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High Dimensional Factor Analysis with Weak Factors

Jungjun Choi, Ming Yuan

Abstract

This paper studies the principal components (PC) estimator for high dimensional approximate factor models with weak factors in that the factor loading ($\boldsymbolΛ^0$) scales sublinearly in the number $N$ of cross-section units, i.e., $\boldsymbolΛ^{0\top} \boldsymbolΛ^0 / N^α$ is positive definite in the limit for some $α\in (0,1)$. While the consistency and asymptotic normality of these estimates are by now well known when the factors are strong, i.e., $α=1$, the statistical properties for weak factors remain less explored. Here, we show that the PC estimator maintains consistency and asymptotical normality for any $α\in(0,1)$, provided suitable conditions regarding the dependence structure in the noise are met. This complements earlier result by Onatski (2012) that the PC estimator is inconsistent when $α=0$, and the more recent work by Bai and Ng (2023) who established the asymptotic normality of the PC estimator when $α\in (1/2,1)$. Our proof strategy integrates the traditional eigendecomposition-based approach for factor models with leave-one-out analysis similar in spirit to those used in matrix completion and other settings. This combination allows us to deal with factors weaker than the former and at the same time relax the incoherence and independence assumptions often associated with the later.

High Dimensional Factor Analysis with Weak Factors

Abstract

This paper studies the principal components (PC) estimator for high dimensional approximate factor models with weak factors in that the factor loading () scales sublinearly in the number of cross-section units, i.e., is positive definite in the limit for some . While the consistency and asymptotic normality of these estimates are by now well known when the factors are strong, i.e., , the statistical properties for weak factors remain less explored. Here, we show that the PC estimator maintains consistency and asymptotical normality for any , provided suitable conditions regarding the dependence structure in the noise are met. This complements earlier result by Onatski (2012) that the PC estimator is inconsistent when , and the more recent work by Bai and Ng (2023) who established the asymptotic normality of the PC estimator when . Our proof strategy integrates the traditional eigendecomposition-based approach for factor models with leave-one-out analysis similar in spirit to those used in matrix completion and other settings. This combination allows us to deal with factors weaker than the former and at the same time relax the incoherence and independence assumptions often associated with the later.
Paper Structure (71 sections, 32 theorems, 269 equations, 2 tables)

This paper contains 71 sections, 32 theorems, 269 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Factor Model and Method of PC
  3. Assumption A.
  4. Eigenecomposition and Rotation.
  5. Alternative Matching Matrix.
  6. Leave-one-out Analysis.
  7. Independent Noise
  8. Assumption B.
  9. Assumption C.
  10. Assumption D.
  11. Dependent Noise
  12. Leave-neighbor-out Analysis.
  13. Temporal Dependence
  14. Assumption B'.
  15. Assumption D'.
  16. ...and 56 more sections

Key Result

Theorem 3.1

Suppose that Assumptions A and B are satisfied. If $\max\{N, T \}=o(N^\alpha T)$, then

Theorems & Definitions (39)

  • Theorem 3.1: Convergence rate of PC estimator
  • Theorem 3.2: CLT for PC estimator
  • Lemma 4.1: Examples of Assumption B'(iv)
  • Theorem 4.2: Convergence rate of PC estimator
  • Theorem 4.3: CLT for PC estimator
  • Theorem 4.4: Convergence rate of PC estimator
  • Theorem 4.5: CLT for PC estimator
  • Theorem 4.6: Convergence rate of PC estimator
  • Theorem 4.7: CLT for PC estimator
  • Proposition A.1
  • ...and 29 more