# 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. – Ram Ahluwalia Jul 21 '11 at 4:37