Impact of non-stationarity on estimating and modeling empirical copulas of daily stock returns

All too often measuring statistical dependencies between financial time series is reduced to 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 particularly to which extent the non-stationarity of financial time series affects both the estimation and the modeling of empirical copulas. We estimate empirical copulas from the non-stationary, original return time series and stationary, locally normalized ones. Thereby we are able to explore the empirical dependence structure on two different scales: a global and a local one. 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 non-stationarity of dependencies as a model parameter, and with the skewed Student's t-copula. The K-copula covers the empirical dependence structure on the local scale most adequately, whereas the skewed Student's t-copula best captures the asymmetry of the empirical copula on the global scale.