Operational time and in-sample density forecasting

Abstract

In this paper, we consider a new structural model for in-sample density forecasting. In-sample density forecasting is to estimate a structured density on a region where data are observed and then reuse the estimated structured density on some region where data are not observed. Our structural assumption is that the density is a product of one-dimensional functions with one function sitting on the scale of a transformed space of observations. The transformation involves another unknown one-dimensional function, so that our model is formulated via a known smooth function of three underlying unknown one-dimensional functions. We present an innovative way of estimating the one-dimensional functions and show that all the estimators of the three components achieve the optimal one-dimensional rate of convergence. We illustrate how one can use our approach by analyzing a real dataset, and also verify the tractable finite sample performance of the method via a simulation study.