[Paper Review] Statistical inference for large-dimensional tensor factor model by iterative projections

Matteo Barigozzi, Yong He+2 more|arXiv (Cornell University)|Jun 20, 2022

Tensor decomposition and applications5 citations

TL;DR

This paper proposes a novel iterative projection estimator for large-dimensional tensor factor models based on Tucker decomposition, offering a least-squares interpretation analogous to PCA in vector factor models. The method enhances signal-to-noise ratio by simultaneously reducing signal dimensionality and idiosyncratic component magnitude, achieving faster convergence rates than naive PCA estimators under weak cross- and auto-correlation in idiosyncratic errors.

ABSTRACT

Tensor Factor Models (TFM) are appealing dimension reduction tools for high-order large-dimensional tensor time series, and have wide applications in economics, finance and medical imaging. In this paper, we propose a projection estimator for the Tucker-decomposition based TFM, and provide its least-square interpretation which parallels to the least-square interpretation of the Principal Component Analysis (PCA) for the vector factor model. The projection technique simultaneously reduces the dimensionality of the signal component and the magnitudes of the idiosyncratic component tensor, thus leading to an increase of the signal-to-noise ratio. We derive a convergence rate of the projection estimator of the loadings and the common factor tensor which are faster than that of the naive PCA-based estimator. Our results are obtained under mild conditions which allow the idiosyncratic components to be weakly cross- and auto- correlated. We also provide a novel iterative procedure based on the eigenvalue-ratio principle to determine the factor numbers. Extensive numerical studies are conducted to investigate the empirical performance of the proposed projection estimators relative to the state-of-the-art ones.

Motivation & Objective

• Address the lack of statistical inference tools for large-dimensional tensor factor models (TFM) in time series contexts.
• Develop a projection-based estimator that improves signal-to-noise ratio by reducing both signal and idiosyncratic component magnitudes.
• Establish theoretical convergence rates for loadings and common factor tensors under weak dependence in idiosyncratic components.
• Propose a novel iterative eigenvalue-ratio procedure to consistently estimate the number of factors in each mode.
• Provide a least-squares interpretation of the estimator, paralleling PCA in vector factor models.

Proposed method

• Propose a projection estimator for Tucker-decomposed tensor factor models using iterative projections to reduce dimensionality and noise.
• Apply least-squares minimization to the projected tensor data to estimate loadings and common factors, analogous to PCA in vector models.
• Use iterative refinement: at each step, project onto the space spanned by estimated loadings and add one extra component if needed.
• Employ the eigenvalue-ratio principle in an iterative algorithm to determine the number of factors per mode, improving consistency.
• Utilize Weyl’s inequality and matrix perturbation theory to derive convergence rates for eigenvalues of projected sample covariance matrices.
• Establish theoretical bounds under mild dependence assumptions (weak cross- and auto-correlation) on idiosyncratic components.

Experimental results

Research questions

• RQ1 Can a projection-based estimator for Tucker-decomposed tensor factor models achieve faster convergence rates than naive PCA estimators?
• RQ2 How can the signal-to-noise ratio be improved in high-dimensional tensor time series through joint dimensionality reduction of signal and noise?
• RQ3 What is the theoretical convergence rate of the proposed projection estimator for loadings and common factor tensors?
• RQ4 Can an iterative eigenvalue-ratio procedure consistently estimate the number of factors in each mode of a tensor factor model?
• RQ5 Under what conditions does the least-squares interpretation of the projection estimator hold, and how does it parallel PCA in vector factor models?

Key findings

• The proposed projection estimator achieves faster convergence rates for loadings and common factor tensors compared to the naive PCA-based estimator.
• The method improves the signal-to-noise ratio by simultaneously reducing the dimensionality of the signal component and the magnitude of the idiosyncratic component tensor.
• Theoretical convergence rates are established under mild conditions allowing weak cross- and auto-correlation in the idiosyncratic components.
• The iterative eigenvalue-ratio procedure consistently estimates the number of factors per mode, even in high-dimensional settings.
• The estimator admits a least-squares interpretation analogous to PCA in vector factor models, providing a principled statistical foundation.
• Numerical studies confirm superior empirical performance of the projection estimator relative to state-of-the-art methods in finite samples.