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Partial correlation analysis: Applications for financial markets
Dror Y. Kenett, Xuqing Huang, Irena Vodenska, Shlomo Havlin, H. Eugene Stanley
TL;DR
The paper addresses the limitation that Pearson correlations do not reveal whether other stocks influence a pairwise relationship. It applies statistically robust partial-correlation analysis across four markets, identifying validated influence links and using them to study market stability and cross-sector effects. The applications show differences in market-structure stability and that stocks can be influenced by sectors beyond their primary classification.
Problem
Pearson correlation captures similarity in two stocks’ price changes but does not measure whether other stocks influence their relationship, leaving many-body financial interactions insufficiently examined.
Method
The paper uses partial correlations and statistically validated filtering to extract underlying influence relationships between financial assets across four markets.
Results
The applications find higher market-structure stability in the US, UK, and Japan than in India, and show that a stock can be influenced by sectors outside its primary classification.
Takeaways & Limitations
The methodology provides information about implicit asset dependencies relevant to investors, traders, regulators, and policy makers.
Takeaways & Limitations
The Fisher transformation used for significance testing relies on assumptions including bivariate normality, independent observation pairs, and sufficiently large samples with correlations that are not too large.
Abstract
from arXiv · showhide
The presence of significant cross-correlations between the synchronous time evolution of a pair of equity returns is a well-known empirical fact. The Pearson correlation is commonly used to indicate the level of similarity in the price changes for a given pair of stocks, but it does not measure whether other stocks influence the relationship between them. To explore the influence of a third stock on the relationship between two stocks, we use a partial correlation measurement to determine the underlying relationships between financial assets. Building on previous work, we present a statistically robust approach to extract the underlying relationships between stocks from four different financial markets: the United States, the United Kingdom, Japan, and India. This methodology provides new insights into financial market dynamics and uncovers implicit influences in play between stocks. To demonstrate the capabilities of this methodology, we (i) quantify the influence of different companies and, by studying market similarity across time, present new insights into market structure and market stability, and (ii) we present a practical application, which provides information on the how a company is influenced by different economic sectors, and how the sectors interact with each other. These examples demonstrate the effectiveness of this methodology in uncovering information valuable for a range of individuals, including not only investors and traders but also regulators and policy makers.
1. Introduction
Financial-market correlations reveal shared price movements but do not show whether other assets control pairwise relationships. The paper develops a statistically validated partial-correlation framework to identify meaningful influence links without averaging across all pairs.
- Financial assets commonly exhibit synchronous cross-correlations, which researchers use to study market structure and dynamics.
- Existing measures such as CoVaR, Granger causality, and conditional correlation largely examine relationships between pairs of variables over a given period.
- Pearson correlation indicates similarity between two stocks but cannot determine whether another stock controls their observed relationship.
- Partial correlation measures the relationship remaining between two variables after accounting for their correlations with a third variable, enabling influence estimation.
- The proposed method filters statistically significant influence links before averaging them, rather than averaging influence across all pairs before validation.
- The framework is intended to quantify how companies, sectors, markets, or macroeconomic factors influence assets for applications including risk management and portfolio optimization.
2. Quantifying underlying relationships between financial assets
The paper uses partial correlations conditioned on market indices and third stocks to extract statistically significant underlying relationships among financial assets. Daily stock-return data from four markets support an influence measure that separates index effects, tests stock-to-pair links, and selects significant interactions.
- Data: Daily adjusted closing prices from the U.S., U.K., Japan, and India are converted into log returns after filtering illiquid stocks.The sample covers stocks active from January 2000 through December 2010; stocks with no price movement for more than 6% of 2700 trading days are removed.
- Stock raw and partial correlation: Partial correlation measures the relationship between two stock returns after removing their associations with a market index.The method obtains residuals by regressing both stock returns on the index, then correlates those residuals.
- Stock raw and partial correlation: Conditioning on the index produces lower partial correlations than raw correlations for all plotted stock pairs, indicating positive index influence.After additionally conditioning on a third stock, influence values occur on both sides of the diagonal, with many negative values at low index-conditioned correlations.
- Stock raw and partial correlation: The influence of stock Z on the pair X and Y is d(X, Y : Z) = ρ(X, Y : M) − ρ(X, Y : M, Z).A large value indicates that a substantial fraction of the index-conditioned correlation can be explained by Z.
- Statistical significance testing: The method evaluates up to N(N−1)(N−2)/2 partial-correlation interactions and filters them using Fisher-transformation and empirical significance tests.The empirical test shuffles return sequences to destroy correlations, estimates null influence thresholds, and retains significant links; the paper uses the two-tailed threshold z > 1.6449.
- Statistical significance testing: The empirical test recommends two-tailed significance because significant negative influences are also important.For the S&P 500 example, the reported two-tailed thresholds include z > 1.6449 at 2%, z > 1.2816 at 10%, and z > 0.8416 at 20%.
3. Market structure and its stability
The paper represents market structure by ranking stocks according to their average partial-correlation influence, then compares these rankings across quarterly periods. Rank similarity generally declines with time, revealing different stability patterns and crisis-related structural changes across markets.
- Market structure: Stocks are ranked by their average influence on all other stocks, and Kendall τ compares these market-structure rankings across 44 quarterly periods.The investigated 11-year period is divided into 44 quarters.
- Market stability: Longer intervals generally produce lower rank correlations, indicating decreasing similarity between market structures over time.The comparison covers every possible quarter pair for the four investigated markets.
- Cross-market comparison: S&P 500, FTSE 350, and Nikkei 500 rankings show strong stability patterns, whereas Indian BES 200 rankings show almost no stable pattern.The paper relates this contrast to developed markets retaining their structure longer than fast-developing markets.
- Crisis-related changes: The US market shows structural changes after the 2000 dot-com crisis and 2008 credit crunch, marked by crisis-period similarity followed by low subsequent rank correlations.The low values represent changes in structure after the crisis periods.
- Crisis-related changes: The 2008 financial crisis produces observable structural changes in the UK, but not in Japan or India.These market-specific differences are identified through the quarter-pair ranking correlations.
- Quantifying stability: An approximate exponential decay, τ = τ0e^-t/λ, summarizes how ranking correlations decline with time and provides two parameters for market stability.τ0 measures consecutive-quarter consistency, while λ measures the decay period or speed.
- Practical implication: The τ0 and λ parameters can help monitor structural changes and their persistence, supporting regulatory and policy monitoring of market stability and robustness.The paper presents this methodology as a tool for regulators and policy makers.
4. Quantifying the influence of economic sectors
The paper aggregates partial-correlation influence across sectors to characterize how sectors affect individual stocks and interact across markets. Sector-based influence generally aligns with official classifications while revealing cross-sector dependencies and concurrent sector effects.
- Sector influence for a stock is computed by averaging influence from stocks within each sector, then normalizing the averages into sector-attribution coefficients.The average influence dS_X represents the influence stock X receives from sector S; the normalized coefficients attribute stock performance to sector performance.
- Example stock profiles: Alcoa shows substantial influence from energy, materials, and industrials, whereas Franklin Templeton is influenced most by financials and GE mainly by materials, utilities, and financials.
- Example stock profiles: Apple exhibits a more homogeneous distribution of sector influence, possibly indicating greater diversity across economic activities than the other example stocks.
- Validation: Partial-correlation rankings retain high sector-prediction accuracy across most S&P 500 sectors and agree with GICS classifications, except telecommunications, possibly because it has few stocks.
- Sector interactions: Financials and energy appear internally cohesive, while industrials and materials show stronger dependencies on other sectors; sector-influence pairs also reveal correlated effects across markets.
5. Summary
The paper develops a statistically robust dependency-network framework based on partial correlations and applies it to financial markets. Its applications compare market-structure stability across countries and identify cross-sector influences on individual stocks, with relevance for investors, regulators, and policy makers.
- The framework extends dependency-network methodology with statistically robust theoretical or empirical filtering of influence relationships.
- Developed markets such as the US, UK, and Japan exhibit greater market-structure stability than developing India in the presented comparison.
- A stock can be influenced by multiple economic sectors beyond its primary sector classification.
- The methodology provides information about interactions among assets and economic sectors that is valuable to investors, practitioners, regulators, and policy makers.