## Measurement Error in Beta

Roll's critique tells us that CAPM tests are handicapped from the outset. But suppose that we could get past Roll's problem by obtaining data on the returns of the true market portfolio. We still would have to deal with the statistical problems caused by measurement error in the estimates of beta from the first-stage regressions.

It is well known in statistics that if the right-hand-side variable of a regression equation is measured with error (in our case, beta is measured with error and is the right-hand-side variable in the second-pass regression), then the slope coefficient of the regression equation will be biased downward and the intercept biased upward. This is consistent with the findings cited above, which found that the estimate of 70 was higher than predicted by the CAPM and that the estimate of 71 was lower than predicted.

Indeed, a well-controlled simulation test by Miller and Scholes6 confirms these arguments. In this test a random-number generator simulated rates of return with covariances similar to observed ones. The average returns were made to agree exactly with the CAPM expected return-beta relationship. Miller and Scholes then used these randomly generated rates of return in the tests we have described as if they were observed from a sample of stock returns. The results of this "simulated" test were virtually identical to those reached using real data, despite the fact that the simulated returns were constructed to obey the SML, that is, the true 7 coefficients were 70 = 0, 71 = rM - rf and 72 = 0.

6 Miller and Scholes, "Rate of Return in Relation to Risk: A Reexamination of Some Recent Findings."

CHAPTER 13 Empirical Evidence on Security Returns 389

This postmortem of the early test gets us back to square one. We can explain away the disappointing test results, but we have no positive results to support the CAPM-APT implications.

The next wave of tests was designed to overcome the measurement error problem that led to biased estimates of the SML. The innovation in these tests, pioneered by Black, Jensen, and Scholes (BJS),7 was to use portfolios rather than individual securities. Combining securities into portfolios diversifies away most of the firm-specific part of returns, thereby enhancing the precision of the estimates of beta and the expected rate of return of the portfolio of securities. This mitigates the statistical problems that arise from measurement error in the beta estimates.

Obviously, however, combining stocks into portfolios reduces the number of observations left for the second-pass regression. For example, suppose that we group our sample of 100 stocks into five portfolios of 20 stocks each. If the assumption of a single-factor market is reasonably accurate, then the residuals of the 20 stocks in each portfolio will be practically uncorrelated and, hence, the variance of the portfolio residual will be about one-twentieth the residual variance of the average stock. Thus the portfolio beta in the first-pass regression will be estimated with far better accuracy. However, now consider the second-pass regression. With individual securities we had 100 observations to estimate the second-pass coefficients. With portfolios of 20 stocks each we are left with only five observations for the second-pass regression.

To get the best of this trade-off, we need to construct portfolios with the largest possible dispersion of beta coefficients. Other things being equal, a sample yields more accurate regression estimates the more widely spaced are the observations of the independent variables. Consider the first-pass regressions where we estimate the SCL, that is, the relationship between the excess return on each stock and the market's excess return. If we have a sample with a great dispersion of market returns, we have a greater chance of accurately estimating the effect of a change in the market return on the return of the stock. In our case, however, we have no control over the range of the market returns. But we can control the range of the independent variable of the second-pass regression, the portfolio betas. Rather than allocate 20 stocks to each portfolio randomly, we can rank portfolios by betas. Portfolio 1 will include the 20 highest-beta stocks and Portfolio 5 the 20 lowest-beta stocks. In that case a set of portfolios with small nonsystematic components, eP, and widely spaced betas will yield reasonably powerful tests of the SML.

Fama and MacBeth8 used this methodology to verify that the observed relationship between average excess returns and beta is indeed linear and that nonsystematic risk does not explain average excess returns. Using 20 portfolios constructed according to the BJS methodology, Fama and MacBeth expanded the estimation of the SML equation to include the square of the beta coefficient (to test for linearity of the relationship between returns and betas) and the estimated standard deviation of the residual (to test for the explanatory power of nonsystematic risk). For a sequence of many subperiods they estimated for each subperiod, the equation ri = 70 + 71 Pi + 72P2 + 73°- (e) (13.5)

The term 72 measures potential nonlinearity of return, and 73 measures the explanatory power of nonsystematic risk, °(ei). According to the CAPM, both 72 and 73 should have coefficients of zero in the second-pass regression.

7 Fischer Black, Michael C. Jensen, and Myron Scholes, "The Capital Asset Pricing Model: Some Empirical Tests," in Michael C. Jensen, ed., Studies in the Theory of Capital Markets (New York: Praeger, 1972).

8 Eugene Fama and James MacBeth, "Risk, Return, and Equilibrium: Empirical Tests," Journal of Political Economy 81 (March 1973).

Table 13.1 Summary of Fama and MacBeth (1973) Study (all rates in basis points per month)

Period |
- |
- |
- |
- |

Av. r. |
13 |
2 |
9 |
26 |

Av. 7„ - r, |
8 |
10 |
8 |
5 |

Av. - r,) |
0.20 |
0.1 1 |
0.20 |
0.10 |

- |
130 |
195 |
103 |
95 |

Av. 7, |
114 |
118 |
209 |
34 |

Av. |
1.85 |
0.94 |
C.. O Î3 |
0.34 |

Av. 7 |
- |
- |
- |
0 |

Av. t(y2) |
- |
- |
- |
0 |

Av. 73 |
516 |
817 |
- |
960 |

Av. |
1.1 1 |
0.94 |
- |
1.1 1 |

Av. ff-SQR |
0.31 |
0.31 |
0.32 |
0.29 |

Fama and MacBeth estimated equation 13.5 for every month of the period January 1935 through June 1968. The results are summarized in Table 13.1, which shows average coefficients and i-statistics for the overall period as well as for three subperiods. Fama and MacBeth observed that the coefficients on residual standard deviation (nonsystematic risk), denoted by y3, fluctuate greatly from month to month and were insignificant, consistent with the hypothesis that nonsystematic risk is not rewarded by higher average returns. Likewise, the coefficients on the square of beta, denoted by y2, were insignificant, consistent with the hypothesis that the expected return-beta relationship is linear.

With respect to the expected return-beta relationship, however, the picture is mixed. The estimated SML is too flat, consistent with previous studies, as can be seen from the fact that 70 - rf is positive, and that 7 is, on average, less than rM - rf. On the positive side, the difference does not appear to be significant, so that the CAPM is not clearly rejected.

CONCEPT CHECK ^ QUESTION 3

According to the CAPM, what are the predicted values of 70, 7b 72, and 73 in the Fama-MacBeth regressions for the period 1946 — 1955?

In conclusion, these tests of the CAPM provide mixed evidence on the validity of the theory. We can summarize the results as follows:

1. The insights that are supported by the single-factor CAPM and APT are as follows:

a. Expected rates of return are linear and increase with beta, the measure of systematic risk.

b. Expected rates of return are not affected by nonsystematic risk.

2. The single-variable expected return—beta relationship predicted by either the risk-free rate or the zero-beta version of the CAPM is not fully consistent with empirical observation.

Thus, although the CAPM seems qualitatively correct in that (3 matters and a(e;) does not, empirical tests do not validate its quantitative predictions.

CONCEPT CHECK ^ QUESTION 4

What would you conclude if you performed the Fama and MacBeth tests and found that the coefficients on p2 and a(e) were positive?

CHAPTER 13 Empirical Evidence on Security Returns 391

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