高維因子模型中基於資訊準則的秩估計方法

Abstract

本研究探討因子模型中因子數估計的問題。由於模型具有多重解釋性，因子數並不存在嚴格且唯一的定義。然而，在樣本數n與維度p同時發散的漸近情境下，透過隨機矩陣理論（RMT），可以在未假設特定因子模型的情況下，將因子數以「秩」的形式，重新給予更嚴謹且抽象化的定義。本研究以秩估計為核心目標，特別關注基於資訊準則的秩估計量之理論性質。本研究的主要貢獻有二。

第一，在RMT的假設下，我們針對包括AIC、BIC、GIC，以及PC類與IC類等資訊準則秩估計量，建立了統一選擇一致性定理。既有研究已討論過AIC、BIC與GIC達成選擇一致性的充要條件，亦即間隙條件，但我們進一步推導了PC類與IC類共六種估計量的間隙條件，並將其與既有結果整合，得到統一選擇一致性定理。該條件由最小信號特徵值強度與最大可容許雜訊特徵值強度之間的關係式所構成，揭示了各估計量在信號偵測能力與雜訊穩健性之間的權衡關係。我們發現，這種權衡關係在有限樣本情境下，構成了基於資訊準則的秩估計之實質性限制。

第二，基於此，作為第二項貢獻，本研究提出了一種新的資訊準則，稱為「擴展廣義資訊準則（eGIC）」。eGIC的推導動機源自近年關於橢圓分佈族之隨機矩陣理論的發展，透過在GIC的推導過程中引入橢圓分佈作為工作分佈而得。所得之eGIC具有根據資料生成分佈的尾部厚度來縮放懲罰項的結構。此一設計使得eGIC不僅依賴譜資訊，亦能反映分佈的幾何特徵。透過數值模擬實驗，我們證明了eGIC與傳統GIC一樣，能捕捉到微弱的訊號特徵值，並大幅提升了雜訊穩健性，從而驗證了其在有限樣本下克服既有方法限制的有效性。

This study investigates the problem of estimating the number of factors in factor models. Due to the multiple interpretability of models, a strict and unique definition of the number of factors generally does not exist. However, in the asymptotic regime where both the sample size n and dimension p diverge, Random Matrix Theory (RMT) allows for a rigorous reformulation of the number of factors as the "rank," without assuming factor models. In this study, we focus on rank estimation methods, and specifically investigate the theoretical properties of rank estimators based on information criterion. The main contributions of this study are as follows.

First, under RMT assumptions, we establish a unified selection consistency theorem for information criterion-based rank estimators, including AIC, BIC, and GIC, as well as the PCs and ICs. Existing literature has discussed the necessary and sufficient conditions, specifically, the "gap conditions," for AIC, BIC, and GIC to achieve selection consistency. In this study, we further derive the gap conditions for six rank estimators based on PC and IC-type criterion. By integrating our findings with existing results, we obtain a unified selection consistency theorem. The gap conditions specify the minimum strength of signal eigenvalue and the maximum tolerable noise eigenvalue, revealing a trade-off between sensitivity to signal eigenvalues and robustness against noise eigenvalues. We find that this trade-off emerges as a substantial limitation for information criterion-based rank estimators in practical finite-sample settings.

Second, based on these findings, we propose a new information criterion, namely the "extended Generalized Information Criterion (eGIC)." The derivation of eGIC is motivated by recent developments in RMT concerning elliptical distribution families. The eGIC is constructed by introducing elliptical distributions as the working distribution within the derivation process of GIC. The resulting eGIC is designed to scale the penalty term according to the tail heaviness of the data-generating distribution. This design enables eGIC to reflect not only spectral information but also the geometric characteristics of the data-generating distribution. Through numerical simulations, we demonstrate that eGIC, like the standard GIC, is capable of detecting weak signal eigenvalues, while significantly improving noise robustness. This validates its effectiveness in overcoming the limitations of existing methods encountered in finite-sample scenarios.