Journal of Risk

Impact of nonstationarity on estimating and modeling empirical copulas of daily stock returns

Marcel Wollschläger and Rudi Schäfer

  • Daily stock returns exhibit asymmetric tail dependence.
  • Removing time-varying drifts and volatilities yields dependence on a local scale.
  • The K-copula captures the empirical dependence structure on the local scale.
  • Asymmetry is stronger on the global scale, captured by skewed Student t-copula.


All too often, measuring statistical dependencies between financial time series is reduced to the study of a linear correlation coefficient. However, this may not capture all facets of reality. We study empirical dependencies of daily stock returns by their pairwise copulas. Here, we investigate in particular the extent to which the nonstationarity of financial time series affects both the estimation and the modeling of empirical copulas. We estimate empirical copulas from the nonstationary, original return time series and stationary, locally normalized ones, and we are thereby able to explore the empirical dependence structure on two different scales: globally and locally. Additionally, the asymmetry of the empirical copulas is emphasized as a fundamental characteristic. We compare our empirical findings with a single Gaussian copula, with a correlation-weighted average of Gaussian copulas, with the K-copula directly addressing the nonstationarity of dependencies as a model parameter and with the skewed Student t-copula. The K-copula covers the empirical dependence structure on the local scale most adequately, whereas the skewed Student t-copula best captures the asymmetry of the empirical copula found on the global scale.

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