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I want to test the CAPM. I have collected monthly data from 1-9-2012 to 1-9-2017. I have data from the return of Standard&Poor's 500 and data from various US companies. The beta's seem right, but my goodness of fit ($R^2$) are low: Ameren 0.0614, Apple 0.2456. My question is: Can this be right? If you're testing the CAPM or am I doing something wrong. I used the CAPM regression $R_{i,t}-R_{f,t}=\alpha_i+\beta_i(R_{M,t}-R_{f,t})+e_i$.

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    $\begingroup$ What estimation procedure are you using? Are you running regressions in two stages: one to find the betas and the other to test the CAPM? If this is the case, which regression are you talking about when you talk about the $R^2$ ? $\endgroup$ – jmbejara Sep 26 '17 at 2:19
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    $\begingroup$ I am using OLS. I just one regression: the regression descriped above for 20 stocks. Fama-Macbeth had in their 1973 paper beautifull R^2's, so I was wondering if I did something wrong. But my question was answered splendidly by a number of people below. My thanks! $\endgroup$ – Cardinal Sep 27 '17 at 13:22
  • $\begingroup$ OK. Sounds good. But just to clarify, the Fama-Macbeth procedure uses two regressions. The first is a time series regression to obtain the betas. The second is a cross-sectional regression to regress the risk-premia of each security on their betas. The CAPM hypothesis puts restrictions on the second regression, not the first. So, since your question is about interpreting the first regression, your question doesn't really have anything to do with testing the CAPM. This confusion is why I claimed that the question was probably a little ambiguous. $\endgroup$ – jmbejara Sep 27 '17 at 22:02
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$\beta$ is the measure of the sensitivity of stock returns to market returns. This has nothing to do with the value of $R^2$.

Your results appear to be fine, you can get significant beta estimates but low $R^2$. Why?

As measured by $R^2$, 24.56% of variation in Apple returns is accounted for by the variation in the market index, $S$&$P 500$. Clearly, there must be other factors that determine the remaining 75.44$% of variation in the Apple returns. Other factors that you could include in the model are copmany size and market to book value (Fama and French Three-Factor Model). By including these omitted variables in your new model, you may achieve higher r-squared value.

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First note the variables $\alpha_i$ and $e_i$ are completely degenerated, there's no way to tell one apart from the other, unless you have a strong prior on them. The model should look something like

$$ R_i = R_f + \beta (R_{m,i} - R_f) + e_i \tag{1} $$

which can be rewritten in the form

$$ R_i = \beta_0 + \beta R_{m,i} + e_i \tag{2} $$

For the example below I selected Apple as my asset (with returns labeled rasset) and S&P 500 as the market index (with returns labeled rmarket), this for the window 2012-09-01 to 2017-09-01.

This is a summary of OLR applied to Eq (2) enter image description here

You're right, the correlation between the two is low, which you can actually already see from a plot of the returns

enter image description here

However, you can see from the table above that

$$ P(>|t|) = 0 $$

for the coefficient $\beta$, which means you're almost 100\% sure that $\beta\not = 0$

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  • $\begingroup$ This question doesn't (at least directly) address the OP's concern over the $R^2$. $\endgroup$ – jmbejara Sep 26 '17 at 19:54
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Note: I'm adding this simply to supplement the other answers.

Also, I'm going to assume that you're talking about the $R^2$ of the time series regression of security $i$ on the market portfolio $M$ $$ R_{it} = \alpha_i + \beta_i R_{Mt} + \epsilon_{it}. $$ As @london described in his answer, the $\beta_i$ is simply the sensitivity of security $i$ to the market. In the Sharpe-Lintner CAPM, as well as in other variations, systematic risk is then defined as $\beta^2 \cdot \sigma^2_{R_M}$ and idiosyncratic risk is $\sigma^2_{\epsilon_i}$.

Thus, the $R^2$ (r-squared) of this regression should not be interpreted so much as "goodness of fit." Rather, it is simply a measure of the ratio of ratio of systematic risk to total risk (systematic plus idiosyncratic risk). As @london described, total risk includes all the other things that drive security $i$'s returns. To see this, note that the definition of $R^2$ in the population sense is $$ R^2 = \frac{\beta_i^2 \sigma^2_{M}}{\beta_i^2 \sigma^2_M + \sigma_\epsilon^2}. $$

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