Fama–MacBeth Risk Premia and Errors in Estimated Betas

# Fama/Macbeth Regression - negative estimate for market premium


# Fama/Macbeth Regression - negative estimate for market premium












I just conducted a Fama-Macbeth regression to estimate the risk premia of Mkt-Rf, HML and SMB. As a result, I got a negative risk premium for Mkt-Rf which makes no sense in my opinion. As I couldn't find any mistakes in the regression I did it again with the specification of no constant resulting in risk premia as I would expect them. As nice as these results are I don't think holding the constant at zero is correct, so does anyone of you have an idea what went wrong?

Thanks!

## Answer by kurtosis (score 4)

https://quant.stackexchange.com/a/57501

It sounds like you have estimated a bunch of $\beta$s for excess returns $R_i=r_i-r_f$ and $R_M=r_M-r_f$ above the risk-free rate $r_f$ -- and then run the following model: $$ \bar{R}_i = \gamma_0 + \gamma_M \hat\beta_{iM} + \gamma_{SMB} \hat\beta_{iSMB} + \gamma_{HML} \hat\beta_{iHML} + \eta_i. $$

This is similar to the CAPM "testing" setup of Lintner (1965), Miller and Scholes (1972), and Fama and Macbeth (1973). Note that all of those find attenuated values for $\hat\gamma_M$ ("market risk premium").

The problem is that your $\beta$s are random variables: any $\hat\beta$ has noise in it since you have estimated it. This leads to a classic errors-in-variables problem where $\hat\gamma_0$ will tend to be biased away from 0 while $\hat\gamma_M$, $\hat\gamma_{SMB}$, and $\hat\gamma_{HML}$ will be biased toward 0. (With multiple $\beta$s, this is less clean since collinearity might bias one much closer to 0 while the other is estimated farther from 0.)

This setup may also has an issue inherent to some uses of Fama-Macbeth: if you try to create portfolios with maximal beta dispersion, you are inherently sorting the data which induces a mechanical reversion to the mean effect. That can yield spurious results as well.

I do not have high hopes for you fixing this. Kandel and Stambaugh (1987) tried fixing the CAPM tests and they were able to do a little better using a zero-beta approach; however, they found that $\hat\gamma_0$ and $\hat\gamma_M$ were biased proportional to the efficiency of the market proxy. Given that SMB is often significant, we have evidence that a more broad-based index than the S&P 500 could be useful -- which means your market index is not efficient and $\hat\gamma_M$ should be expected to be biased.

## Answer by QuantStats (score 0)

https://quant.stackexchange.com/a/49886

By negative risk premium, I am assuming you are referring to a negative $\beta_i$, the slope parameter for $r_{m,t}-r_f$.

For simplicity, I am going to use the simple CAPM model without the augmented Fame-French three factors here. The interpretation will be the same for the augmented model. The simple CAPM model is as follows, $$r_{i,t} = \alpha_i+\beta_i (r_{m,t}-r_f) +\varepsilon_{i,t}.$$

By the simple linear regression formula, it can be shown that $$\hat{\beta_i}=\hat\rho_{r_i,r_m}\frac{\hat\sigma_{r_i}}{\hat\sigma_{r_m}}.$$

Since $\hat\sigma_{r_i},\,\hat\sigma_{r_m}>0$, a negative $\hat{\beta_i}$ implies a negative $\hat\rho_{r_i,r_m}$, i.e. a negative correlation between the asset and the market return.

Does it make sense to have a negative correlation, and how can one interpret this?

A negative correlation means the asset return tends to be higher (lower) when the market return is doing poorly (well). For example, an investor may hold this asset as an insurance against a recession.

Finally, you should not impose an intercept term of zero for a linear regression, unless all of your variables ($y$ and $X$) are already demeaned, otherwise, the regression will return misleading results.