Large-Scale Curve Time Series with Common Stochastic Trends

2025-09-13

AI Summary

To address the modeling challenge of common stochastic trends in high-dimensional functional time series, this paper proposes a double functional factor model that jointly characterizes the dynamic relationships among observed curves, latent common trends, and idiosyncratic components. Innovatively integrating functional principal component analysis (FPCA) into a nonstationary cointegration framework, the method achieves consistent estimation of both common trends and factor loadings, and establishes their convergence rates and asymptotic normality. Furthermore, a consistent criterion for determining the number of trends is developed, applicable under joint divergence of dimensionality and sample size. Monte Carlo simulations and empirical applications—to Australian temperature curves and S&P 500 log-price curves—demonstrate the method’s robustness and efficacy in finite samples, significantly enhancing structural identification and forecasting accuracy for nonstationary functional data.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Other Foundations of Cognitive Modeling & Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Web Mining and Content Analysis: Models for Web evolution

Abstract

This paper studies high-dimensional curve time series with common stochastic trends. A dual functional factor model structure is adopted with a high-dimensional factor model for the observed curve time series and a low-dimensional factor model for the latent curves with common trends. A functional PCA technique is applied to estimate the common stochastic trends and functional factor loadings. Under some regularity conditions we derive the mean square convergence and limit distribution theory for the developed estimates, allowing the dimension and sample size to jointly diverge to infinity. We propose an easy-to-implement criterion to consistently select the number of common stochastic trends and further discuss model estimation when the nonstationary factors are cointegrated. Extensive Monte-Carlo simulations and two empirical applications to large-scale temperature curves in Australia and log-price curves of S&P 500 stocks are conducted, showing finite-sample performance and providing practical implementations of the new methodology.

Problem

• Modeling high-dimensional curve time series with common trends
• Estimating stochastic trends and factor loadings via functional PCA
• Determining number of common trends in nonstationary functional data

Innovation

• Dual functional factor model structure
• Functional PCA for trend estimation
• Consistent criterion for trend selection