This paper presents a method to identify causal interactions between two time series. The largest eigenvalue follows a Tracy-Widom distribution, derived from a Coulomb gas model. This defines causal interactions as the pushing and pulling of the gas, measurable by the variability of the largest eigenvalue’s explanatory power. The hypothesis that this setup applies to time series interactions was validated, with causality inferred from time lags. The standard deviation of the largest eigenvalue’s explanatory power in lagged correlation matrices indicated the probability of causal interaction between time series. Contrasting with traditional methods that rely on forecasting or window-based parametric controls, this approach offers a novel definition of causality based on dynamic monitoring of tail events. Experimental validation with controlled trials and historical data shows that this method outperforms Granger’s causality test in detecting structural changes in time series. Applications to stock returns and financial market data show the indicator’s predictive capabilities regarding average stock return and realized volatility. Further validation with brokerage data confirms its effectiveness in inferring causal relationships in liquidity flows, highlighting its potential for market and liquidity risk management.

Keywords: Tracy-Widom distribution, Coulomb gas model, Causal interactions, Lagged correlation matrices, Time series analysis, Stock returns / Liquidity flows

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 5.5/10
• Quadrant: Holy Grail
• Why: The paper employs advanced random matrix theory (Tracy-Widom, Coulomb gas) and derives novel causal indicators, demonstrating high mathematical sophistication. It also shows strong empirical validation with synthetic data, historical financial data, and brokerage data, comparing performance to Granger causality and outlining practical applications in risk management.

  flowchart TD
    A["Research Goal: Identify causal interactions between time series<br>using variations in the first eigenvalue"] --> B["Methodology: Lagged Correlation Matrices & Coulomb Gas Model"]
    B --> C{"Calculate Largest Eigenvalue λ₁(t) for each lag"}
    C --> D["Statistical Analysis: Apply Tracy-Widom Distribution"]
    D --> E["Compute Explanatory Power Variability<br>Standard deviation of λ₁"]
    E --> F["Define Causal Interaction Indicator<br>Push/Pull dynamics measured by tail event variability"]
    F --> G["Key Findings & Outcomes"]
    
    G --> H1["Outperforms Granger's Causality in detecting structural changes"]
    G --> H2["Validated on stock returns: Predicts avg return & realized volatility"]
    G --> H3["Effective in inferring liquidity flow causality"]
    G --> H4["Novel method for market & liquidity risk management"]