Asymmetric dependence patterns in financial time series

Abstract

This article proposes a new copula-based approach to test for asymmetries in the dependence structure of financial time series. Simply splitting observations into subsamples and comparing conditional correlations lead to spurious results due to the well-known conditioning bias. Our suggested framework is able to circumvent these problems. Applying our test to market data, we statistically confirm the widespread notion of significant asymmetric dependence structures between daily changes of the VIX, VXN, VDAXnew, and VSTOXX volatility indices and their corresponding equity index returns. A maximum likelihood method is used to perform a likelihood ratio test between the ordinary t-copula and its asymmetric extension. To the best of our knowledge, our study is the first empirical implementation of the skewed t-copula to generate meta-skewed Student's t-distributions. Its asymmetry leads to significant improvements in the description of the dependence structure between equity returns and implied volatility changes.

Keywords:

• copulae
• asymmetric dependence concepts

JEL Classification :

• C01

Acknowledgements

We thank the participants of the ‘Conference on Copulae and Multi-Variate Probability Distributions in Finance’ in Warwick, UK, and our anonymous referees for very helpful suggestions and comments.

Notes

See, e.g. Aas and Haff (2006).

See, e.g. Embrechts, McNeil, and Straumann (2002).

See, e.g. Koutmos and Booth (1995), Fortin and Kusmics (2002), and the references therein.

The likelihood is called ‘pseudo’ as it is conditioned on the time series models of the univariate marginal distributions. Genest, Ghoudi, and Rivest (1995) prove the consistency and asymptotic normality of the copula parameter estimators if the data sample is iid. Also see, among others, Demarta and McNeil (2005).

Values of the new VIX have been backfilled to 2 January 1990. As its calculation is only based on observable values, it can be determined in an unbiased manner. For its theoretic construction, see e.g. Carr and Madan (1998), Demeterfi et al. (1999), and Britten-Jones and Neuberger (2000).

Using a similar calculation method to the VIX, values of the VDAXnew have been backfilled to 2 January 1992.

See, e.g. Fleming, Ostdiek, and Whaley (1995), Moraux, Navatte, and Villa (1998), Whaley (2000), Blair, Poon, and Taylor (2001), Corrado and Miller (2005), as well as Simon (2003). This list is by no means exhaustive.

A different method would be to use changes in the logarithm of the closing level. This method leads to conclusions that are very similar to our results.

See, e.g. Demarta and McNeil (2005).

The variable X has the law of an inverse Gamma distribution if it follows the density function

• where x>0, α>0, and β>0.

In the case of the skewed t-copula, the EM algorithm by Demarta and McNeil (2005), ν was also determined by numerically solving

• This procedure requires the calculation of
  and , which depend upon the specification of . In order to avoid this dependence, we decide to directly determine ν by a gradient method, although this requires more computational effort.