Determining the Number of Factors from Empirical Distribution of Eigenvalues

Abstract

We develop a new estimator of the number of factors in the approximate factor models. The estimator works well even when the idiosyncratic terms are substantially correlated. It is based on the fact, established in the paper, that any finite number of the largest "idiosyncratic" eigenvalues of the sample covariance matrix cluster around a single point. In contrast, all the "systematic" eigenvalues, the number of which equals the number of factors, diverge to infinity. The estimator consistently separates the diverging eigenvalues from the cluster and counts the number of the separated eigenvalues. We consider a macroeconomic and a financial application. (c) 2010 The President and Fellows of Harvard College and the Massachusetts Institute of Technology.

Selection of the Number of Factors in Presence of Structural Instability: A Monte Carlo study

“…Information criteria procedures are represented by Bai and Ng (2002) and Amengual and Watson (2007). Onatski (2010) and Ahn and Horenstein (2013) are tests based on the theory of random matrices, while Bai and Ng (2007) exploit the rank of matrices. Finally, Hallin and Liska (2007) build on spectral density representation of factor models.…”

“…Both (limited) cross-sectional and temporal correlations in e are allowed. Onatski (2010) observes that any finite number of the largest idiosyncratic eigenvalues of the sample covariance matrix clusters around a single point, while all the systematic eigenvalues-the number of which equals the number of factor-diverge to infinity. The estimator then separates the diverging eigenvalues from the cluster and counts the number of separated eigenvalues-this is the estimated number of factors.…”

“…Bai and Ng (2002), Hallin and Liska (2007), and Amengual and Watson (2007) made the assumption that the factor's cumulative effect on the N cross-sectional units grows proportionally to N. According to this assumption, with r static factors, r eigenvalues of the data's covariance matrix grow proportionally to N while the rest of the eigenvalues stay bounded. Onatski (2010) estimates the number of factors without making any assumption on the rate of growth of the factor's cumulative effect.…”

“…Let k be the number of factors, and λ j the j largest eigenvalues of X ʹ′ X / T , Onatski (2010) shows that for j > k, the differences λ j − λ j+1 converge to zero while the differences λ k − λ k+1 diverge to infinity. Let k max n ,n ∈ !…”

“…In fact, researchers can face misspecification problems when the number of factors is underestimated, or problems related to power when the number of factors is overestimated. Many methods have been proposed to estimate the number of (static and dynamic) factors (see, among others, Ng, 2002, 2007;Onatski, 2009Onatski, , 2010Alessi, Barigozzi and Capasso, 2010;Ahn and Horenstein, 2013;Hallin and Liska, 2007;Amengual and Watson, 2007).…”