Timeline for CAPM yields very poor fit (low R-squared). Is that normal?
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| Sep 26 at 4:20 | comment | added | mark leeds | Hi Richard: I was referring to the equation right after "or more briefly" in your question. Isn't $\beta_i$ the coefficient to be estimated ? and $r_{mt}$ the regressor ? If not, then I'm even more confused than I thought I was !!!! Thanks. | |
| Sep 25 at 15:38 | comment | added | Richard Hardy | @Kevin, 1. You are describing something that I am not doing. I am using a single period, not an average over many periods. I totally understand why you are recommending what you are recommending, though! 2. In your regression, even if CAPM held exactly, $R^2$ would not be 1 due to sampling variation, as the sample average does not equal the underlying expected value. | |
| Sep 25 at 15:08 | comment | added | Kevin | @RichardHardy I hope not this time :) To test the CAPM, you should regress average returns on betas. So it's a cross-sectional regression: You have, say, 25 portfolios. Then, there are 25 betas (from TS regressions) and 25 average returns (simple TS means). It's now a simple OLS regression by fitting a line through 25 points. In theory, all points should be on that line (deviations from that line are alphas). That line is the SML. The $R^2$ and mean absolute error from that regression are very informative about the fit of the CAPM. | |
| Sep 25 at 14:32 | comment | added | Richard Hardy | @Kevin, might you have jumped the gun again? :) The dependent variable is not the expected value (how could it be if the expected value is never observed?) but a single realization from the random variable, so I do not think the $R^2$ should be 1. | |
| Sep 25 at 13:19 | comment | added | Kevin | @RichardHardy It seems I jumped the gun. Sorry! I was thinking in my comment about the first-stage time series regression of excess stock returns on factors. That $R^2$ is meaningless theoretically (albeit not econometrically). If you then run a cross-sectional regression, then this $R^2$ (ie your $R^2$) is absolutely critical. The CAPM does predict it to be one. A cross-sectional $R^2$ of one would mean there are no alphas/pricing errors (ie, beta explains everything). Your low $R^2$ simply means the CAPM is a bad model for average stock returns. Sorry for rushing in with a comment. | |
| Sep 25 at 6:54 | comment | added | Richard Hardy | @markleeds, in the cross-sectional regression that I am asking about, $\beta_i$ is a regressor. The coefficient I am estimating is $\lambda_\tau$ which can be interpreted as the price of systematic risk. Under $R^2=0$, we have $\lambda_\tau=0$, so the price of systematic risk is 0. Thus, systematic risk does not explain any cross-sectional variation in the expected returns which in my book would suggest the CAPM is useless (I guess?). | |
| Sep 25 at 0:54 | comment | added | mark leeds | Kevin and Marcus and Richard: You're all more into this material than I am but am I correct in saying that, if all volatility is idiosyncratic, then not only does $\alpha_{i} = 0$ but $\beta_{i}$ also. And, in this case, the result is that CAPM does fail because the concept of $\beta$ and its covariance with the market turns out to not exist. | |
| Sep 24 at 20:33 | comment | added | Kevin | @RichardHardy Marcus is absolutely right (of course he is as a top finance researcher!). The CAPM only predicts that alphas are zero: this is a statement about expected returns (means). $R^2$ is about variation in realised returns. A high $R^2$ means a lot econometrically (small standard errors...) but means little economically. Another way to think about it: Size factors or industry portfolios contribute little to explaining alphas but push $R^2$s up. These factors help with performance attribution, risk management, etc but do not necessarily price test assets (explain alphas) | |
| Sep 24 at 10:18 | comment | added | Richard Hardy | Thank you. That makes sense. My question is more about what should be expected in practice for real stock markets. Also, after a point, the lower the $R^2$, the lower the economic importance of the model. (The economic importance under $R^2$ of 0.7 vs. 0.9 is maybe not so different, but under 0.0 vs. 0.2 I think it is substantial.) Since the CAPM is so prominent, I was expecting it to have nonnegligible explanatory power. | |
| Sep 24 at 9:12 | review | Late answers | |||
| Oct 1 at 8:52 | |||||
| Sep 24 at 8:58 | history | edited | Marcus Opp | CC BY-SA 4.0 |
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| S Sep 24 at 8:52 | review | First answers | |||
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| S Sep 24 at 8:52 | history | answered | Marcus Opp | CC BY-SA 4.0 |