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

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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.