Choosing Whether to Include an Intercept in Factor Regressions
Summary
The document considers whether to constrain the intercept to zero when estimating a security’s factor beta from a single-factor time-series regression. Asset-pricing theory implies a zero intercept for excess returns, while practical regressions may produce statistically significant nonzero estimates. The estimated betas may then feed into a cross-sectional risk model.
The responses describe competing considerations rather than resolve them. One notes that commercial risk models often use predicted or fundamental betas, with an approach that appears to include an intercept. Another gives the general statistical recommendation to retain an intercept unless theory strongly justifies removing it. In two-stage asset-pricing tests, the intercept is included and tested, even though APT motivates ignoring it. The document offers no empirical comparison of out-of-sample accuracy, so it does not establish which specification performs better for risk-model beta estimation.
Key ideas
- APT implies a zero intercept in the excess-return time-series regression under its theoretical assumptions.
- Practical regression estimates can yield intercepts that differ statistically from zero.
- Some commercial predicted-beta methods appear to include an intercept.
- The document leaves the accuracy trade-off unresolved and points to out-of-sample comparison as a useful test.
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Full text
# Efficiency vs. Robustness - To use a constant or not in single factor time-series regression? # 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. ## Answer by Tal Fishman (score 2, accepted) https://quant.stackexchange.com/a/1518 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. ## Answer by Ram Ahluwalia (score 2) https://quant.stackexchange.com/a/1575 Here's an answer from a purely statistical point of view: http://www.duke.edu/~rnau/regnotes.htm#constant And another from Cross Validated: https://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...
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