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On Testing Kronecker Product Structure in Tensor Factor Models

Zetai Cen, Clifford Lam

TL;DR

The paper develops a residual-based test for Kronecker product structure in tensor factor models with Tucker decomposition, linking Kron structure to Tucker representations via tensor reshaping. By comparing residuals from full Tucker TFMs with those from reshaped-factor models and grounding the approach in asymptotic normality, the authors derive a practical, robust testing procedure that accommodates weak factors and unknown mode sets. Theoretical results are complemented by simulations and real-data applications (NYC taxi flows and Fama-French portfolios), demonstrating that the method can detect departures from Kronecker structure and guide dimension reduction. This work offers a principled diagnostic for validating low-rank Kron structures in high-dimensional tensor time series, with implications for improved estimation and inference when such structure holds.

Abstract

We propose a test for testing the Kronecker product structure of a factor loading matrix implied by a tensor factor model with Tucker decomposition in the common component. Through defining a Kronecker product structure set, we define if a tensor time series response $\{\mathcal{Y}_t\}$ has a Kronecker product structure, equivalent to the ability to decompose $\{\mathcal{Y}_t\}$ according to a tensor factor model. Our test is built on analysing and comparing the residuals from fitting a full tensor factor model, and the residuals from fitting a (tensor) factor model on a reshaped version of the data. In the most extreme case, the reshaping is the vectorisation of the tensor data, and the factor loading matrix in such a case can be general if there is no Kronecker product structure present. Theoretical results are developed through asymptotic normality results on estimated residuals. Numerical experiments suggest that the size of the tests gets closer to the pre-set nominal value as the sample size or the order of the tensor gets larger, while the power increases with mode dimensions and the number of combined modes. We demonstrate out tests through a NYC taxi traffic data and a Fama-French matrix portfolio of returns.

On Testing Kronecker Product Structure in Tensor Factor Models

TL;DR

The paper develops a residual-based test for Kronecker product structure in tensor factor models with Tucker decomposition, linking Kron structure to Tucker representations via tensor reshaping. By comparing residuals from full Tucker TFMs with those from reshaped-factor models and grounding the approach in asymptotic normality, the authors derive a practical, robust testing procedure that accommodates weak factors and unknown mode sets. Theoretical results are complemented by simulations and real-data applications (NYC taxi flows and Fama-French portfolios), demonstrating that the method can detect departures from Kronecker structure and guide dimension reduction. This work offers a principled diagnostic for validating low-rank Kron structures in high-dimensional tensor time series, with implications for improved estimation and inference when such structure holds.

Abstract

We propose a test for testing the Kronecker product structure of a factor loading matrix implied by a tensor factor model with Tucker decomposition in the common component. Through defining a Kronecker product structure set, we define if a tensor time series response has a Kronecker product structure, equivalent to the ability to decompose according to a tensor factor model. Our test is built on analysing and comparing the residuals from fitting a full tensor factor model, and the residuals from fitting a (tensor) factor model on a reshaped version of the data. In the most extreme case, the reshaping is the vectorisation of the tensor data, and the factor loading matrix in such a case can be general if there is no Kronecker product structure present. Theoretical results are developed through asymptotic normality results on estimated residuals. Numerical experiments suggest that the size of the tests gets closer to the pre-set nominal value as the sample size or the order of the tensor gets larger, while the power increases with mode dimensions and the number of combined modes. We demonstrate out tests through a NYC taxi traffic data and a Fama-French matrix portfolio of returns.
Paper Structure (19 sections, 12 theorems, 131 equations, 6 tables)

This paper contains 19 sections, 12 theorems, 131 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Notations and Tensor Reshape
  3. Notations
  4. Introduction to tensor reshape
  5. Factor Model and Testing its Kronecker Product Structure
  6. Factor model and Kronecker product structure
  7. A test on Kronecker product structure
  8. Constructing the test statistic
  9. Assumptions and Theoretical Results
  10. Assumptions
  11. Main results and practical test design
  12. Numerical Studies
  13. Simulations
  14. Real data analysis
  15. Appendix
  16. ...and 4 more sections

Key Result

Theorem 1

(Tensor Reshape Theorem I) With the notations in Definition def: kron_structure, $\{\mathcal{Y}_t\}$ following eqn: fm_wo_Kron along any given $\mathcal{A} = \{a_1,\ldots,a_\ell\} \subseteq [K]$ with a Kronecker product structure is equivalent to $\{\mathcal{Y}_t\}$ following a Tucker-decomposition where $\mathcal{C}_t$ is the common component, $\mathcal{F}_t\in \mathbbm{R}^{r_1\times \dots \time

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 1
  • Remark 2
  • Example 1
  • Remark 3
  • Remark 4
  • Theorem 2
  • Theorem 3
  • ...and 10 more