Robust estimation of time-varying moments, mutual information, and transfer entropy by means of quantile regression based density forecasts

我们提出了一种方法，使得揭示多变量时间序列和横截面数据的非线性结构和特征变得更容易。我们将基于Baur和Dimpfl(《金融计量学杂志》，即将出版)中估计收益的条件方差的方法，使用分位数回归来估计条件矩和无条件矩、互信息、传递熵以及进一步的基于熵的度量。该过程基于多变量联合密度(计算互信息和传递熵所需的)分解成可通过分位数回归建模的条件和无条件密度函数。我们期望我们提出的方法只需要很少的假设，相对容易实现和计算，不需要大量的数据，并且由于分位数回归的半参数性质，并且基于现有关于分位数估计的渐近理论的文献，允许对单个测量进行一致的解释。此外，到目前为止，对于连续时间序列的传递熵的估计是基于数据的入库，以便能够计算不同入库的相对频率。当使用基于分位数回归的密度预测时，入库变得过时。然而，新的问题出现了，比如这样得到的密度预测是否应该被平滑，或者在数值积分中应该使用多少个锚点。评价该方法论的优缺点是本研究计划的核心。

We propose a methodology which makes uncovering non-linear structures and characteristics of multivariate time series and cross-sectional data easier. We will estimate conditional and unconditional moments, mutual information, transfer entropy, and further entropy-based measures using quantile regression, based on the methodology to estimate the conditional variance of returns in Baur and Dimpfl (Journal of Financial Econometrics, forthcoming). The procedure is based on the decomposition of multivariate joint densities (which are needed to calculate mutual information and transfer entropy) into conditional and unconditional density functions which can be modelled via quantile regression. We expect that our proposed methodology only needs little assumptions, is relatively easy to implement and compute, does not require a huge amount of data, and allows for a consistent Interpretation of the individual measures due to the semp-parametric character of a quantile regression and based on the existing literature which deals with asymptotic theory of quantile estimators. Furthermore, the estimation of transfer entropy for continuous time series is up to now based on binning of the data in order to be able to calculate relative frequencies for the different bins. When using density forecasts based on a quantile regression, the binning becomes obsolete. However, new problems arise like the question whether the so obtained density forecasts should be smoothed or how many anchor points should be used in numerical integration. To evaluate the advantages and disadvantages of the methodology is the core of the proposed research project.