I am interested in fitting a factor analysis-like model on asset returns or other similar latent variable models. What are good papers to read on this topic? I am particularly interested in how to handle the fact that a factor analysis model is identical under a sign change for the "factor loadings".


5 Answers 5


Some references to help you out.

  1. Tipping, M. E. & Bishop, C. M. Probabilistic principal component analysis Journal of the Royal Statistical Society (Series B), 1999, 21, 611-622
  2. Tom Minka. Automatic choice of dimensionality for PCA. NIPS 2000 url:


  3. Šmídl, V. & Quinn, A. On Bayesian principal component analysis Computational Statistics & Data Analysis, 2007, 51, 4101-4123

If you are familiar with information theoretic model selection (MML, MDL, etc.), I highly recommend checking out:

  1. Wallace, C. S. & Freeman, P. R. Single-Factor Analysis by Minimum Message Length Estimation Journal of the Royal Statistical Society (Series B), 1992, 54, 195-209
  2. C. S. Wallace. Multiple Factor Analysis by MML Estimation.

    Tech report: http://www.allisons.org/ll/Images/People/Wallace/Multi-Factor/TR95.218.pdf


Here are a few suggestions, more from the statistics literature, with an eye toward applications in finance:

  1. Geweke, J., & Zhou, G. (1996). Measuring the pricing error of the arbitrage pricing theory. Review of Financial Studies, 9(2), 557. Soc Financial Studies. Retrieved January 29, 2011, from http://rfs.oxfordjournals.org/content/9/2/557.abstract.

You might start here - a detailed discussion of identifiability issues (related to and including the sign indeterminacy you describe)

  1. Aguilar, O., & West, M. (2000). Bayesian Dynamic Factor Models and Portfolio Allocation. Journal of Business & Economic Statistics, 18(3), 338. doi: 10.2307/1392266.

  2. Lopes, H. F., & West, M. (2004). Bayesian model assessment in factor analysis. Statistica Sinica, 14(1), 41â68. Citeseer. Retrieved September 19, 2010, from here.

Good luck!


You should take a look at some of the nonparametric Bayesian approaches (see this paper and this paper) to factor analysis which do not assume the number of factors to be known; the first one can also model the case where the factors have a dependency structure among them.


A decent overview of factor analysis is Latent Variable Methods and Factor Analysis by Bartholomew and Knott. They write about the interpretation of latent factors. This book is not as algorithmically-oriented as I would like, but their description of e.g. partial factor analysis is decent.

  • $\begingroup$ I agree, a very nice book. But I would love to see another refresh (last ed was in 1999). Most recent treatments don't stack up IMO. $\endgroup$
    – JMS
    Apr 30, 2011 at 16:23
  • $\begingroup$ You're in luck! The third edition was published in 2011, as Latent Variable Models and Factor Analysis: A Unified Approach. $\endgroup$
    – Sycorax
    Jul 10, 2013 at 19:48

This article deals with Bayesian estimation of dynamical hierarchical factor model:

E. Moench, S. Ng, S. Potter. Dynamic Hierarchical Factor Models, Federal Reserve Bank of New York, 2009, Report No. 412. link.

Naturally it can be adapted for non-hierarchical case. As usual you will find more references on topic by perusing the references in the article.


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