Forecasting with Interval and Histogram Data. Some Financial Applications

Data sets across many disciplines are becoming consistently large and they bring with them the need of new methods for processing information. We introduce the analysis of interval-valued and histogram-valued data sets as an alternative to classic single-valued data sets and we show the promise of this approach on dealing with economic and …nancial data. Being our current focus the prediction problem we explore two di¤erent venues to produce a forecast with interval time series (ITS) and histogram time series (HTS). For ITS, we adapt classical regression methods and time series strategies for model building and prediction. For ITS and HTS, we implement …ltering techniques, such as smoothing, and nonparametric methods such as the k-NN. We need interval arithmetic in ITS and the concept of a barycentric histogram in HTS to compute the appropiate averages required by smoothing and k-NN. The assessment of the forecast error also requires the introduction of dissimilarity measures like a kernel based distance for ITS and the Wasserstein and Mallows distances for HTS. We apply the proposed methods to predict the daily interval-valued dispersion for the level of SP500 index and the weekly cross-sectional histogram of the returns to the constituents of the SP500 index. Overall, k-NN methods perform very well. Key Words: Interval-valued data, histogram-valued data, interval arithmetic, dissimilarity measures, exponential smoothing, k-NN, Wasserstein distance, Mallows distance. JEL Classi…cation: C22, C53