Tensor PCA for Factor Models

TL;DR

This paper introduces tensor principal component analysis for high-dimensional factor models, providing optimal estimation, improved convergence for weak factors, and new inferential tools, with applications to missing data imputation.

Contribution

It develops a tensor PCA framework for factor models, including optimal estimation, iterative improvements, and a novel test for determining the number of factors.

Findings

Tensor PCA is optimal for strong factor models.

Alternating least-squares improves convergence for weak factors.

New test effectively determines the number of factors.

Abstract

Modern empirical analysis often relies on high-dimensional panel datasets with non-negligible cross-sectional and time-series correlations. Factor models are natural for capturing such dependencies. A tensor factor model describes the $d$-dimensional panel as a sum of a reduced rank component and an idiosyncratic noise, generalizing traditional factor models for two-dimensional panels. We consider a tensor factor model corresponding to the notion of a reduced multilinear rank of a tensor. We show that for a strong factor model, a simple tensor principal component analysis algorithm is optimal for estimating factors and loadings. When the factors are weak, the convergence rate of simple TPCA can be improved with alternating least-squares iterations. We also provide inferential results for factors and loadings and propose the first test to select the number of factors. The new tools are applied to the problem of imputing missing values in a multidimensional panel of firm characteristics.