Empirical likelihood based nonparametric testing for CAPM 109As shown in theorem 1, the empirical log-likehood ratio is asymptotically equivalent
to a StudentisedL 2 -distance betweenm ̃γ(x)andmˆh(x), so it may be compared to the
statistic tests used in [9] and [12]. The main attraction of the test procedure described
here is its ability to automatically studentising the statistic, so we do not have to
estimateV(x;h), contrary to what happens with other nonparametric goodness-of-fit
tests. Based on Theorem 1 and on the assumed independence of the regressors, we
use the following goodness-of-fit test statistic
∑S∗r= 1l{
m ̃γˆ(xr,p)}
, (16)
which is built on a set ofS∗<Spointsxr,p, selected equally spaced in the support of
the regressors. The statistic in (16) is compared with the percentiles of aχ^2 distribution
withS∗degrees of freedom.
4 Empirical results and conclusions
In this section we discuss some results obtained by estimating the modelsA,BandC
presented in equations (4), (5) and (6) and we apply the nonparametric step to test the
linearity of such models. Note that here we consider specifically the linear functions
under the null, but the hypotheses stated in (8) might refer to other functional forms
formγ(x).
The market log-return is given by the S&P500 index, while the asset log-returns
are theS=498 assets included in the S&P stock index. The time series are observed
from the 3rd of January 2000 to the 31st of December2007, for a total of 1509 time
observations. We consider three different rolling window lengths, that isw=22,
66 and 264, which correspond roughly to one, three and twelve months of trading.
The total number of periods in the cross-section analysis (second stage of the Fama
and MacBeth method) is 1487 whenw=22, 1443 whenw=66 and 1245 when
w=264.
For each asset, we estimate the coefficientsβˆj,p,j= 1 ,...,S,p= 1 ,...,T−
w+1 from equation (3). We obtain a matrix of estimated betas, of dimension( 1510 −
w, 498 ). For each resulting period we estimate the cross-section modelsA,BandC
and we apply the nonparametric testing scheme. The assumptions A.1 in [8] are
clearly satisfied for the data at hand. The bandwidth used in the kernel smoothing in
(9), (11) and (12) has been selected automatically for each periodp, by considering
optimality criteria based on a generalised cross-validation algorithm. In (16) we have
consideredS∗=30 equally spaced points. It is well known that kernel estimations
generally suffer from some form of instability in the tails of the estimated function,
due to the local sparseness of the observations. To avoid such problems, we selected
theS∗points in the internal side of the support of the regressors, ranging on the central
95% of the total observed support.
In Table 1 we summarise the results of the two testing procedures (parametric and
nonparametric) described in previous sections.