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)

2) 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 http://citeseerx.ist.psu.edu/viewdoc/download?doi=

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.