# 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.

• 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, 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.

 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