[Paper Review] Estimation and Inference for CP Tensor Factor Models

Bin Chen, Yuefeng Han+1 more|arXiv (Cornell University)|Jun 25, 2024

NMR spectroscopy and applications4 citations

TL;DR

This paper proposes a novel iterative simultaneous projection estimation method for high-dimensional tensor factor models using CP decomposition, enabling consistent estimation and asymptotic normality under weak dependence and weak cross-dimension correlation. The method outperforms existing approaches in simulations and empirical applications, with consistent estimation of the number of factors via eigenvalue ratio-based estimators.

ABSTRACT

High-dimensional tensor-valued data have recently gained attention from researchers in economics and finance. We consider the estimation and inference of high-dimensional tensor factor models, where each dimension of the tensor diverges. Our focus is on a factor model that admits CP-type tensor decomposition, which allows for non-orthogonal loading vectors. Based on the contemporary covariance matrix, we propose an iterative simultaneous projection estimation method. Our estimator is robust to weak dependence among factors and weak correlation across different dimensions in the idiosyncratic shocks. We establish an inferential theory, demonstrating both consistency and asymptotic normality under relaxed assumptions. Within a unified framework, we consider two eigenvalue ratio-based estimators for the number of factors in a tensor factor model and justify their consistency. Simulation studies confirm the theoretical results and an empirical application to sorted portfolios reveals three important factors: a market factor, a long-short factor, and a volatility factor.

Motivation & Objective

• To develop a robust estimation and inference framework for high-dimensional tensor factor models with diverging tensor dimensions.
• To address limitations of autocovariance-based methods by using contemporary covariance matrices, especially under weak or no serial correlation in factors.
• To establish asymptotic theory—consistency, convergence rates, and asymptotic normality—under relaxed assumptions.
• To extend eigenvalue ratio-based estimators for factor number selection to tensor factor models and prove their consistency.
• To demonstrate superior finite-sample performance through simulations and real-world applications in finance and international trade.

Proposed method

• Proposes an iterative simultaneous projection estimation method based on the contemporary covariance matrix, avoiding reliance on autocovariance structures.
• Uses a generalized eigenvalue ratio approach to estimate the number of latent factors, extending the method of Ahn and Horenstein (2013) to tensor data.
• Employs a CP decomposition with non-orthogonal loading vectors to model the low-rank structure of tensor data.
• Applies a projection-based iterative algorithm to estimate loading vectors, ensuring consistency and asymptotic normality.
• Derives convergence rates and limiting distributions under weak dependence and weak cross-dimension correlation in idiosyncratic shocks.
• Validates the method through simulation studies and two empirical applications: sorted portfolios and international trade flows.

Experimental results

Research questions

• RQ1 Can a contemporary covariance-based estimation method achieve consistency and asymptotic normality in high-dimensional tensor factor models with diverging dimensions?
• RQ2 How does the proposed iterative projection method perform under weak dependence among factors and weak cross-dimension correlation in idiosyncratic errors?
• RQ3 Can eigenvalue ratio-based estimators consistently estimate the number of factors in a tensor factor model?
• RQ4 Does the proposed method outperform existing autocovariance-based or PCA-based approaches in finite samples?
• RQ5 What is the theoretical justification for the asymptotic normality of the estimated loading vectors under relaxed moment and dependence assumptions?

Key findings

• The proposed iterative projection estimator achieves consistency and asymptotic normality under relaxed assumptions, including weak dependence and weak cross-dimension correlation.
• The eigenvalue ratio-based estimators for the number of factors are consistent under the proposed framework, extending classical results to tensor data.
• Simulation results show that the proposed method outperforms AC-ISO and TPCA in estimation accuracy, especially under moderate to strong serial correlation in factor processes.
• Empirical applications on sorted portfolios and international trade flows demonstrate the method’s practical advantages in real-world settings.
• Theoretical analysis confirms that the leading term in the estimation error of loading vectors depends on the signal strength and factor structure, with central limit theorem justification under appropriate scaling.
• The method remains robust even when autocovariance-based methods fail due to weak or zero serial correlation in factors.