Selecting the Correct Number of Factors in Approximate Factor Models: The Large Panel Case With Group Bridge Estimators

Abstract

This article proposes a group bridge estimator to select the correct number of factors in approximate factor models. It contributes to the literature on shrinkage estimation and factor models by extending the conventional bridge estimator from a single equation to a large panel context. The proposed estimator can consistently estimate the factor loadings of relevant factors and shrink the loadings of irrelevant factors to zero with a probability approaching one. Hence, it provides a consistent estimate for the number of factors. We also propose an algorithm for the new estimator; Monte Carlo experiments show that our algorithm converges reasonably fast and that our estimator has very good performance in small samples. An empirical example is also presented based on a commonly used U.S. macroeconomic dataset.

KEY WORDS:

• Bridge estimation
• Common factors
• Selection consistency

ACKNOWLEDGMENTS

The authors thank the co-editor Shakeeb Khan, an associate editor, and two anonymous referees who made this article better. The authors also thank Robin Sickles, James Stock, and the participants at the High Dimension Reduction Conference (December 2010) in London, the Panel Data Conference (July 2011) in Montreal, the CIREQ Econometrics Conference (May 2013), and the North American Summer Meeting of the Econometric Society (June 2013) for their comments.

Notes

The definitions of static factors, dynamic factors, and approximate factor models are discussed in the second paragraph of Section 2.

This is why Bai and Ng (2002) defined their estimator for factors as where V^k is a diagonal matrix consisting of the first k largest eigenvalues of XX′/NT in a descending order. Since the kth (k > r) eigenvalue of XX′/NT is , the last p − r columns of their factor matrix are asymptotically zeros and only the first r columns of the factor matrix matter. As their criteria focus on the sum of squared errors, this asymptotically not-of-full-column-rank design does not affect their result. Since we focus on estimating and penalizing the factor loadings, we use to ensure full column rank instead of Bai and Ng’s (2002) as the estimator for F⁰.

The definition of H is given by Lemma 1 in the appendix.

Our Monte Carlo experiments show that the algorithm is reasonably fast and performs very well in terms of selecting the true value of r. We also tried 500 and 1000 instead of 750 to adjust the increment, and the results are very similar and, hence, not reported.

See Onatski (2010) for the details of the computation of ζ.

Ideas similar to the ER estimator have also been considered by Luo, Wang, and Tsai (2009) and Wang (2012)