# Efficiency vs. Robustness - To use a constant or not in single factor time-series regression?

Arbitrage pricing theory states that expected returns for a security are linear combination of exposures to risk factors and the returns on these risk factors. Betas, or the exposures of the security to a given risk factor, can be estimated via a time-series regression of the excess returns of the security on the excess returns of the factor return (single variable time-series regression).

Theory says the constant (or zero-beta excess return) in the time-series regression should be zero. However, practically speaking it is the case that some constants may be estimated that are statistically significant and different from zero. Question: Should one estimate Betas while forcing the constant equal to zero (i.e. theoretically consistent) vs. extracting Betas while allowing the constant term to exist?

What is the trade-off or which decision leads to superior accuracy? Ultimately the estimated factor exposures from the above time-series regression would be used to estimate the factor returns in a cross-sectional risk model.

Attached is a link to John Cochrane's Asset Pricing chapter 12 which describes the theory in fuller detail.

-
The only commercial portfolio optimizer I've used took "error" (excess return) terms in addition to the coefficient matrix of betas. I'm not sure what the optimizer did under-the-hood, but I figure it's better to have more information than less. –  chrisaycock Jul 19 '11 at 14:19
The optimizer might have used the error terms to gauge the certainty of the estimate (similar to Omega - the uncertainty matrix - in Black Litterman procedure). In my scenario, the regression equation is specified as: Excess Return of security = Intercept + Beta * Factor Return + Error. The expected value of the error term is zero. So the question is whether the intercept should be a free variable or suppressed. –  Quant Guy Jul 21 '11 at 4:37

Time-series regression is not a great method for determining betas on individual securities. Rather, the most common method used by the commercial risk model providers is called "predicted beta" or "fundamental beta." The leader in this area is Barra. The way they define the predicted beta, it appears that they include the constant in the regression.

-
I performed a couple rolling regression tests and found that, indeed, using the intercept does improve estimation of Beta out-of-sample –  Quant Guy Oct 1 '11 at 16:40

Here's an answer from a purely statistical point of view: http://www.duke.edu/~rnau/regnotes.htm#constant

And another from Cross Validated: http://stats.stackexchange.com/questions/7948/when-is-it-ok-to-remove-the-intercept-in-lm

The lean in both cases is to include the intercept unless there is a strong theoretical reason.

A more satisfying answer would be from the asset-pricing literature Ironically, when you perform an two-stage cross-sectional asset-pricing test (see Cochrane 2005) the procedure is to include the intercept and test whether it is statistically different from zero. However, APT theory argues that the intercept can be ignored. The best way to answer this would be empirical research on the out-of-sample performance of models with and without the intercept...

-