6
$\begingroup$

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!

$\endgroup$
  • 2
    $\begingroup$ Hi: It would be useful if you wrote the model down so that the question becomes clearer. $\endgroup$ – mark leeds Feb 11 at 17:06
  • 1
    $\begingroup$ What do you think? If you estimate $\beta\_i$ poorly (in time-series regression $R_{it } - R^f_t = \alpha + \beta_i F_t + \epsilon_{it}$ where $F_t$ is some factor), what issues does that cause for estimating the cross-sectional relationship between $\beta_i$ and expected return $\operatorname{E}[R_i - R^f]$? Will increasing the number of test assets help? Why and/or why not? $\endgroup$ – Matthew Gunn Feb 11 at 21:54
2
$\begingroup$

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.
$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.