[Paper Review] Approximate Factor Models with Weaker Loadings

Jushan Bai, Serena Ng|arXiv (Cornell University)|Sep 8, 2021

Economic theories and models4 citations

TL;DR

This paper establishes the consistency and asymptotic normality of principal component estimators in approximate factor models under weaker loading strength, where loadings scale as $N^{-\alpha}$ with $\alpha \in (0,1]$. It shows that estimation remains consistent even when loadings are weak, with convergence rates slowing as $\alpha$ decreases, and provides a unified framework for heterogeneous loadings and simplified proofs applicable to strong factor analysis.

ABSTRACT

Pervasive cross-section dependence is increasingly recognized as a characteristic of economic data and the approximate factor model provides a useful framework for analysis. Assuming a strong factor structure where $\Lop\Lo/N^α$ is positive definite in the limit when $α=1$, early work established convergence of the principal component estimates of the factors and loadings up to a rotation matrix. This paper shows that the estimates are still consistent and asymptotically normal when $α\in(0,1]$ albeit at slower rates and under additional assumptions on the sample size. The results hold whether $α$ is constant or varies across factor loadings. The framework developed for heterogeneous loadings and the simplified proofs that can be also used in strong factor analysis are of independent interest.

Motivation & Objective

• To establish the asymptotic properties of principal component estimators when factor loadings are weak, i.e., $\bm{\Lambda}^{0'\prime}\bm{\Lambda}^{0}/N^{\alpha}$ has a positive definite limit for $\alpha \in (0,1]$.
• To resolve ambiguity about whether PC estimators remain consistent when the strong factor assumption ($\alpha=1$) fails, particularly in light of Onatski (2012) showing inconsistency when $\alpha=0$.
• To develop a general framework for heterogeneous loadings with varying $\alpha_r$ across factors, showing that the weakest loading determines asymptotic behavior.
• To simplify existing proofs in strong factor analysis by using higher-level assumptions and approximations to the rotation matrix $\bm{H}$, enabling more flexible inference.

Proposed method

• The paper introduces a general asymptotic framework where the factor loading strength is indexed by $\alpha \in (0,1]$, allowing for weaker loadings than the classical $\alpha=1$ case.
• It uses a rotation matrix $\bm{H}$ to align estimated factors $\tilde{\bm{F}}$ with true factors $\bm{F}^0$, and derives asymptotic approximations to $\bm{H}$ to simplify the analysis.
• The authors derive error rates for the low-rank component $\tilde{\bm{C}}$ and the factor space, showing $\frac{1}{NT}\sum\|\tilde{C}_{it}-C_{it}^0\|^2 = O_p(N^{-1}) + O_p(T^{-1})$.
• For factor estimation, the error is $\frac{1}{T}\sum\|\tilde{F}_t - \bm{H}'F_t^0\|^2 = O_p(N^{-\alpha}) + O_p((N^{1-\alpha}/T)^2)$, which is $o_p(1)$ when $\alpha > 0$ and $N^{1-\alpha}/T \to 0$.
• The framework handles heterogeneous loadings by allowing $\alpha_1 \geq \cdots \geq \alpha_r > 0$, where the weakest loading $\alpha_r$ governs the asymptotic behavior.
• The paper uses a novel proof technique based on approximating the rotation matrix $\bm{H}$, which leads to simplified asymptotic variance representations and enables inference under user-preferred variance formulations.

Experimental results

Research questions

• RQ1 Are principal component estimators consistent when factor loadings are weak, i.e., when $\alpha \in (0,1)$?
• RQ2 What are the convergence rates of the estimated factors and loadings under weaker loadings, and how do they compare to the strong factor case ($\alpha=1$)?
• RQ3 Does asymptotic normality of the estimators still hold under weaker loadings, and if so, under what conditions on $\alpha$?
• RQ4 How does the asymptotic behavior change when loadings have heterogeneous strength across factors, and which loading strength determines the limiting distribution?
• RQ5 Can the proof techniques be simplified and generalized to apply to both weak and strong factor models?

Key findings

• The principal component estimator for the low-rank component $\tilde{\bm{C}}$ is consistent with error rate $O_p(N^{-1}) + O_p(T^{-1})$, identical to the strong factor case.
• The estimated factor space converges at rate $O_p(N^{-\alpha}) + O_p((N^{1-\alpha}/T)^2)$, which is $o_p(1)$ when $\alpha > 0$ and $N^{1-\alpha}/T \to 0$, ensuring consistent estimation of the factor space.
• Asymptotic normality of $\sqrt{N^\alpha}(\tilde{\bm{F}}_t - \bm{H}'\bm{F}_t^0)$ requires $\alpha > 1/2$, while consistency of $\tilde{\bm{F}}_t$ holds for $\alpha > 1/3$, and consistency of $\tilde{\bm{\Lambda}}_i$ for $\alpha > 0$.
• For heterogeneous loadings with $\alpha_r$ being the weakest, asymptotic normality requires $\alpha_r > 1/2$, but consistency of individual estimates holds even for $\alpha_r \leq 1/2$, with $\alpha_r > 0$ sufficient for consistency.
• The new proof technique simplifies existing derivations and allows for flexible inference by enabling the use of user-preferred asymptotic variance representations.
• Simulation results confirm that the estimator remains consistent and well-performing under weak loadings, with $R^2$ values for factor and loading estimates improving with $T$ and $N$, even when $\alpha < 1$.