# How do I control for a firm's “factor loadings” based on the Fama French model in a regression model?

I asked this question before, but in the wrong community (sorry):

I want to explain stock returns in a regression model. Besides regressing against my main explanatory variables, I want to control for at least the most common risk factors. In a paper (reference below), the authors state that they "[...] control for the firm’s factor loadings based on the Fama-French three-factor model [...]" (p. 13).

This sounds as if they include the factor loadings as explanatory variables in their regression. To me, this does not make sense and I guess, I interprete this wrong. I would be very thankful, if somebody could help me to clarify this. How exactly do they control for the firm's factor loadings? Do they include the factor returns, rather than the factor loadings, as explanatory variables?

Thank you very much in advance!

PS: This is the paper I am talking about:

Lins, Karl V., Henri Servaes, and Ane Tamayo. "Social capital, trust, and firm performance: The value of corporate social responsibility during the financial crisis." The Journal of Finance (2017).

• Are your explanatory variables tradable returns (eg. return on Apple stock, return on MSCI index)? Or are they something other than returns (eg. GDP growth)? – Matthew Gunn Jul 6 '17 at 18:49
• Hello @MatthewGunn! They are something other than tradable returns (corporate social responsibility ratings). Does this cause problems since the factor returns are, more or less tradable, returns? – Hans Leifson Jul 6 '17 at 19:48

It sounds like something reasonable/standard to do would be.

1. Sort your companies into five portfolios based upon quintiles of social responsibility.
• Also make a long-short portfolio of the top quintile portfolio minus the bottom quintile portfolio. (This long-short return will be an excess return so when you run the below regression, you would not subtract the risk free rate.)
2. Regress returns on Fama-French factors (and possibly momentum) to control for those risk factors.

For example, to compute Jensen's alpha relative to the Fama-French three factor model you would run the following regression for portfolio $i$:

$$R_{it} - R^f_t = \alpha_i + \beta_{i,1} \mathit{RMRF}_t + \beta_{i,2} \mathit{SMB}_t + \beta_{i,3} \mathit{HML}_t + \epsilon_{it}$$

Or for five factor model: $$R_{it} - R^f_t = \alpha_i + \beta_{i,1} \mathit{RMRF}_t + \beta_{i,2} \mathit{SMB}_t + \beta_{i,3} \mathit{HML}_t + \beta_{i,4}\mathit{CMA}_t + \beta_{i,5} \mathit{RMW}_t + \epsilon_{it}$$

The $\alpha_i$, Jensen's alpha, is the average return above and beyond what would be expected based upon covariance with the various risk factors.

Factor returns etc... are on Ken French's website.

You're forming portfolios based upon some signal and checking against some asset pricing model by estimating Jensen's alpha. Some call this forming calendar time portfolio sand it naturally corrects standard-errors for cross-sectional correlation in returns. Calculate heteroscedasticity consistent standard errors.

### Computing abnormal returns

The basic idea of abnormal returns is that they are returns minus some expectation of what returns should be given an asset pricing model.

$$\mathit{AR}_{it} = R_{it} - \operatorname{E}[R_{it} \mid \mathcal{F}]$$

For example, under the Fama-French three factor model, the abnormal return would be:

$$\mathit{AR}_{it} = R_{it} - R^f_t - \left( \beta_1 \mathit{RMRF}_t + \beta_2 \mathit{SMB}_t + \beta_3 \mathit{HML}_t \right)$$

where the betas are computed using a time-series regression.

If you regress abnormal returns on stuff, you should cluster standard errors by time because of cross-sectional correlation.

• Thank you very much for this answer @Matthew Gunn! It will certainly help me with my thesis. You even gave me hints regarding appropriate standard errors! However, I came across criticism concerning such kind of portfolio studies, because effects of CSR might be "drowned by noise" (or something like that, I can provide some sources if you like). That's why I thought about regressing returns of single stocks against social responsibility and control variables, including the Fama-French factors. Do you have any idea how I could do so? – Hans Leifson Jul 7 '17 at 16:02

To my knowledge, there are two options which can be utilized in the Fama-French Five-Factor Model.

1. Multivariate multiple regression model
2. Use MANOVA but you should be using a purposive or judgmental sampling technique.it means that before going to select any sample, you should use inclusion and exclusion criteria, which is possible through purposive sampling. also, this is the primary way to control errors and any biasness in the independent variables. As, a result, you should be able to control errors and biasness before data analysis takes place.if you don't do this use a random sampling technique, and then besides MANOVA, you should also utilizing MANCOVA for removing errors and biases in the mean population of the dependent variables. Thus, you should use Covariate as a controlling factor.
3. You should use dummy variables as a covariate in MANCOVA analysis. Examples, psychological factors, active manger, passive managers, etc. Thus, we use 1 and 0 in the inclusion of dummy variables. as 1 for active and 2 for passive.
4. We use covariate for confounding or extraneous factors. These are those factors that don't have any correlation with independent variables but directly correlated with dependent variables. ALSO, in another language, extraneous factors have a correlation with independent variables when we using MANOVA not MANCOVA because that error term first affects independent variables and then indirectly dependent variables.
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