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I have an interpretation problem. As you can see below there's a linear regression output for the CAPM. I don't know how to interpret the significance level. ExIndex has a very low p-value, but the Significance level is 0. So can I reject H0 or not? The same question for the intercept.

Coefficients:
              Estimate     Std. Error  t value  Pr(>|t|)    
(Intercept) - 0.003258     0.001560    -2.089       0.0377 *  
 ExIndex      0.898980     0.106511     8.440     2.3e-15 ***
 Signif. codes:   0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02508 on 258 degrees of freedom Multiple
R-squared:  0.2164,    Adjusted R-squared:  0.2133 
F-statistic: 71.24 on 1 and 258 DF,  p-value: 2.304e-15
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  • $\begingroup$ Welcome to CV. With respect to your questions, lower or smaller values imply greater significance in evaluating the null hypothesis, below some threshold of course. $\endgroup$ – Mike Hunter May 25 '16 at 15:50
  • $\begingroup$ You need to specify exactly what regression you're running and why. Are you estimating market betas? That is, run a time series regression of excess returns (i.e. return minus risk free rate) on excess returns of the overall market to find the coefficient on the market ret? $\endgroup$ – Matthew Gunn May 25 '16 at 16:15
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    $\begingroup$ Are you trying to test the Capital Asset Pricing Model (CAPM)? The CAPM says that expected returns of any tradable security are linear in its market beta (i.e. higher market beta, higher expected returns). The standard ways to do this are either: (1) (easier) test the intercept is equal to zero in the time series regression of excess returns of a security/portfolio/etc... on market excess returns or (2) (don't do this if confusing) run a cross-sectional regression of of average returns on the market betas, correcting for cross-sectional correlation (eg. use Fama-Macbeth or cluster by time). $\endgroup$ – Matthew Gunn May 25 '16 at 16:20
  • $\begingroup$ I am writing my bachelor thesis and have to do the OLS method for the CAPM. I did a linear regression on the one factor model in R with lm(...). I already estimated beta, but I have to interpret this output. So beside the model, what does the significance level mean fin general for the output. Can H0 with beta=0 being rejected? My intention was yes, because the p-value is very low. $\endgroup$ – Ju Li May 25 '16 at 16:48
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p-values are to some extend based on the readers or the analysts perspective, but the general rule is, the closer your value is to zero the more evidence you have against $H_0$. R uses a star rating system according to commonly used significant (cut-off points) levels,so as to help the readers decision. For example the three stars next to the p-value of ExIndex suggest that there is very strong evidence against the hypothesis that $\beta_1 = 0$, while on the other hand the one star next to the p-value of the intercept suggests that the hypothesis that $\beta_0 = 0 $ might be rejected at a cut of point of 5%, while is not rejected at 1% and 0.1% cut of point. It is worth noting that things like practical and statistical significant are to be taken into account when dealing with p-values.

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