Spectral Dynamics and Regularization for High-Dimensional Copulas

TL;DR

This paper develops a high-dimensional time-varying copula model that combines spectral dynamics with non-linear shrinkage to capture asymmetric, tail-dependent dependence across many assets. By parameterizing the copula via a spectral decomposition of the dependence matrix and updating only a few dominant eigenvalues with score-driven dynamics, the approach remains parsimonious and scalable in large $d$. Regularized shrinkage debiases the eigen-spectrum, improving out-of-sample performance, while a BIC-based selection chooses the number of dynamic eigenvalues. Empirically, the model applied to 100 European stocks across 10 countries and sectors reveals pronounced international co-movements during stress, with the first spectral direction driving systemic risk and reduced diversification, outperforming clustering-based factor copulas in many settings.

Abstract

We introduce a novel model for time-varying, asymmetric, tail-dependent copulas in high dimensions that incorporates both spectral dynamics and regularization. The dynamics of the dependence matrix' eigenvalues are modeled in a score-driven way, while biases in the unconditional eigenvalue spectrum are resolved by non-linear shrinkage. The dynamic parameterization of the copula dependence matrix ensures that it satisfies the appropriate restrictions at all times and for any dimension. The model is parsimonious, computationally efficient, easily scalable to high dimensions, and performs well for both simulated and empirical data. In an empirical application to financial market dynamics using 100 stocks from 10 different countries and 10 different industry sectors, we find that our copula model captures both geographic and industry related co-movements and outperforms recent computationally more intensive clustering-based factor copula alternatives. Both the spectral dynamics and the regularization contribute to the new model's performance. During periods of market stress, we find that the spectral dynamics reveal strong increases in international stock market dependence, which causes reductions in diversification potential and increases in systemic risk.

Figures (7)

• Figure 1: Performance of the skewed $t$ copula with regularized spectral dynamics. The left plot shows changes in the (in-sample) BIC of a model with static eigenvalues versus a model with eigenvalues $1,\ldots,i$ being dynamic, as a function of the spectral index $i$. The BIC is minimal for $i=2$, which is also the true number of dynamic eigenvalues in the dgp. The average result of 10 Monte Carlo simulations is shown. The right-hand plot shows the dynamics of the first eigenvalue for a single MC simulation, where the true dynamics of the dgp is compared with the predicted dynamics from the estimated model. The period after 4 years is an out-of-sample forecast.
• Figure 2: Autocorrelation functions for log-returns $|r_{i,t}|$, devolatilized residuals $|y_{i,t}|$, $|y^{\star}_{i,t}|= |G^{-1}(u_{i,t})|$ and spectral projections $|\tilde{y}^{\star}_{i,t}|$ for $i=1,2,10$. The Gaussian copula ($\gamma=\bm{0}_{d}$ and $\nu^{-1}=0$) is used to determine $y^{\star}_{i,t}$ and $\tilde{y}^{\star}_{i,t}$.
• Figure 3: Spectral dynamics of the first spectral eigenvalue, where the ratio of the first eigenvalue and the sum of the higher eigenvalues is plotted over time. Under normal market circumstances, the first eigenvalue explains about 30% of the spectral variance, leading to a ratio below 1/2. When the ratio increases, it signals enhanced spectral variance in the parallel direction (first eigenvector) compared to the total spectral variance in the orthogonal directions, indicating less diversification and higher systemic risk or return. The three most pronounced events are given an economic interpretation. The period from January 2020 to December 2024 represents an out-of-sample forecast.
• Figure 4: Heatmap of estimated eigenvectors weights $\hat{w}_{i,j}$ of the unconditional copula correlation matrix $\hat{\bm{R}}$ from Eq. \ref{['eq:target omega2']}. The first two eigenvectors are shown ($j=1,2$). The stocks $i$ are ordered in terms of countries along the $x$-axis and in terms of sectors along the $y$-axis. The white color indicates that the country-sector combination is absent in the data set. In case of two stocks per country-sector combination, the result for the first stock from Table \ref{['tabIndices100']} is shown. The first eigenvector has positive weights for all stocks, corresponding to a parallel market movement. We compare the second eigenvector with optimal cluster assigments using Oh2023 giving 17 clusters, indicated by the numbers in each sector-country cell. The colors relate to the value of the corresponding element of the eigenvectors in the spectral copula specification. The bottom panel contains the same information as the right-hand panel, but the stocks are ordered per cluster (in order of their average value of the second eigenvector elements).
• Figure 5: Estimated eigenvalues excluding and including shrinkage, $\hat{\lambda}_{i}$ and $\hat{\mu}_{i}$, of the unconditional copula correlation matrix for the dynamic skew $t$ copula. The sample eigenvalues are ordered from high to low. The left plot shows the first 10 eigenvalues and the middle plot the other 90 eigenvalues. The sample eigenvalues are shown in blue, while the eigenvalues after applying shrinkage are shown in red. The right plot shows the out-of-sample log-likelihood improvement when the first $i$ sample eigenvalues are replaced with shrinkage eigenvalues. The largest improvements stem from the highest spectral indices, whose sample eigenvalues are most severely biased.

Theorems & Definitions (1)

• Proposition 1: score equations for spectral dynamics