Singularity-Based Consistent QML Estimation of Multiple Breakpoints in High-Dimensional Factor Models

2025-03-09

AI Summary

This paper addresses the estimation problem in high-dimensional factor models when factor loadings exhibit multiple unknown structural breaks, particularly focusing on cases where breaks induce an expansion of the factor space—rendering the pseudo-factor covariance matrix singular. We propose a framework that reduces high-dimensional break detection to a low-dimensional problem and, for the first time, systematically distinguish and handle two distinct break types: “singular breaks” (which alter the number of factors) and “rotation breaks” (which only affect the orthogonal representation of loadings). We construct an information criterion that consistently estimates the number of breaks, relaxing the restrictive single-break assumption. Leveraging quasi-maximum likelihood (QML) estimation and asymptotic analysis of small eigenvalues, we establish consistency and stability of the estimators under both break types. Monte Carlo simulations confirm excellent finite-sample performance. An empirical application to the FRED-MD dataset (1959–2024) successfully identifies five genuine loading breaks.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

This paper investigates the estimation of high-dimensional factor models in which factor loadings undergo an unknown number of structural changes over time. Given that a model with multiple changes in factor loadings can be observationally indistinguishable from one with constant loadings but varying factor variances, this reduces the high-dimensional structural change problem to a lower-dimensional one. Due to the presence of multiple breakpoints, the factor space may expand, potentially causing the pseudo factor covariance matrix within some regimes to be singular. We define two types of breakpoints: {f a singular change}, where the number of factors in the combined regime exceeds the minimum number of factors in the two separate regimes, and {f a rotational change}, where the number of factors in the combined regime equals that in each separate regime. Under a singular change, we derive the properties of the small eigenvalues and establish the consistency of the QML estimators. Under a rotational change, unlike in the single-breakpoint case, the pseudo factor covariance matrix within each regime can be either full rank or singular, yet the QML estimation error for the breakpoints remains stably bounded. We further propose an information criterion (IC) to estimate the number of breakpoints and show that, with probability approaching one, it accurately identifies the true number of structural changes. Monte Carlo simulations confirm strong finite-sample performance. Finally, we apply our method to the FRED-MD dataset, identifying five structural breaks in factor loadings between 1959 and 2024.

Problem

• Estimating multiple structural breaks in high-dimensional factor models
• Distinguishing between singular and rotational changes in factor loadings
• Proposing an information criterion to identify true breakpoint number

Innovation

• Defines singular and rotational breakpoint types
• Proposes information criterion for breakpoint number estimation
• Uses QML estimators with stable error bounds