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.
 A: 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-Shanken (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...
