Impact of External Shocks on Global Major Stock Market Interdependence: Insights from Vine-Copula Modeling

3.1 Copula

Copula functions establish a connection between marginal and joint distributions and are used to describe the dependence relationships among random variables. Copula functions effectively characterize the dependencies between random variables, encompassing not only linear and nonlinear correlations but also symmetric and asymmetric correlations, as well as upper and lower tail dependencies.

The concept of Copula functions stems from Sklar’s theorem (1959): For a p -dimensional random variable x = ( x 1 , x 2 , … , x p ) T , where F represents the marginal distributions F 1 ( x 1 ) , F 2 ( x 2 ) , … , F p ( x p ) , there exists a Copula function C ( ⋅ ) such that:

C ( F 1 ( x 1 ) , F 2 ( x 2 ) , … , F p ( x p ) ) represents the probability density function of the Copula function, where f k ( x k ) denotes the probability density function of the marginal distribution of the variable x k ( k = 1 , 2 , … , p ) . The Vine-Copula model is an extension of the Copula function model.

3.2 Pair-Copula

To address the challenge of modeling complex dependence structures, particularly those exhibiting asymmetry and non-linearity, scholars Aas et al. (2009) proposed the Pair-Copula model. The Pair-Copula model decomposes the multivariate density into a product of several bivariate copula densities. The mathematical form and tail dependence characteristics of each Pair-Copula are summarized in Table 1.

The Normal copula, Student-t copula, and the Frank copula model present symmetric dependencies. However, the Student-t copula can capture heavy-tailed behavior, making it more sensitive to extreme co-movements. For asymmetric dependencies, the Gumbel and Survival Gumbel copulas are employed. The Gumbel copula captures upper tail dependence, highly sensitive to joint positive extreme events (stronger correlation in bull markets). Conversely, the Survival Gumbel copula, which is a 180-degree rotation of the Gumbel, captures lower tail dependence (stronger correlation in bear markets). The capacity to specify these distinct asymmetric structures for different pairs of variables is fundamental to constructing a robust Vine-Copula model that accurately reflects the changes of market shocks.

3.3 Vine-Copula

The Vine-Copula model, developed based on the Pair-Copula model and graph theory, adopts a tree structure to combine multiple bivariate Copula functions. The tree structure created by Vine-Copula, known as the “Vine graph,” was introduced by Bedford and Cooke (2002). The Vine structure includes nodes, branches, and trees. Each node signifies a variable, each branch represents the Copula function between variables, and each tree consists of branches connecting two nodes. The Vine-Copula model is a method used to model the joint distribution of multivariate random variables, to describe the dependencies between variables.

The primary reason for selecting the Vine-Copula in our study is its unparalleled flexibility. Unlike standard multivariate models, it allows for the selection of different bivariate copula families for each pair of variables, enabling an accurate representation of heterogeneous dependence patterns. For example, it can present symmetric dependence between one pair of markets and asymmetric tail dependence between another. This flexibility is crucial for accurately modeling the non-linear and heterogeneous risk spillovers across diverse international markets, providing a more interpretable and realistic depiction of the underlying dependence structure.

Assuming X = ( X 1 , X 2 , … , X d ) is a d -dimensional random vector with marginal distribution functions F 1 , F 1 , … , F d , C ( u 1 , u 2 , … , u d ) is its Copula function. The Vine-Copula model represents the Copula function using the product form conditional probability distributions in a tree-like structure, given by:

Here, C i , j ∣ p i , j represents the conditional Copula function between X i and X j on the tree structure, where p i , j indicates their parent node on the tree. In the tree structure, nodes represent variables, edges represent dependencies between variables, and each node’s edge represents its dependency on the parent node. The Vine-Copula model can accurately describe the dependency relationships among multidimensional random variables due to the flexibility and interpretability of the tree structure. Vine structures are commonly categorized into three types: C-Vine, D-Vine, and R-Vine.

3.4 Tail Dependence

The tail dependence coefficient refers to the probability of the occurrence of extreme values in one or more random variables when another or other random variables are in extreme states within a multivariate random variable setting. In other words, tail dependence explores the degree of correlation among different assets when extreme losses or extreme gains occur simultaneously. This study employs the limit form of conditional probabilities to measure tail dependence, with specific definitions for the upper tail dependence coefficient τ U and the lower tail dependence coefficient τ L as follows:

where X and Y are two random variables, and F ( X ) , G ( Y ) are the distribution functions of X and Y.

The time-varying Copula function modifies the static Copula function by replacing the constant correlation coefficient τ with a time−varying correlation coefficient τ t . This modification is used to measure the dynamic upper and lower tail dependence between two variables. τ t U represents the upper tail correlation coefficient, quantifies the limiting conditional probability that one variable will assume an extremely high value, conditional on the other variable having also done so. Conversely, the lower tail correlation coefficient τ t L measures the corresponding probability for two variables to jointly assume extremely low values. Asymmetric tail dependence is present if these two measures differ ( τ t U ≠ τ t L ).

The parameter evolution equations are given by:

In equation (10), Λ ( x ) = 1 1 + e − x is the modified logistic function, and u and v are from Symmetrized Joe-Clayton (SJC) Copula model. Patton (2002) proposed the SJC-Copula model, which is an extension of the Joe-Clayton Copula model, the time-varying expression for the Joe-Clayton Copula given by:

In equation (11), κ = 1 log 2 ( 2 − τ t U ) , γ = − 1 log 2 τ t L .

The expression for the time-varying SJC Copula model is:

5.1 Results of Fitting Marginal Distributions

To capture the volatility clustering and asymmetry in the returns, we employ the ARMA-GARCH framework to fit the marginal distributions. We utilize the ARMA model to determine the optimal lag order for the conditional mean equation. Then, we compare the results of standard residual distributions with normal distribution, skewed normal distribution, T-distribution, skewed T-distribution, GED distribution, and skewed GED distribution. The ARMA(p,q)-GARCH(1,1)-skewed Student’s t-distribution for the residuals is selected as the optimal model based on parameter significance and the minimum AIC principle. To capture changes in the dependence structure across distinct market phases, we independently construct and estimate parameters for Vine-Copula models over three delineated subperiods: Phase 1 (Stable Period), Phase 2 (COVID-19 Period), and Phase 3 (Russia–Ukraine Conflict Period). We use R Studio to fit the marginal distributions of stock index returns in different stages, the model parameters are estimated and summarized in Table 4.

The marginal distribution model parameters reveal that the ARCH ( α ) and GARCH ( β ) values for each series are statistically significant and generally satisfy the stationarity constraint α + β < 1 . The estimated β values across all return series exceed 0.5, signifying substantial historical influence on their yield rates and suggesting volatility clustering phenomena, which is particularly noteworthy for the SPX index. To validate the adequacy of the model specification, we conduct a series of diagnostic tests on the standardized residuals, including the Ljung-Box test for autocorrelation in both residuals (LB) and squared residuals (LB²), and the Lagrange Multiplier (ARCH-LM) test for autoregressive conditional heteroscedasticity. We utilized a lag order of five for diagnostic tests – commonly adopted in similar studies, the results indicate no evidence of significant autocorrelation or remaining ARCH effects (heteroscedasticity). Combining these diagnostic results with the AIC values and likelihood estimates, we conclude that the GARCH(1,1)-Skewed Student’s t model provides a good fit for the return series.

5.2 Vine-Copula Estimation Results

We employ the Vine-Copula method described in Section 2 to analyze the dependence structure among stock markets by fitting probability integral transformed residuals separately for three sub-periods. Kendall’s rank correlation coefficient serves as the measure of pairwise dependence, and we utilize the maximum spanning tree (MST) algorithm to construct the Vine-Copula model (specifically, the first tree, T1). The copula parameters are estimated using the maximum likelihood method. Subsequently, we determine the optimal R-Vine structure and select the best-fitting pair-copula families based on the minimum Akaike Information Criterion (AIC) criterion. If the chosen copula exhibits asymmetric tail characteristics (e.g., Gumbel, or their rotated forms), this indicates an asymmetric dependence structure. Otherwise, if the chosen copula is a T-Copula, Normal-Copula, or F-Copula, it indicates a symmetric dependence structure between the two markets. Asymmetry can also be assessed by comparing the upper tail dependence coefficient (utd) and the lower tail dependence coefficient (ltd). A difference between the two indicates the presence of asymmetric tail dependence. Table 5 presents the parameters and selected pair-copula families of the first tree for each stage. The results indicate that while some countries exhibit consistent interdependence, there is also considerable variability in the chosen pair-copula families, the strength of dependence, and implied tail dependence.

We utilize the graphical representation of the R-Vine structure (Vine trees) to provide a more intuitive visualization of the dependency structure, including the dependence strength (τ) and selected pair-copula families, across the three stages, as illustrated in Figure 3. Table 6 summarizes stock market dependence characteristics among different regions across the three stages: Stage 1 (pre-pandemic stable period), Stage 2 (COVID-19 crisis), and Stage 3 (Russia–Ukraine war). For each regional grouping (intra-European, intra-Asian, Europe–USA, Europe–Asia, and RU-Euro), the table highlights the strength of interdependence (high/moderate/low), and the structure of interdependence (symmetry/asymmetry/mixed).

Figure 3

The first tree structure of the Vine-Copula for each stage. Note: This figure illustrates the Vine-Copula tree structures for three distinct stages. On each edge (connecting branch), the Kendall rank correlation coefficient (τ) and the Copula type are presented. Each node represents a country’s stock market. Stage 1: Stable Period; Stage 2: COVID-19 Pandemic Period; Stage 3: Russia–Ukraine War Period.

The overall structure exhibits a star-chain configuration, wherein the Netherlands acts as a central hub of stock market volatility transmission, linking European, USA, and Asian markets. This pattern reveals pronounced regional clustering (Hypothesis 1); specifically, the interconnectedness between European and US markets is notably tighter compared to their connection with Asian markets. The Kendall rank correlation within the EU stock markets exceeded 0.55 across all periods, initially presenting a star configuration with symmetrical tail dependencies, as evidenced by Table 5’s copula types. However, during the COVID-19 pandemic in Stage 2, this symmetric structure evolved into an asymmetric chain structure in Stage 2, with a shift from T-Copula (NL-FR; τ = 0.7, utd = ltd = 0.62) to SG-Copula ((NL-FR; τ = 0.65, ltd = 0.73), reflecting a marked change. This proves our hypothesis 2: heightened contagion during downturns has stronger lower tail dependence. Despite the UK’s exit from the EU, its dependency remained significant, with an asymmetric structure. In Stage 3, as the pandemic subsided and economic revival, the EU’s central node shifted to Switzerland, presenting a star-shaped structure with enhanced dependencies, as seen in the CH-NL (τ = 0.66) and CH-DE (τ = 0.74) pair. Some dependencies revert to symmetry, e.g., CH-UK and CH-FR.

The interdependence between the USA and European equities generally maintained a symmetric dynamic. In Stage 2, the US stock market experienced a series of trading halts, and average interconnectedness with European markets decreased by approximately 12.8% compared to the first stage (e.g., τ dropped from 0.39 to 0.34 for US-NL). According to investor behavior theory, global crises heighten market uncertainty, driving investors to seek refuge in safe-haven assets like gold. This behavior diminishes market liquidity and amplifies volatility, thereby reinforcing the volatility contagion effect. This dependency rebounded towards pre-crisis levels in Stage 3. Notably, the interdependence between the USA and Europe has maintained symmetry throughout all stages.

Prior to the Russia–Ukraine conflict, Russia maintained a moderate level of symmetric dependency with European markets (RU-NL, τ = 0.27), while the pandemic heightened this dependency (RU-NL, τ = 0.39) and changed it into asymmetry. However, subsequent political factors and the onset of hostilities led to the severance of economic ties between Russia and Europe, leaving Russia with only a weak dependency on the USA in Stage 3 (RU-USA, τ = 0.08).

Asian markets sustained a stable dependency on European markets, with an uptick during recent years. The profound military and political linkages between South Korea, Japan, and the USA, alongside their capitalist frameworks, have cemented their integration with Western markets. Inter-market connectivity between South Korea and Japan approximately doubled amid the pandemic (SKR-JP, τ = 0.21 in stage 1; τ = 0.42 in stage 2), likely influenced by geographic proximity and pandemic-related lockdowns. Conversely, the dependency between China and other countries transitioned from asymmetric (CN-SKR, τ = 0.37 in stage 1) to symmetric (CN-SKR, τ = 0.24 in stage 2; τ = 0.23 in stage 3), exhibiting a weakening trend in dependence strength.

5.3 Robust Check

We further examine the robustness of our findings by analyzing both static and dynamic tail dependencies within Copula frameworks across stock markets, alongside exploring variations in transmission strength. We compute static and dynamic tail dependence coefficients for ten pairs of interrelated stock indices, with Table 7 presenting the static lower and upper tail dependence coefficients for these pairs. The market pairs are: US&NL, US&UK, RU&NL, RU&US, FR&DE, NL&UK, NL&DE, JP&SKR, SKR&NL, and CN&JP. The selection of these pairs is guided by the following considerations: (1) they exhibit strong or representative dependence relationships in the prior Vine-Copula analysis (as depicted in the first tree structure in Figure 3); (2) they encompass key intra-regional linkages (e.g., FR&DE within Europe, JP&SKR within Asia) and inter-regional linkages (e.g., US&NL between the Americas and Europe, SKR&NL between Asia and Europe); (3) they include market pairs with varying dependence strengths and characteristics to comprehensively examine variations in tail dependence.

European stock markets exhibit a pronounced high level of dependence. Notably, the pair between the German DAX and French CAC indices demonstrates the highest upper (0.7564) and lower (0.8183) tail dependence coefficients, indicating strong mutual influences in both market upswings and downturns. The US demonstrates a moderate dependency on European markets, with minimal difference between its upper and lower tail dependence coefficients, suggesting predominantly symmetric volatility spillovers. Interdependencies between Asian and European markets are less pronounced than those between the USA and Europe. Within Asia, South Korea and Japan exhibit the highest level of co-movement, while China’s market interdependence with other markets is markedly low. Overall, our analysis reveals that the lower tail dependence among stock market pairs generally exceeds the upper tail dependence. This implies that markets are more correlated during downturns (bear markets) than during upturns (bull markets), suggesting that negative news spreads faster, inciting market panic and leading to steeper declines in stock prices.

To model dynamic dependence accurately, we consider a set of candidate copula models – including Normal, Clayton, Frank, Gumbel, t, and SJC Copulas. The kernel density estimation method is used to estimate the marginal distributions, and then, the copula models are fitted to the probability integral transformed data. The optimal dynamic copula model is selected based on the minimum AIC value, which identifies the time-varying SJC Copula as the optimal specification for our analysis, as specified in equations (10)–(12).

Subsequently, we calculate the dynamic upper and lower tail dependence coefficients for the ten pairs of dependent stock indices under study. The results are visualized through line plots, as depicted in Figures 4–7.

Figure 4

The dynamic tail dependence of US stock index returns.

Figure 5

The dynamic tail dependence of Russian stock index returns.

Figure 6

The dynamic tail dependence of European stock index returns.

Figure 7

The dynamic tail dependence of Asian stock index returns.

Figure 4 illustrates the dynamic dependence changes between the stock markets of USA and Europe, revealing that the frequency of fluctuations in lower tail dependence is higher than that in upper tail dependence. A sharp decrease in dependency is observed at the start of the second stage, followed by a gradual recovery.

Figure 5 displays the dynamic dependency fluctuations between the Russian market and other stock markets. At the onset of the Russia–Ukraine War, its dependency on European markets sharply decreased, maintaining levels below the average throughout the third stage. The dependency between Russia and the USA, though initially reduced, swiftly reverted to its historical average level.

Figure 6 demonstrates the dynamic dependency changes within European stock markets. The dependency among these markets remains high, with minimal fluctuation over time. Similarly, the pandemic and the Russia–Ukraine conflict did not appear to significantly disrupt the internal dependency structure within Europe. The Brexit event in 2020 did not significantly impact the dependency between the UK and other European stock markets.

Figure 7 presents the dynamic dependency between Asian stock markets and others. Within Asia, Japan and South Korea exhibit the highest dependency, which significantly increased during the pandemic. China’s dependency on other markets is the lowest and shows a downward trend in recent years. South Korea serves as a link between Asian and European markets, with its dependency not being high but stable, and less affected by major events. These observations align with the results from Vine-copula analysis.