Which approach to estimating fundamental factor models is better, cross-sectional (unobservable) factors or time-series (observable) factors?

There are many approaches to estimating fundamental factor equity models. I would like to focus on two traditional methods:

1. The time-series regression approach of Fama and French. Factors are defined ex ante. Betas to the factors are estimated in the time-series.
2. The Barra cross-sectional regression approach described in Menchero, Orr, and Wang (2011), Grinold and Kahn (2000) and Sheikh (1995). Factor realizations are derived ex post. Factors are estimated independently in each time period in the cross-section.

I'll sketch the methodology of each and the high-level pros/cons. I'm curious if anyone has experience or links to research regarding which approach is better for the purposes of hedging, optimal portfolio construction, and alpha generation.

Fama-French time-series regression approach:

1. Build a design matrix where each column is a time-series of economic factor returns. These factors could be traditional economic factors but also may include "spread" returns such as Fama-French factors SMB, HML, MKT generated from portfolio sorts.

2. Perform N time-series regressions (one per security). In particular, for each security regress the security returns on the economic factor returns and estimate the beta.

In this approach, the betas are constant and the factors are time-varying depending on the regression window. The advantage here is that estimation of beta is diversified away across securities, so it seems to be this approach would be superior for portfolio construction. The disadvantage is that betas are slower to respond to changes that change the risk profile of a firm (for example, in a sudden shift in Debt-to-Equity ratio).

Barra cross-sectional regression approach:

1. Assume that fundamental factor characteristics are Betas. For example, create z-scores of the fundamental factor characteristics thereby generating Betas for each time slice for each security.

2. Perform T cross-sectional regressions (one per factor). In particular, at each time slice regress the panel of security returns on the normalized Betas to estimate the un-observable factor realizations.

The advantage here is that the betas respond instantaneously to changes in firm characteristics. The disadvantage is that there is a potential errors-in-variable bias -- in fact, the errors from model mis-specification do not diversify away. Intuition suggests this approach may be better suited for alpha signal generation.

Is there research comparing the out-of-sample performance of these two methods for various applications (i.e. risk decomposition, portfolio construction, hedging, alpha signal generation)?

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I hope you don't mind but your question was rather long so I tried to make it a bit more readable. Also, do you have a link for the Sheikh paper? I'm not sure what you're referring to there. – Tal Fishman Sep 14 '11 at 16:25
I tried to find a link for Sheikh paper but could not find it on google. I'll check Jstor - thanks for cleaning it up - I appreciate it! – Ram Ahluwalia Sep 14 '11 at 19:36
I did some digging on MSCI Barra's web site, and Axioma (I believe they use a Fama-French type approach, not sure yet). I couldn't find a research piece where they compare their approach to competitors, but I did find an excellent summary of the state of the art according to Barra, which I added to your question. I also think your terminology regarding unobservable factors may be confusing, as they are observable ex post, but derived. I tried to re-cast in terms of ex ante / ex post, not sure if this helps, still looking for best way to describe the difference. – Tal Fishman Sep 15 '11 at 2:43
Barra's state of the art is the "Eigenfactor" methodology. (It seems that Barra's desire to serve multiple users each who have conflicting objectives has led them to add one contraption or ad-hoc adjustment on top of another. I have seen some simple approaches that out-perform BARRA out-of-sample.) BARRA's original procedure (a la Rosenberg) follows the cross-sectional regression procedure given industry loadings and standardized factors loadings. I am surprised that there are not published empirical results on the performance of cross-sectional vs time-series methods. – Ram Ahluwalia Sep 15 '11 at 3:20
Maybe there are, but I just haven't found them yet. A customized model will often outperform off-the-shelf models like Barra, but sometimes it just isn't worth the effort, particularly if you need to keep the model updated frequently and you work in a relatively small team that focuses mostly on alpha generation. – Tal Fishman Sep 15 '11 at 11:25

Jennifer Bender of MSCI Barra has a paper from 2007 entitled:

To Beta or Not to Beta: A Comparison of Historical Versus Fundamental Betas for Hedging Market Risk

She deals specifically and exclusively with which method is superior for hedging long-only portfolios. Not surprisingly, she finds that Barra's approach is better. She tests long-only and long-short portfolios separately, using random portfolios with asymmetric bets.

What you call "cross-sectional" or "unobservable" factors she calls "fundamental betas," and "time-series" or "observable" factors are termed "historical betas." Here is the abstract:

Fundamental betas provide several conceptual advantages to historical betas--they reflect information on a timelier basis and are less likely to confuse noise for information. This paper revisits the advantages of using fundamental beta for hedging systematic risk in the U.S. Fundamental beta appears to be a more consistent measure for hedging market risk, particularly for investors who care about downside risk and tail risk.

BTW, I have met her IRL and she seems pretty smart, so I trust she did the research right, but of course you must always be skeptical of company research that makes their product look better than their competitors'.

• For hedging, my impression is that there is not a great difference between the two approaches, although fundamental betas win out slightly over historical betas.
• For portfolio construction, where stability of betas is much more important, she makes a convincing case for fundamental betas. Fundamental betas are more stable during stable periods and adjust more quickly to large shifts in a stock's expected behavior due to M&A or spinoff activity. She also argues risk estimates will adapt more quickly to a regime change with fundamental versus historical betas, particularly as she finds the best historical beta overall to be a 60-month regression.
• Fundamental betas also provide a distinct advantage over historical for risk decomposition due to the rapidly changing nature of cross-correlations and the lower propensity for assigning idiosyncratic moves coincident with a particular risk factor to the beta for that factor.
• Alpha signal generation is not really the goal of either approach, so I'm not sure if it is fair to evaluate based on this.

The fact that the two major commercial risk model providers essentially use the same approach should also tell you something. Barclays Capital's US Equity Risk Model (for institutional POINT clients) also uses cross-sectional regressions. In fact, in the face of all this industry research, the real question is why do academics still use the historical time-series regression approach?

Gregory Connor published an article The Three Types of Factor Models: A Comparison of Their Explanatory Power in 1995 in the Financial Analysts Journal. However, his goal is to compare statistical and fundamental models, not the two types of fundamental models to each other (he also throws in macroeconomic models, which he finds to be close to useless). The research is somewhat dated but still interesting for those looking at risk models in general.

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John Cochrane provides a nice conceptual reconciliation of time-series and cross-sectional models in his paper "Discount Rates". Dennis Chaves (his former PHD) student goes on to elaborate the distinctions between time-series and cross-sectional models "What explains the variances of prices and returns". – Ram Ahluwalia Sep 26 '11 at 20:39
One of the distinguishing features of fundamental factor models that hasn't been mentioned is their dimensionality reduction: calculating correlations that are statistically robust for 1000 assets would require data for at least 1000 periods for each asset, which simply may not be available at monthly or weekly horizons. Cross-sectional models are therefore very useful in forecasting correlations for a large universe of assets, as is the case for most equity models (almost 40k securities in the case of Axioma's global equity model, for example). – michaelv2 Sep 28 '11 at 18:33
@michaelv2 that is not really an advantage of cross-sectional over time-series models, but rather of fundamental models in general. – Tal Fishman Sep 28 '11 at 19:02