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!
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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
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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
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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.
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$\begingroup$ many thanks for pointing out the potential issues here! Reading through the error in variables issue from the hyperlink, it seems to me that there would be an underestimation issue in the cross-sectional regression if the beta coefficient from the time-series regression is estimated with error. So, for my case, does it mean that the beta estimates could be higher if there is a measurement error in the beta? If so, then the 'true' effect of that beta risk is likely much greater? Then, the p-value of the risk premium will be even smaller (or even more significant)? $\endgroup$ – Janys Feb 20 at 19:39
