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，使发现非线性结构和特征的多变量时间序列和横截面数据更容易。我们将使用分位数回归估计条件和无条件矩、互信息、传递熵以及进一步的基于熵的度量，基于Baur和Dimpfl（Journal of Financial Econometrics，即将出版）中估计收益率条件方差的方法。该过程是基于多变量联合密度（这是需要计算互信息和传递熵）分解成条件和无条件的密度函数，可以通过分位数回归建模。我们希望我们提出的方法只需要很少的假设，是相对容易实现和计算，不需要大量的数据，并允许一致的解释的个别措施，由于半参数性质的分位数回归和现有的文献，其中涉及渐近理论的分位数估计。此外，到目前为止，连续时间序列的传递熵的估计是基于数据的分箱，以便能够计算不同箱的相对频率。当使用基于分位数回归的密度预测时，分组变得过时。然而，出现了新的问题，如这样获得的密度预测是否应平滑或有多少锚点应用于数值积分的问题。评价该方法的优缺点是本研究项目的核心。

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.