Testing Whether a Carbon Factor Adds to a Fama–French Model

# How do we know if an additional factor has improved the predictive power of a Fama French 3 factor equity model?


# How do we know if an additional factor has improved the predictive power of a Fama French 3 factor equity model?












We're building a multi-factor model for climate risk by adding an additional factor for carbon risk on top of a Fama and French 3 factor model. This is open source at: https://github.com/opentaps/open-climate-investing

We've tested the additional carbon risk factor and found that it increases the R2 of the regressions: https://github.com/opentaps/open-climate-investing/issues/4

Is this sufficient to say that the additional factor has improved the predictive power of the model? Or should there be other tests done?

## Answer by skoestlmeier (score 2)

https://quant.stackexchange.com/a/85469

One standard method to address your question is spanning regressions, commonly applied in empirical asset pricing. Fama/French (2015) use this method to discuss if the value factor HML is potentially redundant in their five-factor asset pricing model. You may take a look at Section 7 'HML: a redundant factor' of the paper. I will outline the method adjusted for your use case.

Let's start with an extended version of the three-factor model including the market portfolio MKTRF (value-weighted market excess return), the size-factor SMB, and the value-factor HML. Additionally, you have the carbon risk-factor CRF (carbon risk factor), which is the long-short return of tradable assets sorted on their exposure to a carbon characteristik at the firm-level.

To assess if CRF adds to the explanation of average returns, you regress CRF on all other factors:

$$CRF_t = \alpha + \beta_{0} MKTRF_t + \beta_{1} SMB_t + \beta_{2} HML_t + \epsilon_t.$$

You may drop CRF from your model, if $\alpha$ is close to zero, i.e., not significantly distinguishable from zero. Similarly, you may test all other factors by regressing each of them on all others.

However, even if CRF may be irrelevant in case of $\alpha \approx 0$, there could be a large carbon risk premium (i.e., the average return of CRF is significantly different from zero) targeted by funds or money managers, captured by exposures to others factors (e.g., CRF could be correlated with HML).

To test the above argument, define CRFO (orthogonal CRF) as the sum of the intercept and residual from the regression of CRF on all other factors, i.e.:

$$CRFO_t = \alpha + \epsilon_t$$

from the above regression. In essential, you take the component of CRF orthogonal to all other factors. You then run the regression

$$r_{i,t} - rf_t = a_i + b_i MKTRF + s_i SMB + h_i HML + c_i CRFO + \epsilon_{i,t}$$

using portfolio returns $r_{i,t}$ of appropriate test portfolios (e.g. the intersection of $5\times5$ independently sorted portfolios) in excess of the risk free rate $rf_t$. If the estimated slope coefficient $\hat{c}_i$ is insignificant, CRF does not add to the asset pricing model.

### References

Fama/French (2015), A five-factor asset pricing model, Journal of Financial Economics 116, p. 1-22.

## Answer by phdstudent (score 1)

https://quant.stackexchange.com/a/82495

Edit answer ... it seems that the right thing is to compute GRS statistics.

- Run $N$ separate regressions for each asset $i$

- Get the sample vector of residuals $\varepsilon_{iT}$

- Calculate the sample covariance matrix of the residuals $\hat{\Sigma}$

- The covariance matrix of the $\alpha'$s is the covariance matrix of the sample means of the residuals. Thus, the covariance matrix of the $\alpha'$s is $\hat{\Sigma} / T$

If we know the residual covariance matrix, the multivariate test is $$ \frac{T}{\left(1 + \hat{\theta}_p^2\right)} \hat{\alpha}_T' \Sigma^{-1} \hat{\alpha}_T \sim \chi^2(N) $$

Gibbons, Ross, and Shanken (1987) show us that when we have to estimate the covariance matrix, the joint test becomes $$ \frac{T - N - 1}{N} \cdot \frac{1}{1 + \hat{\theta}_p^2} \hat{\alpha}_T' \hat{\Sigma}^{-1} \hat{\alpha}_T \sim F(N, T - N - 1) $$

$\hat{\Sigma}$ is the estimated covariance matrix of the residuals without adjusting for the degrees of freedom. Estimating the covariance matrix is hard when there are many assets $N$ relative to the number of observations $T$.

The noise introduced in the estimation of $\hat{\Sigma}$ makes it harder to reject the model. Since the residuals are estimated by minimizing their variance, in-sample spurious correlation makes them too small on average, and that’s why the small sample adjustment comes about.

As $T \to \infty$, the statistic converges to the chi-squared version stated for the case with known covariance matrix.