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Testing for common structures in high-dimensional factor models

Marie-Christine Düker, Vladas Pipiras

Abstract

This work proposes a novel procedure to test for common structures across two high-dimensional factor models. The introduced test allows to uncover whether two factor models are driven by the same loading matrix up to some linear transformation. The test can be used to discover inter individual relationships between two data sets. In addition, it can be applied to test for structural changes over time in the loading matrix of an individual factor model. The test aims to reduce the set of possible alternatives in a classical change-point setting. The theoretical results establish the asymptotic behavior of the introduced test statistic. The theory is supported by a simulation study showing promising results in empirical test size and power. A data application investigates changes in the loadings when modeling the celebrated US macroeconomic data set of Stock and Watson.

Testing for common structures in high-dimensional factor models

Abstract

This work proposes a novel procedure to test for common structures across two high-dimensional factor models. The introduced test allows to uncover whether two factor models are driven by the same loading matrix up to some linear transformation. The test can be used to discover inter individual relationships between two data sets. In addition, it can be applied to test for structural changes over time in the loading matrix of an individual factor model. The test aims to reduce the set of possible alternatives in a classical change-point setting. The theoretical results establish the asymptotic behavior of the introduced test statistic. The theory is supported by a simulation study showing promising results in empirical test size and power. A data application investigates changes in the loadings when modeling the celebrated US macroeconomic data set of Stock and Watson.
Paper Structure (27 sections, 15 theorems, 156 equations, 8 tables)

This paper contains 27 sections, 15 theorems, 156 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Hypothesis testing problem
  3. Two-subject factor model
  4. Structural changes over time
  5. Hypothesis testing problem
  6. Testing procedure
  7. Motivation
  8. The test statistic
  9. Assumptions
  10. Limiting distribution under the null
  11. Power against the alternative hypothesis
  12. Extensions and other observations
  13. Different numbers of factors
  14. Supremum approach
  15. Discussion on alternative approaches
  16. ...and 12 more sections

Key Result

Proposition 3.1

Recall the definition of $W(\cdot)$ in eq:def:Wald and suppose Assumptions ass:1--ass:3, ass:C1--ass:C3, ass:9,ass:10, and $\frac{\sqrt{T}}{d} \to 0$, as $d,T \to \infty$. Then, under the null hypothesis eq:H01, where $\chi^{2}(r(r+1)/2)$ denotes the chi-squared distribution with $r(r+1)/2$ degrees of freedom.

Theorems & Definitions (32)

  • Example 2.1
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 4.1
  • proof : Proof of Proposition \ref{['prop1']}
  • proof : Proof of Proposition \ref{['prop2']}
  • proof : Proof of Proposition \ref{['prop3']}
  • ...and 22 more