# Alpha estimation from factor models

This question makes reference to section 8.4 - Application to performance measure - of the 2007 publication "Performance Measurement for Traditional Investment" by Véronique Le Sourd. You can find the paper here.

In her article, the author states that the implementation of factor models is carried out in two stages (if I understand correctly, she refers to the Fama Macbeth methodology). First, betas are estimated by means of a series of time series regressions of asset returns (one for each asset $$i$$) on factor returns:

$$(1)$$ $$R_{it} = \beta_{i0} + \sum_{k=1}^K\beta_{ik}F_{kt}+\epsilon_{it}$$

Then, lambdas are estimated running a cross-sectional regression at each date $$t$$:

$$(2)$$ $$R_{it} - R_f = \hat\alpha + \sum_{k=1}^K\hat\beta_{ik}\hat\lambda_{kt}+\zeta_{it}$$

After calculating the average risk premiums as:

$$(3)$$ $$\lambda_k = \frac{1}{T}\sum_{t=1}^T\lambda_{kt}$$

She states that fund performance is given by:

$$(4)$$ $$\alpha_i = \bar{R_i} - \bar{R_f} - \sum_{k=1}^K\hat{\beta_{ik}}\lambda_k$$

What is not clear to me is why one would estimate alpha as in equation $$(4)$$ instead of simply considering as alpha the estimate of the coefficient called $$\beta_{i0}$$ in equation $$(1)$$.

For instance, if I were to employ the Fama-French three-factor model to estimate alpha, should I follow the procedure above or simply estimate alpha as the intercept of the following regression?

$$R_{it} - R_{ft} = \alpha_i + \beta_i(R_{mt}-R_{ft}) + s_iSMB_t + h_iHML_t + \epsilon_{it}$$

The advantage of the method is that you can use it regardless of whether the factor is traded or non-traded. If the factor is traded, you are correct, you can use time-series tests and test whether the intercept is zero ($$\beta_{i0}$$) in your case.

We are testing whether $$\beta_{i0}=$$ in:

$$R_{it} = \beta_{i0} + \sum_{k=1}^K\beta_{ik}F_{kt}+\epsilon_{it}$$

In other words, we are testing whether the pricing errors are simply a product of "normal" sample variation or in fact a result of a misspecified model.

The steps to perform time-series regressions (instead of Fama-McBethe) are:

1. Run $$N$$ separate regressions for each asset $$i$$
2. Get the sample vector of residuals $$\epsilon_{it}$$
3. Calculate the variance covariance matrix of residuals $$\hat{\Sigma}$$
4. Gibbons, Ross, and Shanken (1987) show us that when we have to estimate the covariance matrix, the joint test become:

$$\frac{T-N-1}{N}\frac{1}{\theta_p^2} \hat{\beta_{0}}' \hat{\Sigma}^{-1} \hat{\beta_{0}} \sim F(N,T-N-1)$$ where:

$$\theta_p^2 = \frac{\bar{F}^2}{Var_T(F_t)}$$

Check GRS original paper for an economic intuition of this test.