Approximate factor models: Finite sample distributions

Abstract

In the growing literature of factor analysis, little is done to understand the finite sample properties of an approximate factor model solution. In empirical applications with relatively small samples, the asymptotic theory might be a poor approximation and the resulting distortions might affect the estimation (bias in the point estimate and the standard errors) and the statistical inference. The present paper uses the estimation method of Bai and Ng [Bai, J. and Ng, S., 2002, Determining the number of factors in approximate factor models. Econometrica, 70, 191–221.] and assesses the sampling behavior of the estimated common components, common factors and factor loadings. The study compares the empirical distributions to the asymptotic theory of Bai [Bai, J., 2003, Inference on factor models of large dimension. Econometrica, 71, 135–171.]. Simulation results suggest that the point estimates have a Gaussian distribution for panels with relatively small dimensions. However, these estimates have a significant finite sample bias and the dispersion of their sampling distribution is severely underestimated by the asymptotic theory.

Keywords:

• Approximate factor models
• Principal components
• Latent factors
• Eigenvalues; Eigenvectors
• Orthogonal rotation
• Sampling distribution

Acknowledgements

Support from CAER and SRG grants at the School of Economics, The University of New South Wales, Sydney, is acknowledged. The author is indebted to the Associate Editor and the referee for their helpful comments and their contributions to the exposition. The author would also like to thank Jushan Bai and Serena Ng for their invaluable input at the start of this research.

Notes

^†CAPM uses only one factor (regressor), the risk premium of the market as a whole, to explain excess returns.

^‡In this paper, the author shows that the world cycle is statistically significant and persistent in the growth rates of output, consumption and investment for the G7 countries.

^†The author finds that indexes based forecasts outperformed the traditional VARMA models.

^‡Bai 8 shows that, under normality assumption, the principal component method is asymptotically equivalent to the maximum likelihood method.

^† See ref. 6.

^‡ See ref. 9 for the proof.

^§See ref. 6 for the proof and a thorough discussion.

^†See Proof 1 in Appendix.

^‡The following results are from ref. 8.

1. If N/T→∞, then α=√T and .

2. If T/N→θ, then α=√ N and Q _it =V _it +θ W _it .

3. If N/T→0, then α=√ N and , where Σ_Λ=plim _N→∞ (Λ^0′ Λ⁰/N).

^†For a thorough discussion, see ref. 7.

^‡See ref. 12.

^†Notice that in the simulation experiment, t=T/2. The covariance matrix is r×r, a scalar in the experiment where, r=1.

Additional information

Notes on contributors

Rachida Ouysse

Email: [email protected]