106 P. Coretto and M.L. Parrella
2.3 The parametric step
We complete the first step by using the same methodology developed by Fama and
MacBeth [7]. In order to apply this methodology we need to make some further as-
sumptions. The CAPM is a model of expected returns in a one-period economy. What
we actually observe, though, is a time series of asset prices and other variables from
which we can compute the realised returns over various holding periods. We need
to assume that investors know the return distribution over one particular investment
period. In order to estimate the parameters of that distribution it is convenient to
assume that the latter is stationary. In addition, we assume that returns are drawn
independently over time. Although the last assumption appears to be too strong, sev-
eral empirical studies proved that this cannot seriously affect the first-step estimation
(see [7]). The latter comment applies in particular when short sequences of daily
returns are used to estimate the betas (see below).
What about the “market portfolio”? Can the market portfolio be easily identified?
It is worth remembering that the CAPM covers all marketable assets and it does not
distinguish between different types of financial instruments. This is the focus of Roll’s
Critique [14]. As a market proxy we will use the S&P500 index. The CAPM provides
us with no information about the length of the time period over which investors choose
their portfolios. It could be a day, a month, a year or a decade.
Now we describe the Fama-MacBeth estimation methodology. We have a time
series of assets’ prices recorded in some financial market. Let us assume thatRj,tis
the log-return at timetfor the assetj,wherej= 1 , 2 ,...,Sandt= 1 , 2 ,...,T.
LetRM,tbe the market log-return at timet. The relation (1) has to hold at eachtfor
each asset. We have to estimate the CAPM for eacht. To do the latter we need a time
series ofβs.
The first stage is to obtain a time series of estimated betas based on a rolling
scheme. For each assetj= 1 , 2 ,...,S,andforfixedwandp= 1 , 2 ,...,T−w+1,
we take the pairs{Rj,t,RM,t}t=p,p+ 1 ,...,p+w− 1 and we estimate the market equation
Rj,t=αj,p+βj,pRM,t+εj,t, (3)where{εj,t}t=p,p+ 1 ,...,p+w− 1 is an i.i.d. sequence of random variables with zero mean
and finite variance. The (3) is estimated for eachj= 1 , 2 ,...,S, to obtainβjfor
periodsp= 1 , 2 ,...,T−w+1. The estimatedβˆj,pis the estimate of the systematic
risk of thejth asset in periodp. From this first regression we also store the estimated
standard deviation of the error term, sayσˆj,p=
√
Va rˆ (ˆεj,t). The latter is a measure
of the unsystematic risk connected to thejth asset in periodp. The use ofσˆj,pwill
be clear afterwards.
In the second stage for each periodp= 1 , 2 ,...,T−w+1weestimatethe
linear model implied by the CAPM applying a cross-section (acrossj= 1 , 2 ,...,S)
linear regression of assets’ returns on their estimated betas. For each periodp=
1 , 2 ,...,T−w+1 the second-stage estimation is:
Rj,p+ 1 =γ 1 A+γ 2 Aβˆj,p+ξAj,p+ 1 , (4)