# Humboldt-Universität zu Berlin - High Dimensional Nonstationary Time Series

B3 - Copulae

## B3 - Copulae

Understanding the joint distribution of high dimensional data is fundamental to applied statistics. The conventional procedure in multivariate data analysis is to approximate them with normal distributions. That implies, however, that the dependence structure is reduced to a fixed type and that univariate marginal distributions are also normal. Predetermining a multivariate normal distribution means that the tails of the distribution are not too heavy, the distribution is symmetric and the dependence between variables are linear. Empirical evidence for these assumptions is questionable and alternative models, with more flexible dependence structure and arbitrary marginal distributions being in demand. Copulae are very useful for modelling and estimating multivariate distributions, their flexibility basically follows on from Sklar's Theorem (Sklar, 1959), which says that each joint distribution can be decomposed into its marginal distributions and a copulae responsible for the dependence structure. The copulae approach gives more freedom than the normality assumptions, marginal distributions with asymmetric heavy tails (typical for e.g. financial returns) can be combined with different dependence structures, resulting in multivariate distributions (far different from the multivariate normal) that describe the empirical characteristics of high dimensional data in a more satisfactory manner. A list of flexible copulae has been developed recently and the most flexible seems to be the so called hierarchical Archimedean copulae (HAC), see Okhrin, Okhrin, Schmid (2010a, 2010b). Unfortunately copulae theory today is not so well advanced for modelling time series as it could be and only a few papers have been published or submitted to date, e.g. Lee and Long (2009), Patton (2006), Härdle, Okhrin, Okhrin (2010c), Giacomini, Härdle, Spokoiny (2009). One of the research aims of our group is based on developing fully flexible high-dimensional time series models based in particular on HAC. Some preliminary work has been done in extending the DCC-GARCH model to the case of copulae based residuals, and even in its simplest case we outperform classical models. As mentioned above, the modelling of high-frequency data recently became extremely important, and a new copulae based model for high-frequency data will also be proposed and developed by PhD students in our team. The use of these modern techniques allows us a better understanding of market behaviour.

Coordination

• Ostap Okhrin: Ostap Okhrin’s main research interests lie in developing and studying methods of construction multivariate copulae. Recent research include time series analysis based on copulae (GARCH type models, adaptive estimation), asymptotic theory of the estimators, and application to different fields like: collateralised debt obligations, crop insurance, and portfolio management.

• Yuanyuan Lin: Her interests are Probability and Statistics, Survival data analysis, Statistical machine learning. Her research includes work on least absolute relative error estimation and large sample statistical analysis.

• Linlin Niu: Her interests are Macro Finance, International Economics, Applied Econometrics.

• Wolfgang Härdle: His main interests are non- and semiparametrics statistics and econometrics. His research includes work in nonparametric modelling, local adaptive models, reduction techniques, stationary models, quantile regression.

• Zongwu Cai: His main interests are Econometrics, Quantitative Finance, Nonlinear Time Series. His research includes work in Theoretical and applied econometrics, quantitative finance and risk management, nonparametric curve estimation problems, nonlinear and non-stationary time series, panel data analysis.

• Ying Fang: His main interests are Econometrics, Applied Econometrics, Economy of China. His research includes work in nonparametric and semi-parametric method, panel data analysis, and instrumental variable selection.

• Ming Lin: His main interests are Monte Carlo Methods, self selection, dimension reduction methods. His research includes work in Monte Carlo algorithm, Bayesian statistics, nonparametric statistics.

• Nikolaus Hautsch: His main interests are in Econometric models and Empirical Finance. His recent research concentrates on linear and nonlinear time series models, latent factor models, econometrics for financial transaction data, market microstructure analysis, information processing and analysis of limit order book markets.