Invariant Inference and Efficient Computation in the Static Factor Model

ABSTRACT

Factor models are used in a wide range of areas. Two issues with Bayesian versions of these models are a lack of invariance to ordering of and scaling of the variables and computational inefficiency. This article develops invariant and efficient Bayesian methods for estimating static factor models. This approach leads to inference that does not depend upon the ordering or scaling of the variables, and we provide arguments to explain this invariance. Beginning from identified parameters which are subject to orthogonality restrictions, we use parameter expansions to obtain a specification with computationally convenient conditional posteriors. We show significant gains in computational efficiency. Identifying restrictions that are commonly employed result in interpretable factors or loadings and, using our approach, these can be imposed ex-post. This allows us to investigate several alternative identifying (noninvariant) schemes without the need to respecify and resample the model. We illustrate the methods with two macroeconomic datasets.

KEYWORDS:

• Bayesian
• Markov chain Monte Carlo
• Reduced rank regression

Supplementary Materials

The supplementary materials provide additional simulation results and technical proofs. More specifically, Part A presents additional simulation results to illustrate the impact of the ordering assumption. Part B provides a comparison in terms of computational time and efficiency with the algorithm in Hoff (2007). Part C gives the detailed derivations of the correction factor for the Savage-Dickey density ratio. Part D presents a detailed proof of Theorem 3.1. And Part E provides a simulation study to illustrate the impact of the number of factors on variance decompositions.

Acknowledgment

The authors thank the associate editor and anonymous referees for useful comments and suggestions that helped us improve the article. The authors thank Robert Kohn and Jim Berger for their encouraging comments. The authors also thank seminar participants at the University of Sydney, the University of Strathclyde, the Multivariate Time Series Modelling and Forecasting Workshop Monash University 2013, the International Workshop on Bayesian Model Selection Shanghai 2013, Workshop, the 18th Australasian Macroeconomics Workshop, and Workshop on Empirical Methods in Macroeconomic Policy Analysis Bucharest for helpful comments and suggestions. Roberto Leon-Gonzalez acknowledges financial support from the Japan Society for the Promotion of Science (#26780135) and the Nomura Foundation (BE-004). All errors are, of course, our own.

Notes

1 In fact, Bai and Ng used $\sqrt{T}$ times the eigenvalue of yy′. This proportional term is not important for the discussion here so we ignore it.

2 If κ were not of this structure then the restriction Λ^pcΛ^pc′ = S²₁ would be destroyed by the transformation. Thus this restriction implies identification against such a transformation.