Multiple Linear Regression 77
security from its characteristic line. The betas from the first-pass regression
are then used to form portfolios of securities ranked by portfolio beta.
The portfolio returns, the return on the risk-free asset, and the portfolio
betas (denoted by βp) are then used to estimate the second-pass regression.
Then the following second-pass regression which is the empirical analogue
of the CAPM is estimated:
Rp − RF = b 0 + b 1 βp + εp (3.20)
where the parameters to be estimated are b 0 and b 1 , and εp is the error term
for the regression.
Unlike the estimation of the characteristic line which uses time series
data, the second-pass regression is a cross-sectional regression. The return
data are frequently aggregated into five-year periods for this regression.
According to the CAPM, the following should be found:
- b 0 should not be significantly different from zero. This can be seen by
comparing equations (3.19) and (3.20). - b 1 should equal the observed risk premium (RM − RF) over which the
second-pass regression is estimated. Once again, this can be seen by
comparing equations (3.19) and (3.20). - The relationship between beta and return should be linear. That is, if,
for example, the following multiple regression is estimated,
Rp − RF = b 0 + b 1 βp + b 2 (βp)^2 + εp
the parameters b 0 and b 2 should not be significantly different from zero.
4. Beta should be the only factor that is priced by the market. That is,
other factors such as the variance or standard deviation of the returns,
and variables such as the price-earnings ratio, dividend yield, and firm
size, should not add any significant explanatory power to the equation.
The general results of the empirical tests of the CAPM are as follows:- The estimated intercept term b 0 , is significantly different from zero and
consequently different from what is hypothesized for this value. - The estimated coefficient for beta, b 1 , has been found to be less than
the observed risk premium (RM − RF). The combination of this and the
previous finding suggests that low-beta stocks have higher returns than
the CAPM predicts and high-beta stocks have lower returns than the
CAPM predicts. - The relationship between beta and return appears to be linear; hence
the functional form of the CAPM is supported.