76 The Basics of financial economeTrics
The CAPM states that, given the assumptions, the expected return on
asset is a positive linear function of its index of systematic risk as measured
by beta. The higher the βi or beta is, the higher the expected return. There
are no other factors that should significantly affect an asset’s expected
return other than the index of systematic risk. A stock’s beta is estimated
from the characteristic line that we described and illustrated in the previ-
ous chapter.
The beta for an asset can be estimated using the following simple linear
regression:
rit − rft = αi + βi[rMt − rft] + εit
where rit=observed return on asset i for time t
rft=observed return on the risk-free asset for time t
rMt=observed return on the market portfolio for time t
εit=error term for time t
The above regression equation is called the characteristic line. Since there is
only one independent variable, rMt − rft, there is a simple linear regression.
If
xt = rMt − rft
and
yt = rit − rft
then the characteristic line can be rewritten as
yt = αi + βixt + εit
The parameters to be estimated are the coefficients αi and βi and the stan-
dard deviation of the error term, εi. The parameter βi is the focus of interest
in this section. Later in this chapter, when we provide an illustration of how
regression analysis is used in performance measurement, we will see the
economic meaning of the intercept term, αi.
To estimate the characteristic line for an asset using regression analysis,
we consider three time series of returns for (1) the asset, (2) the market index,
and (3) the risk-free rate. The beta estimates will vary with the particular
market index selected as well as with the sample period and frequency used.
Typically, a methodology referred to as a two-pass regression is used
to test the CAPM. The first pass involves the estimation of beta for each