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I am studying econometrics from the third edition of 'Introduction to Econometrics' by James H. Stock and Mark W. Watson.

On page 166 it digresses into the beta of the stock. It says

Those betas typically are estimated by OLS regression of the actual excess return on the stock against the actual excess return on a broad market index.

My understanding by the language is that the beta of the stock is the coefficient of the regressor, which is the market index's excess return. That is:

$$(R - R_{f}) = \beta_{0} + \beta(R_{m}-R_{f})+u.$$

Thus to estimate the return of a stock

$$\hat{R}-R_{f}=\hat{\beta_{0}}+\hat{\beta}(R_{m}-R_{f}).$$

However for some odd reason when I do homework problems it uses the following equation to estimate returns:

$$\hat{R}-R_{f}=\hat{\beta}(R_{m}-R_{f}).$$

That is it imposes $\beta_{0} = 0$.

I have no doubt my understanding is incorrect. Any assistance would be greatly appreciated.

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  • $\begingroup$ This is essentially a finance question, not a statistical question. Should it be migrated anywhere? $\endgroup$ – Richard Hardy Sep 1 '16 at 13:25
  • $\begingroup$ I suggest quant.stackexchange.com $\endgroup$ – mic Sep 16 '16 at 9:38
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As Stephen mentions, the confusion is between: (1) the CAPM vs. (2) the market model.

Let $R^f$ denote the risk free rate. We often work with excess returns, which involves subtracting of the risk free rate.

Some simple models for expected returns

  1. ``Market model" $$ R_t - R^f = \alpha + \beta\left(R^m_t - R^f \right) + \epsilon_t $$ $$ E\left[ R_t \right] - R^f = \alpha + \beta\left(E[R^m_t] - R^f \right) $$ The market model is a simple, statistical model and can be justified by assuming that the joint distribution of monthly stock returns is multivariate normal.

  2. Capital Asset Pricing Model (CAPM) $$ E\left[ R_t\right] - R^f = \beta\left(E[R^m_t] - R^f \right) $$ The CAPM is an economic theory that expected excess returns of a stock are linear in the excess return of the market, that $\alpha = 0$ from the market model regression.

Be aware that the CAPM doesn't work. It's all over MBA corporate finance, but asset pricing people find it useless. Something less crazy to use would be the Fama-French 3 Factor Model.

Example of how to use the CAPM (or any of these factor asset pricing models).

  1. Compute excess returns: $ R_{i,t} - R^f_t$
  2. Regress excess returns on excess returns of the market and a constant (i.e. run the market model regression). $$ R_{i,t} - R^f_t = \alpha_i + \beta_i \left( R^m_t - R^f_t \right) + \epsilon_{i,t}$$
  3. Ignore the estimated $\hat{\alpha}$.
  4. Your estimated expected excess return according to the CAPM is $\hat{\beta_i} E[R^m_t - R^f_t] $.
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    $\begingroup$ Thank you for your answer, Matthew. When running regressions is the risk free rate assumed to be constant or is it too a random variable? $\endgroup$ – Gustavo Louis G. Montańo Sep 1 '16 at 23:26
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    $\begingroup$ @GustavoLouisG.Montaño Frankly, you see it done both ways, but it's better to treat it as time varying. In some sense, the risk free rate has it's own dynamics, and these equity pricing models are just building a model of expected returns relative to that. A homework problem might treat $R^f$ as constant. If you have discretion, something easy is to grab the 1 month risk free rate from: mba.tuck.dartmouth.edu/pages/faculty/ken.french/… It's the last item in the 3 factor csv files. It's mostly important for historical data < 1980s when inflation rates were higher. $\endgroup$ – Matthew Gunn Sep 1 '16 at 23:44
  • $\begingroup$ Eg. for recent data, the 1 month risk free rate is basically 0, and subtracting 0 doesn't change much of anything. Back in the 1970s though when inflation was huge and volatile, it's quite a different story. $\endgroup$ – Matthew Gunn Sep 1 '16 at 23:49
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If I recall correctly, it has something to do with a constant called Jensen's alpha and the extension to something called a multi-factor model. During my classes (and also wikipedia), the CAPM was stated as follows: \begin{align} \mathbb{E}[R_i] = R_{f} + \hat{\beta_{1}}(\mathbb{E}[R_{m}] - R_{f}) \end{align} When taking excess returns, one would simply have: \begin{align} \mathbb{E}[R_i] - R_{f} = \hat{\beta_{1}}(\mathbb{E}[R_{m}] - R_{f}) \end{align}

And this is mostly fine. However, this relation assumes that all information can be taken from just the market portfolio in excess to the risk free rate (i.e. $\hat{\beta_0} =0$). However, it could be the case that your specific return $R_i$ could deviate from the returns seen on the market portfolio. In this case, let us add a constant $\beta_0$ to the model, so we have:

\begin{align} R_i - R_f= \beta_0 + \beta_{1}(R_{m} - R_{f}) + \epsilon \end{align} and when taking expectations \begin{align} \mathbb{E}[R_i] - R_{f} =\hat{\beta_0}+ \hat{\beta_{1}}(\mathbb{E}[R_{m}] - R_{f}) \end{align} Now, suppose you calculate the estimates of $\hat{\beta_0} , \hat{\beta_1}$ using OLS. You will thus now have 2 estimates for the parameters. The way you will generally use the $\hat{\beta_0}$ is by performing a simple t-test test on it:

1: Before testing you have to make sure that the standard errors are correct. Note that the t-test relies directly on the standard errors being homoskedastic, so keep in mind that you will probably have to use robust standard errors, such as Eicker-Huber-White or Newey-West.

2: After you made sure the standard errors from your regression equation are in order you can simply look at the individual t-test for $\hat{\beta_0}$. On the basis of the hypothesis of \begin{align} H_0 : \beta_0 = 0 \quad vs \quad H1: \beta_0 \ne 0 \end{align} On the basis of rejecting the null-hypothesis or failing to reject it, we can distinguish between three cases.

1: $\hat{\beta_0} =0$. In this case, your expected return $\mathbb{E}[R_i] - R_f$ neither outperforms or under performs with respect to the market (which is what your homework is assuming here).

2:$\hat{\beta_0} > 0$. Here, your return is outperforming the market. In this case, the market portfolio can not fully explain your return and additional factors are needed (three-factor model or multi factor models).

3:$\hat{\beta_0}< 0$ . Same as above, only this time the return is under performing with respect to the market portfolio.

All in all, whether or not to include this term depends on whether you assume that no additional factors are needed. If the question states that $R_i$ can be completely explained by just the market portfolio you can simply set this term to zero. In practice, you would perform a statistical hypothesis to test whether it is significantly different from zero or not (Jensen's alpha).

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  • $\begingroup$ Thank you for going out of your time to answer my question, @Stephan. Quick question: If $\hat{\beta_{0}} = 0$ and $\hat{\beta_{1}} > 0$ doesn't the security outperform the market? That is, doesn't the security obtain excess returns higher than the market's? $\endgroup$ – Gustavo Louis G. Montańo Sep 1 '16 at 22:57
  • $\begingroup$ When $\hat{\beta_0} = 0$ it is said that the security shows normal performance (in line with the CAPM). When $\hat{\beta_0} > 0$ the security shows out-performance. I've included this in my answer in the three cases. I'll also update on how this $\beta_0$ is generally treated. $\endgroup$ – Stephan Sep 1 '16 at 23:22

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