Estimating Cross-Sectional Factor Premiums with Noisy Betas
Summary
The document addresses whether factor betas that are imprecisely estimated in time-series regressions can still produce a statistically precise risk premium in a cross-sectional regression. The answer is yes in principle, but beta measurement error works against reliable premium estimation. Short samples and noisy returns weaken beta estimates and also undermine the cross-sectional relationship; measurement error in estimated betas creates an errors-in-variables problem and can distort the fitted premium.
The proposed framework first estimates each asset’s beta from its excess returns and a factor, then relates average excess returns across assets to those estimated betas. Cross-sectional dependence in residuals calls for time clustering or a Fama–MacBeth procedure. Having more assets may help, though correlated returns limit the gain. The discussion is conceptual rather than an empirical demonstration, and it does not provide a full correction for beta estimation error. It also cautions against treating arbitrary significance thresholds as the only measure of evidence.
Key ideas
- Noisy time-series beta estimates can, in principle, coexist with a well-measured cross-sectional factor premium.
- Beta estimation error creates an errors-in-variables problem that can bias or weaken premium estimates.
- Cross-sectional residual dependence can be addressed with time-clustered errors or Fama–MacBeth estimation.
- Adding assets may improve precision, but correlated returns limit the benefit.
- The answer outlines challenges without supplying a complete beta-error correction or empirical example.
Tags
Full text
# How to perform cross-sectional asset pricing regression?
# How to perform cross-sectional asset pricing regression?
I'm wondering is that possible to get insignificant beta estimates in the time-series context, but highly significant risk premium associated with that beta in the cross-sectional regression?
Any help would be greatly appreciated!
## Answer by Matthew Gunn (score 7, accepted)
https://quant.stackexchange.com/a/44061
I prefer thinking in terms of well measured vs. poorly measured rather than significant vs. insignificant: arbitrary p-value cutoffs and ignoring sensible priors can both be problematic. On the question, "can poorly measured betas from time-series regressions give rise to well measured factor premiums from cross-sectional regression?" The abstract answer is yes, but you have several problems working against you.
- The large error terms, short samples in the time-series context that lead to poorly estimated $\beta$s will also hurt you in trying to estimate the cross-sectional relationship.
- The worse you measure $\beta$ the worse your error in variables problem is.
- More assets should help estimate the cross-sectional relationship but the high cross-sectional correlation of returns will limit how much this will help.
### Background:
Let $F_t$ denote some factor. You can estimate the beta for a return $R_i$ on the factor with a time-series regression:
$$ R_{it} -R^f_t = \alpha_i + \beta_i F_t + \epsilon_{it}$$
The key idea behind all these factor models is that expected returns should be linearly increasing in the regression beta on the factor. To estimate factor premium $\gamma_F$, you'd like to run the regression:
$$R_{it} - R^f_t = \gamma_0 + \gamma_F \beta_i + u_{it} $$
To do this sensibly, you need to confront several problems:
- Cross-sectional correlation of $u_{it}$. To account for this, cluster standard errors by time or do Fama-Macbeth procedure.
- You don't have $\beta_i$, you have estimate $\hat{\beta}_i$. This creates an error in variables problem. Also as Bayesian logic might suggest, high $\hat{\beta}_i$ tend to be overestimated and low $\hat{\beta}_i$ tend to be underestimated. Confronting this problem is a longer discussion. The worse you measure $\beta_i$, the larger the problem.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.