Vine copula base modelling and forecasting of multivariate realized volatility time-series

可靠的股票市场波动预测是投资组合管理和风险评估的必要条件。由于高频数据的可用性越来越高，使用平方收益来估计事后实现波动率（RV）已成为经验金融中的标准方法之一。在该项目中，我们将解决已实现方差的单变量时间序列和已实现协方差矩阵的矩阵变量时间序列的建模方法。目前最流行的单变量模型是HAR回归。模型的表现是令人信服的，但模型中解释因素的选择（以每日频率测量）仅仅是启发式的。在项目的第一部分，我们解决了非线性建模的时间依赖性的因素。在项目的第二阶段，我们将专注于统计方法，如主成分分析、因子分析和神经网络，这将帮助我们确定用于因子构建的历史数据的最佳聚合。如果我们考虑最初的日内信息，同样的问题仍然存在。在最简单的情况下，使用日内收益的平方和来估计已实现的波动率。解决日内收益的适当汇总和转换问题也很重要。在项目的第一阶段，我们提倡的实现协方差矩阵的方法是基于偏相关的。它显示出良好的表现，但我们希望进一步改善它的表现，并确保更可靠的预测。为此，我们想研究一种新的选择方法，即根据与单变量偏相关时间序列相关的边际模型的预测性能，选择标准相关和偏相关的子集，形成偏相关藤。我们期望这种方法能够提高预测性能。到目前为止，我们还没有考虑到部分相关藤的模型稀疏性。一般情况下，藤联结模型的模型稀疏性可以通过将某些对联结设置为独立联结来实现。这对应于偏相关曲线中的偏相关值为零。所谓截尾藤，是指将一定高度以上的树木的所有对交轴都设置为独立交轴。我们将研究设置截断水平的几种选择。

Reliable forecasts of stock market volatility are necessary in portfolio management and the evaluation of risks. Due to the increasing availability of high-frequency data, the use of squared returns to estimate the ex-post realized volatility (RV) has become one of the standard methods in empirical finance.Within the project we will tackle both the modeling approaches for univariate times series of realized variances and for matrix variate time series of realized covariance matrices. The currently most popular univariate model for realized volatility modeling is the HAR regression. The performance of the model is convincing, but the choice of the explanatory factors (measured at daily frequency) in the model is merely heuristic. Within the first part of the project we addressed the nonlinear modeling of temporal dependence of the factors. In the second phase of the project we will concentrate on statistical approaches such as principle component analysis, factor analysis and neural networks, which will help us to determine the optimal aggregation of historical data for factor building. Similar problem remains if we consider the initial intraday information. The realized volatilities are estimated in the simplest case using the sum of squared intra-day returns. It is of key interest to address the proper aggregation and transformation of intra-day returns too.The method we advocated for the realized covariance matrices in the first phase of the project was based on partial correlations. It shows good performance, but we would like to improve further its performance and ensure more robust forecasts. For this we want to investigate a new selection method which chooses the subset of standard and partial correlations, which form the partial correlation vine, based on the forecasting performance of the marginal models associated to the univariate partial correlation time-series. We expect that this way of proceeding will lead to an improved forecasting performance. So far we have not considered model sparsity of the partial correlation vines. In general, model sparsity in a vine copula model can be achieved by setting certain pair-copulas to the independence copula. This corresponds to a partial correlation value of zero in a partial correlation vine. So-called truncated vines set all pair-copulas of trees above a certain level to the independence copula. We will study several choices of setting this truncation level.