# How to test the 5 Factor CAPM of Fama & French (2014)?

I would like to conduct a study testing the 5 factor CAPM, using UK stocks.

Does anyone have any suggestions of how I can do this?

Could this task be as simple as regressing average returns for a stock with its different factors?

I'd be grateful for any specific pointers/articles or books I can read!

• Are you an R user or are you more comfortable using Excel? @harry – Rime Mar 6 '15 at 1:38
• Very comfortable with Excel, moderate with Stata and weak with R. I'd ideally use Stata out of the three. – Harry Mar 6 '15 at 7:46
• well it would be useful to provide an example but I believe it will be kind of hard to do if I use excel (as far as pasting the example here) . I am more of an R user and don't know Stata. Essentially you are correct by saying that this can be done by linear regression but as far as I know, these factors aim to explain returns on portfolios. I can provide an example using R if you'd like? @harry – Rime Mar 6 '15 at 12:24
• It'll give me something to look into; I'd be very grateful if you could. – Harry Mar 7 '15 at 0:14

## 1 Answer

Don't just run simple time-series regression to see if you get statistically significant betas. This procedure will not tell you if the factors are actually priced. You run a high risk of finding spurious correlations.

There is a fairly well established standard program to test factor models, called the Fama-MacBeth method. It is based on two sets of regressions. A factor model essentially gives you two (linear) equations that you can estimate using regression analysis: $$r_i = a_i + \beta_{i1} f_1 + … + \beta_{iK} f_K + \epsilon_i \ (1)$$ $$\mu = \lambda_0 + \beta_{i1} \lambda_1 + … + \beta_{iK} \lambda_K \ (2)$$ where $r_i$ is the (realized) return of security $i$, $a_i$ is the intercept of security I (often taken to be the risk-free rate), $\beta_{ij}$ are the factor sensitivities of security $i$ to factor $j$, and $f_j$ are the factor realizations of the relevant factors. Given this structure, the APT implies that given absence of arbitrage, the expected return of asset i, $\mu_i$, is given by the second equation where the $\lambda_j$ are the factor risk premia and $\lambda_0$ is the model’s zero beta parameter.

The standard way to test the factors is to first run time-series regression of equation (1) using rolling windows to obtain the beta parameters. With monthly data, for example, one would usually estimate the regression using 5 years of data and take the estimated betas of that regression as the beta observation for the last date of those 5 years. Then you repeat the same setup by moving the estimation window one month further, and so on. This will give you a time series of beta estimates for each stock.

In the next step, you run cross-sectional regressions for each time $t$ using the returns on the securities at time $t$ as the dependent variable and the beta estimates as the independent variables. This will give you estimates of the factor risk premia for each time. Finally, you take the average of the factor risk premia over time to see whether they are statistically different from zero.

You will have to adjust your standard errors appropriately because of the fact that the betas in the cross sectional regressions are estimated. You can read up on the technical details of the method in Cochrane’s book on asset pricing. More details on some other methods are given in Campbell, Lo, MacKinley’s book on econometrics. You should also check out the paper by Peterson which contains a detailed discussion of the statistical issues with this method. He also published STATA and R codes for this paper on his website.

Campbell, J., Lo, A., MacKinley, A. (1997) The Econometrics of Financial Markets. Princeton University Press

Cochrane, J. (2001) Asset Pricing (Revised Edition). Princeton University Press

Fama, Eugene F.; MacBeth, James D. (1973). "Risk, Return, and Equilibrium: Empirical Tests". Journal of Political Economy 81 (3): 607–636.

Petersen, Mitchell (2009). "Estimating Standard Errors in Finance Panel Data Sets: Comparing Approaches". Review of Financial Studies 22 (1): 435–480.

• This is excellent... Upvote – Rime Mar 6 '15 at 19:04
• @pbr142 After doing some research, I've found Fama-Macbeth regression is more useful for APT based models, whereas Fama-French is more useful for equilibrium tests. Can you offer any expertise on this? quant.stackexchange.com/questions/10065/… and in particular papers.ssrn.com/sol3/papers.cfm?abstract_id=1271935 – Harry Mar 23 '15 at 13:28
• To expand my concern, would the Fama French method be better for my needs as the 5 Factor CAPM is not an APT model? – Harry Mar 23 '15 at 14:15
• Maybe this post will help you. If not, write me again quant.stackexchange.com/questions/17125/… – pbr142 Apr 1 '15 at 16:07