Modifying copulas for improved dependence modelling

Abstract

Copulas allow a joint probability distribution to be decomposed such that the marginals inform us about how the data were generated, separately from the copula which fully captures the dependency structure between the variables. This is particularly useful when working with random variables which are both non-normal and possibly non-linearly correlated. However, when in addition, the dependence between these variables change in accordance with some underlying covariate, the model becomes significantly more complex. This research proposes using a Gaussian process conditional copula for this dependence modelling, focusing on time as the underlying covariate. Utilising a Bayesian non-parametric framework allows the simplifying assumptions often applied in conditional dependency computation to be relaxed, giving rise to a more flexible model. The importance of improving the accuracy of dependency modelling in applications such as finance, econometrics, insurance and meteorology is self-evident, considering the potential risks involved in erroneous estimation and prediction results. Including the underlying (conditional) variable reduces the chances of spurious dependence modelling. For our application, we include a textbook example on a simulated dataset, an analysis of the modelling performance of the different methods on four currency pairs from foreign exchange time series and lastly we investigate using copulas as a way to quantify the coupling efficiency between the solar wind and magnetosphere for the three known phases of geomagnetic storms. We find that the Student’s t Gaussian process conditional copula outperforms static copulas in terms of log-likelihood, and performs particularly well in capturing lower tail dependence. It further gives additional information about the temporal movement of the coupling between the two main variables, and shows potential for more accurate data imputation.