Bayesian Estimation for Factor Models, Shrinkage, and Portfolio Returns
Summary
The discussion explains that Bayesian versions of Fama-French or Carhart factor models can be approached as Bayesian regression. It recommends learning Markov chain Monte Carlo and Gibbs sampling, while noting that Hamiltonian Monte Carlo software can reduce the implementation burden. It also points to Bayesian econometrics references and research on factor pricing models.
The replies connect Bayesian regression to covariance shrinkage and regularized regression: under suitable assumptions, an L2 penalty corresponds to a normal prior on regression weights, while shrinkage can reduce overfitting in covariance estimates. The Black-Litterman approach is offered as another portfolio application, combining prior return estimates from reverse optimization with investor views. The discussion gives no head-to-head performance study. It cautions that Bayesian computation may be substantially slower than frequentist alternatives, which can complicate repeated backtests, although some models, such as stochastic volatility, may be more tractable in a Bayesian framework.
Key ideas
- Bayesian factor models can be estimated using Bayesian regression methods.
- MCMC and Gibbs sampling are foundational techniques, while Hamiltonian Monte Carlo can simplify implementation.
- Under appropriate assumptions, L2 regularization corresponds to a normal prior on regression coefficients.
- Covariance shrinkage can reduce overfitting in regression and portfolio estimation.
- Bayesian methods can be computationally costly, though they may suit some complex models.
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Full text
# Bayesian estimation of asset pricing models
# Bayesian estimation of asset pricing models
I am interested in Bayesian methods in the context of financial economics and quantitative finance and have been looking for research which uses Bayesian parameter estimation on asset pricing models, and in particular multi factor models such as the Fama-French model or its Carhart extension.
I have been searching quite a bit but not really found what I am looking for and would therefore greatly appreciate any recommendations of journal articles or textbooks demonstrating this particular application.
## Answer by John (score 6)
https://quant.stackexchange.com/a/16032
Fitting Fama-French or Carhart is as simple as learning how to perform Bayesian regression. Pretty much every introductory book Bayesian estimation will cover this. There are analytic formulae under certain assumptions, but I would definitely try to learn the basics of MCMC and Gibbs sampling before trying this out in practice. Here are two papers. The book Bayesian Methods in Finance by Rachev et al covers quite a bit. Some googling reveals a book coming out next year (2015) titled Bayesian Inference in Factor Asset Pricing Models.
After having some basic understanding, you might find that implementing MCMC is a bit of a hassle if you're programming each on your own. Stan can implement a Hamiltonian Monte Carlo that requires quite a bit less work to set up. There are other programs (WinBugs), but I don't have much familiarity with them.
As to whether there is any benefit to fitting these with Bayesian or frequentist techniques, I'm of a split mind. I find Bayesian statistics far more intuitive, but that they are so ridiculously slow compared to frequentist techniques. It makes it harder to backtest things, unless you've got a some nice equipment and spend a lot of time thinking about parallelism. There are some problems, like fitting stochastic volatility models, that are much easier to do with Bayesian techniques. On the other hand, with daily data over 20+ years, the Bayesian methods will still be much slower than some alternative Garch model fit with MLE.
## Answer by Quantopik (score 3)
https://quant.stackexchange.com/a/18485
I suggest you to read:
> Lancaster, Tony. An introduction to modern Bayesian econometrics. Oxford: Blackwell, 2004.
I studied that to learn Bayesian Regression Model; the book is very clear and well-done and it is a good reference, IMHO, for who is a newbie, but, at the same time, is interested to the topic.
Hope this helps.
## Answer by Mark Horvath (score 3)
https://quant.stackexchange.com/a/18935
To many statistical questions you can get frequentist and Bayesian answers which actually coincide. Such a subject is covariance matrix shrinkage and Bayesian regression.
Have a look at the article "Honey, I Shrunk the Sample Covariance Matrix" from Lediot and Wolf. They introduce a transformation of the covariance matrix, so that the diagonals become more pronounced, hence less (over) fitting happens on the correlation matrix when doing regression.
Similar -- shrinked -- covariance matrix can be used in regression and you will arrive to ridge regression (L2 regularization) or Lasso, LARS (L1 regularization). These regression methods are expressed as minimization of the quadratic error term plus an L1 or L2 penalty term on regression weights. Consider L2 regularization for example, which minimizes:
$$\|A\mathbf{x}-\mathbf{b}\|^2+ \lambda\| \mathbf{x}\|^2$$
While at first this looks like frequentist formula aimed at reducing over-fitting, is is equivalent to the Bayesian solutions of regression with Normal prior on the regression weights (the penalty coefficient $\lambda$ is a function of variance of the Normal prior).
Many statistical packages are equipped with L1 and L2 regressions and covariance shrinkage (e.g. scikit-learn in Python), so you won't have to get your hands very dirty.
An other interesting Bayesian model in portfolio optimizaiton is the Black-Litterman model, where they use a reasonable prior on expected asset returns (derived from reverse optimization in portfolio theory), and Bayesian theory to arrive to a mixed estimation of future asset returns.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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