[Paper Review] Stock market comovements: nonlinear approach for 48 countries

Paulo Ferreira, Andreia Dionísio+1 more|arXiv (Cornell University)|Feb 1, 2015

Complex Systems and Time Series Analysis3 citations

TL;DR

This study analyzes stock market comovements across 48 countries using linear (cointegration, Granger causality) and nonlinear methods (mutual information, MF-DFA, MF-DXA), revealing strong nonlinear dependencies primarily between emerging and frontier markets. It confirms multifractal cross-correlations across all pairs, with significant deviations from linear relations at negative moments, and quantifies persistent cross-correlation via $σ_{DCCA}$. The results highlight the dominance of nonlinear dynamics and multifractal structure in global equity market linkages.

ABSTRACT

This paper examines the stock market comovements using basically three different approaches. Firstly, we used the most common linear analysis, based on cointegration and Granger causality tests; secondly we applied a nonlinear approach, using mutual information to analyze nonlinear dependence. Since underlying data sets are affected by non-stationarities, we also applied MF-DFA and MF-DXA in order to examine the multifractality nature of data and to analyze the relationship and mutual interaction between pairs of series, respectively. The overall results are quite interesting, since we found only 170 pair of stock markets cointegrated, and according to the Granger causality and mutual information we realized that the strongest relations lies between emerging markets, and between emerging and frontier markets. According to scaling exponent given by MF-DFA, $h(q=2)>1$, we found that all underlying data belong to non-stationary process. There is no cross-over in the fluctuation functions determined by MF-DFA method confirmed that mentioned approach could remove trends embedded in the data sets. The nature of cross-correlation exponent based on Mf-DXA is almost multifractal for all stock market pairs. The empirical relation, $h_{xy}(q)=[h_{xx}(q)+h_{yy}(q)]/2$ was confirmed just for $q>0$, while for $q<0$ there was a deviation from this relation. Width of singularity spectrum is in the range $\Delta \alpha_{xx}\in [0.304,0.905]$ which is another confirmation about multifractality nature of underlying data sets. The singularity spectrum for cross-correlation is in the range $\Delta \alpha_{xy}\in [0.246,1.178]$ confirming more complex relation between stock markets. The value of $\sigma_{DCCA}$ which is a measure for quantifying degree of cross-correlation indicates that all stock market pairs in the underlying time interval belong to cross-correlated series.

Motivation & Objective

• To investigate the extent and nature of comovement across global stock markets beyond linear dependencies.
• To assess the presence of multifractality in individual and cross-correlated market time series using advanced scaling techniques.
• To evaluate the robustness of linear relationships (cointegration, Granger causality) against nonlinear dependencies captured by mutual information and cross-correlation analysis.
• To quantify the degree and structure of cross-correlations between stock market pairs using $σ_{DCCA}$ and singularity spectra.

Proposed method

• Employed linear cointegration and Granger causality tests to detect long-term equilibrium and predictive relationships between stock market indices.
• Applied mutual information to detect nonlinear dependence structures not captured by linear methods.
• Used Multifractal Detrended Fluctuation Analysis (MF-DFA) to assess the multifractal nature of individual market return series, with $h(q=2) > 1$ indicating non-stationarity.
• Applied Multifractal Detrended Cross-Correlation Analysis (MF-DXA) to examine cross-correlations between pairs of markets, estimating $h_{xy}(q)$ and singularity spectra.
• Calculated the cross-correlation degree measure $σ_{DCCA}$ to quantify the strength of cross-correlations across all pairs.
• Validated the empirical relation $h_{xy}(q) = [h_{xx}(q) + h_{yy}(q)]/2$ for $q > 0$, and analyzed deviations for $q < 0$.

Experimental results

Research questions

• RQ1 To what extent do linear methods like cointegration and Granger causality detect meaningful comovements in global stock markets?
• RQ2 How do nonlinear dependencies, as measured by mutual information, compare to linear relationships in explaining market comovements?
• RQ3 What is the degree of multifractality in individual stock market return series, and how does it vary across countries?
• RQ4 How do cross-correlation structures between market pairs differ from linear expectations, especially at different moments ($q$)?
• RQ5 To what extent are the observed cross-correlations persistent and quantifiable across all 48 country pairs?

Key findings

• Only 170 out of all possible pairs of stock markets were found to be cointegrated, indicating limited long-term equilibrium relationships.
• Mutual information and Granger causality results revealed the strongest dependencies between emerging and frontier markets, highlighting nonlinear interdependence.
• The scaling exponent $h(q=2) > 1$ confirmed that all individual market return series are non-stationary, supporting the use of detrending methods.
• The absence of a crossover in MF-DFA fluctuation functions indicates that the method effectively removed trends from the data.
• The singularity spectrum width for individual markets, $Δ\alpha_{xx} \in [0.304, 0.905]$, confirms the presence of multifractality in all return series.
• $\sigma_{DCCA}$ values indicate that all stock market pairs are cross-correlated, with the cross-correlation spectrum $\Delta\alpha_{xy} \in [0.246, 1.178]$ showing higher complexity than individual series.