Empirical likelihood based nonparametric testing for CAPM 107where{ξjA,p}j= 1 , 2 ,...,Sis an i.i.d. sequence of random variables having zero mean
and finite variance. Notice that we regressRj,p+ 1 onβˆj,p; this is because it is
assumed that investors base current investment decisions on the most recent available
β. The Fama-MacBeth testing procedure consisted in testing the linear relation (4) for
each periodp. If the linear model (4) holds in periodp, that means that the model (1)
statistically holds in periodp. For a long time it has been thought that when the CAPM
fails this is due to the fact that unsystematic risk affects returns as well as possible
nonlinearities in betas. Two further second-stage equations have been considered to
check for the aforementioned effects. The first alternative to (4) is:
Rj,p+ 1 =γ 1 B+γ 2 Bβˆj,p+γ 3 Bσˆj,p+ξBj,p+ 1 , (5)where we add a further regressor which is the unsystematic risk measure. The second
alternative is:
Rj,p+ 1 =γ 1 C+γ 2 Cβˆ^2 j,p+γ 3 Cσˆj,p+ξCj,p+ 1 , (6)
where the betas enter in the regression squared. In (5) and (6) we assume that the
errors{ξBj,p}j= 1 , 2 ,...,Sand{ξCj,p}j= 1 , 2 ,...,Sare two i.i.d. sequence of random variables
with zero mean and finite variance. As for (4), (5) and (6) are also estimated for each
periodp= 1 , 2 ,...,T−w+1. The modelsA,BandCrepresented by equations
(4),(5) and (6) are estimated and tested in the famous paper by Fama and MacBeth [7].
3 The nonparametric goodness-of-fit test
In this section we apply the method proposed by H ̈ardle et al. [8] for testing the linear
specification of the CAPM model, that is, modelsA,BandCdefined in the second
stage of the previous step. This goodness-of-fit test is based on the combination of
two nonparametric tools: the Nadaraya-Watson kernel estimator and the Empirical
Likelihood of Owen [13]. Here we briefly describe the testing approach, then, in the
next section, we apply it to the CAMP model estimated on the S&P500 stock market.
Let us consider the following nonparametric model
Rj,p+ 1 =m(
Xj,p)
+ej,p+ 1 ,
j= 1 , 2 ,...,S
p= 1 , 2 ,...,T−w+ 1(7)
whereRj,p+ 1 is the log-return of periodpfor the assetj,Xj,p∈Rdis the vector
ofdregressors observed in periodpfor the assetjandej,p+ 1 is the error, for which
we assume thatE(ej,p+ 1 |Xj,p)=0forallj. We also assume that the regressors
Xj,pand the errorsej,pare independent for differentjs, but we allow some condi-
tional heteroscedasticity in the model. The main interest lies in testing the following
hypothesis
H 0 :m(x)=mγ(x)=γTx versus H 1 :m(x)=mγ(x), (8)wheremγ(x)=γTxis the linear parametric model andγis the vector of unknown
parameters belonging to a parameter space∈Rd+^1. Let us denote withmγˆ(x)the
estimate ofmγ(x)given by a parametric method consistent underH 0.