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

2025-05-21

AI Summary

This paper addresses the challenge of modeling the dynamic evolution of state density functions in two distinct data structures: repeated cross-sections (e.g., monthly stock return distributions) and high-frequency intra-period time series (e.g., intraday GBP/USD return distributions). We propose the first unified functional dynamic framework compatible with both. Methodologically, we embed density functions into a Hilbert space and formulate a functional autoregressive (FAR) model, integrating kernel density estimation with asymptotic statistical inference. Theoretically, we establish asymptotic theory for density forecasting and distributional moment dynamics, and prove strong consistency of the estimators. Empirically, our approach significantly outperforms conventional methods on GBP/USD and NYSE datasets, delivering high-accuracy density forecasts. The core innovation lies in overcoming data-type barriers—enabling, for the first time, unified modeling and theoretical analysis of distributional evolution across both inter-period cross-sectional and intra-period temporal dimensions.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Kernel Methods
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

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.

Problem

• Analyzing dynamics of state distributions in cross-sectional and intra-period data
• Estimating autoregressive operator for functional elements in Hilbert space
• Forecasting state density and moment dynamics of distributions

Innovation

• Functional autoregressive model for state distributions
• Hilbert space analysis of density dynamics
• Consistent estimator for autoregressive operator