This paper describes a forecasting exercise of close-to-open returns on major global stock indices, based on high-frequency price patterns that have become available in foreign markets overnight. Generally speaking, out-of-sample forecast performance depends on the forecast method as well as the information that the forecasts are based on. In this paper both aspects are considered. The fact that the close-to-open gap is a scalar response variable to a functional variable, namely an overnight foreign price pattern, brings the prediction exercise in the realm of functional data analysis. Both parametric and non-parametric functional data analysis are considered, and compared with a simple linear benchmark model. The information set is varied by dividing global markets into three clusters, Asia-Pacific, Europe and North-America, and including or excluding price patterns on a per-cluster basis. The overall best performing forecast is nonparametric using all available information, suggesting the presence of nonlinear relations between the overnight price patterns and the opening gaps.

In economics we often face a system which intrinsically imposes a structure of hierarchy of its components, i.e., in modeling trade accounts related to foreign exchange or in optimization of regional air protection policy. A problem of reconciliation of forecasts obtained on different levels of hierarchy has been addressed in the statistical and econometric literature many times and concerns bringing together forecasts obtained independently at different levels of hierarchy. This paper deals with this issue with regard to a hierarchical functional time series. We present and critically discuss a state of art and indicate opportunities of an application of these methods to a certain environment protection problem. We critically compare the best predictor known from the literature with our own original proposal. Within the paper we study a macromodel describing the day and night air pollution in Silesia region divided into five subregions.