How to perform cross-sectional asset pricing regression?

Question

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 1

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.