# Using Regression to Determine whether the CAPM holds

I am taking a beginners course in econometrics and have some questions regarding regressions and the CAPM.

Using data from Yahoo Finance and Kenneth French I run a regression according to the specification:

$$R_{i,t}-r_{f} = \alpha_{i}+\beta_{1}(R_{m,t}-r_{f})+\varepsilon_{i,t}$$

Main results are:

Intercept: -0.0151181
Mkt-RF: 0.016925581


Assume for now I have calculated an estimator for HAC standard errors.

Under the first OLS assumption and assuming the risk free rate is non-random

$$\mathbb{E}(R_{i,t})= \alpha_{i} + r_{f} + \beta_{1}\mathbb{E}(R_{m,t}-r_{f})$$

Therefore $\alpha_{i}$ is the error/deivation from the CAPM.

Question 1: How do I determine whether this error is large?

Ideas 1: Using a t-test I can establish the null hypothesis to be $H_{0}:\ \alpha_{i}=0$. If I reject the null then it is statistically different from 0. However, does this constitute in a large pricing error?

Question 2: How do I determine whether the CAPM holds in this regression?

Ideas 2: I want to apply some sort of a joint test that $H_{0}:\ \alpha_{i}=0,\ \beta_{1} \neq 0$. Is this possible? As far as I know, I can only apply an F-test if all restrictions in the null equate to 0.

Thank your for your time and help.

• Daily data? Monthly? The estimated market beta is 0.017? If your return is from a stock, that beta is almost certainly not right (betas should be near 1). Double check you are matching the dates properly? – Matthew Gunn Nov 2 '16 at 14:50
• @MatthewGunn. The data is monthly. 5 years; equating to 60 months of data. Stock data is General Motors. Market data is from Kenneth French and specifically using mba.tuck.dartmouth.edu/pages/faculty/ken.french/ftp/… There is a little problem interpreting the data from this file. I am not sure if it is my Excel but the dates do not render properly. – Gustavo Louis G. Montańo Nov 2 '16 at 20:14
• @MatthewGunn. I have uploaded my working file if it helps. It is located here: imath.com.au/main.xlsx – Gustavo Louis G. Montańo Nov 2 '16 at 20:40
• Kenneth French data is in %, transform it to decimal before computations. Betas are around 1. – Robert Nov 3 '16 at 3:22
• @Robert. Oh wow! I did not realise this. Thank you! Yes the values look far more realistic now. Do you have any ideas about my original question? – Gustavo Louis G. Montańo Nov 3 '16 at 4:49

Let's say we have 25 portfolios $i=1, \ldots, 25$.

Consider the time-series regressions for each portfolio $i$.

$$R_{it} - R^f_t = \alpha_i + \beta_i \left( R^m_t - R^f_t \right) + \epsilon_{it}$$

If all the right hand side variables in your time-series regression are tradeable[1], then the $\alpha_i$ in your time series regression are equivalent to the residuals in your cross-sectional regression of expected returns on market betas.

$$\mathrm{E}[R_i - R^f] = \gamma \beta_i + \alpha_i$$

Recall that the CAPM theory implies that expected excess returns $\mathrm{E}[R_i - R^f]$ are linear in their market betas $\beta_i$. To test the CAPM, you want to test whether all the $\alpha_i$ are jointly zero. In statistics, this is called an F-test, and in finance, its fancy name is the Gibbons-Ross-Shaken (GRS) test.

You could also run the cross-sectional regression (estimating $\gamma_0$ and $\gamma_1$) $$\mathrm{E}[R_i - R^f] = \gamma_0 + \gamma_1 \beta_i + \alpha_i$$ and see if $\gamma_1$ even has a positive slope.

Basically, the CAPM doesn't work. Checkout Fama-French CAPM Theory and Evidence if you want to go deep.

You can download the 25 Fama-French size, book to market portfolios and test the CAPM on them. It will do horribly in predicting the cross-sectional variation of average returns.

### Basic summary:

You want to run the time series regressions $R_{it} - R^f_t = \alpha_i + \beta_i \left( R^m_t - R^f_t \right) + \epsilon_{it}$ on portfolio/security $i$ and then jointly test whether all the $\alpha_i$ are zero.

There's the classic quote of Box that all models are wrong but some are useful. In some sense, the question is whether the $\alpha$s are large, not whether they can be statistically distinguished from zero.

If you're running this on monthly data (which is standard), $\alpha_i$ will be in units of abnormal monthly returns. 1 percent per month would be absolutely huge. 0.05 percent per month is rather meh.

[1] The market portfolio of all stocks etc... is tradeable. GDP growth would be an example of something that is not tradeable.

For references, search for John Cochrane asset pricing time-series regression cross-sectional regressions etc...

• Thank you for your response @MatthewGunn. I have calculated OLS regressions for my 10 stocks. They all have relatively low alpha's. Do you know how I can create a joint hypothesis on the alpha's across regressions on STATA? – Gustavo Louis G. Montańo Nov 5 '16 at 8:51
• @GustavoLouisG.Montańo I don't know how to do that in Stata, but what you want to run what's known in finance lingo as a GRS test. It's basically a standard F-test for the joint significance of the alphas. – Matthew Gunn Feb 11 '17 at 3:21