[Paper Review] The eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility model

Anja Janßen, Thomas Mikosch+2 more|arXiv (Cornell University)|May 9, 2016

Financial Risk and Volatility Modeling35 references3 citations

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues and eigenvectors of the sample covariance matrix in a multivariate heavy-tailed stochastic volatility model. Using large deviations for regularly varying time series, it derives multivariate α-stable limit distributions, showing that heavy-tailed innovations lead to independent-like limits, while heavy-tailed volatility sequences induce dependent limiting structures determined by the volatility's dependence structure.

ABSTRACT

We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-sample behavior of its sample covariance matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $α$-stable limit distributions of the sample covariance matrix. While we show that in the case of heavy-tailed innovations the limiting behavior resembles that of completely independent observations, we also derive that in the case of a heavy-tailed volatility sequence the possible limiting behavior is more diverse, i.e. allowing for dependencies in the limiting distributions which are determined by the structure of the underlying volatility sequence.

Motivation & Objective

• To understand the limiting behavior of eigenvalues and eigenvectors of the sample covariance matrix in a multivariate heavy-tailed stochastic volatility model.
• To distinguish between two cases: heavy-tailed innovations versus heavy-tailed volatility sequences.
• To derive multivariate α-stable limit distributions for the sample covariance matrix under infinite variance conditions.
• To examine how dependence in the volatility sequence affects the limiting distributional structure of eigenvalues and eigenvectors.

Proposed method

• Application of large deviations techniques for regularly varying time series to analyze tail behavior in multivariate stochastic processes.
• Derivation of multivariate α-stable limit distributions for the sample covariance matrix under infinite variance assumptions.
• Separate analysis of two distinct cases: when i.i.d. innovations are regularly varying and when the volatility sequence is regularly varying.
• Use of spectral analysis to study ordered eigenvalues and corresponding eigenvectors in the limit.
• Explicit characterization of limiting distributions depending on the source of heavy tails—innovations vs. volatility.
• Leveraging properties of regularly varying processes to establish convergence in distribution to α-stable limits.

Experimental results

Research questions

• RQ1 How do the eigenvalues of the sample covariance matrix behave asymptotically in a multivariate heavy-tailed stochastic volatility model?
• RQ2 What is the limiting distribution of the sample covariance matrix when the innovations are regularly varying?
• RQ3 How does the limiting behavior differ when the volatility sequence is regularly varying rather than the innovations?
• RQ4 To what extent do dependencies in the volatility sequence affect the limiting distribution of eigenvalues and eigenvectors?
• RQ5 Can the limiting distribution of the sample covariance matrix be characterized as multivariate α-stable under infinite variance conditions?

Key findings

• When the innovations are heavy-tailed and i.i.d., the limiting behavior of the sample covariance matrix resembles that of independent observations, with eigenvalues converging to a multivariate α-stable distribution.
• In the case of heavy-tailed volatility sequences, the limiting distribution exhibits dependence structures that reflect the temporal dependence in the volatility sequence.
• The limiting eigenvalues and eigenvectors converge in distribution to a multivariate α-stable distribution, with the dependence structure determined by the volatility's serial dependence.
• The large deviations technique for regularly varying time series enables the derivation of these α-stable limits even in the presence of long-range dependence.
• The results show that the source of heavy tails—innovations vs. volatility—leads to fundamentally different limiting behaviors in the spectral properties of the sample covariance matrix.
• The analysis confirms that the limiting distribution is not degenerate and retains non-trivial dependence when the volatility sequence is regularly varying.