Analysis of Distributional Dynamics for Repeated Cross-Sectional and Intra-Period Observations

Abstract: This paper introduces a novel approach to investigate the dynamics of state distributions, which accommodate both cross-sectional distributions of repeated panels and intra-period distributions of a time series observed at high frequency. In our approach, densities of the state distributions are regarded as functional elements in a Hilbert space, and are assumed to follow a functional autoregressive model. We propose an estimator for the autoregressive operator, establish its consistency, and provide tools and asymptotics to analyze the forecast of state density and the moment dynamics of state distributions. We apply our methodology to study the time series of distributions of the GBP/USD exchange rate intra-month returns and the time series of cross-sectional distributions of the NYSE stocks monthly returns. Finally, we conduct simulations to evaluate the density forecasts based on our model.