# 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!

• Hi: It would be useful if you wrote the model down so that the question becomes clearer. – mark leeds Feb 11 at 17:06
• 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? – Matthew Gunn Feb 11 at 21:54

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.