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Universal Factor Models

Songnian Chen, Junlong Feng

Abstract

We propose a new factor analysis framework and estimators of the factors and loadings that are robust to certain weak factors in a large $N$ and large $T$ setting. Our framework, by simultaneously considering all quantile levels of the outcome variable, induces standard mean and quantile factor models, but the factors can have an arbitrarily weak influence on the outcome's mean or quantile at most quantile levels. Our method estimates the factor space at the $\sqrt{N}$-rate as long as each factor is strong at some unknown quantile level, and achieves $\sqrt{N}$- and $\sqrt{T}$-asymptotic normality for the factors and loadings based on a novel sample splitting approach that handles incidental nuisance parameters. We also develop a weak-factor-robust estimator of the number of factors and consistent selectors of factors of any tolerated level of influence on the outcome's mean or quantiles. Monte Carlo simulations demonstrate the effectiveness of our method.

Universal Factor Models

Abstract

We propose a new factor analysis framework and estimators of the factors and loadings that are robust to certain weak factors in a large and large setting. Our framework, by simultaneously considering all quantile levels of the outcome variable, induces standard mean and quantile factor models, but the factors can have an arbitrarily weak influence on the outcome's mean or quantile at most quantile levels. Our method estimates the factor space at the -rate as long as each factor is strong at some unknown quantile level, and achieves - and -asymptotic normality for the factors and loadings based on a novel sample splitting approach that handles incidental nuisance parameters. We also develop a weak-factor-robust estimator of the number of factors and consistent selectors of factors of any tolerated level of influence on the outcome's mean or quantiles. Monte Carlo simulations demonstrate the effectiveness of our method.
Paper Structure (30 sections, 15 theorems, 159 equations, 1 figure, 4 tables, 2 algorithms)

This paper contains 30 sections, 15 theorems, 159 equations, 1 figure, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Model
  3. A Relaxed Assumption on the Strength of Factors
  4. Comparison with the QFM
  5. Comparison with the AFM
  6. The Baseline Estimator
  7. Discretize $\mathcal{U}$
  8. Smooth Quantile Regression
  9. Rate of Convergence
  10. The Inverse Density Weighted Estimator
  11. The Value of Inverse Density Weighting
  12. Inverse Density Estimation
  13. Inverse Density Weighted Universal Factor Analysis
  14. Asymptotic Theory of IDW-UFA
  15. Estimating the Mean Factor Loadings
  16. ...and 15 more sections

Key Result

Lemma 3.1

Assumptions assum.normalization and assum.lip imply that for sufficiently large $M,N$ and $T$, the eigenvalues of $\sum_{m=1}^{M}\Lambda^{*'}_{0}(\tau_{m})\Lambda^{*}_{0}(\tau_{m})/MN$ are bounded away from 0, and the eigenvalues of matrix $F_{0}^{*}\sum_{m=1}^{M}\Lambda^{*'}_{0}(\tau_{m})\Lambda^{*

Figures (1)

  • Figure 1: Histograms of the standardized factor and common component estimates. Row 1 to row 4 plots the histograms of $\tilde{f}_{t}^{std}$, $\tilde{L}_{it}^{std}(0.5)$, $\tilde{L}_{it}^{std}(0.2)$, $\tilde{L}_{it}^{std}(0.8)$, respectively.

Theorems & Definitions (41)

  • Example 2.1: Example 4 in chen2021quantile, p.879
  • Example 2.2: A scale model with diminishing location shift
  • Lemma 3.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • Remark 4.1
  • Remark 4.2
  • Theorem 4.1
  • Remark 4.3
  • ...and 31 more