Empirical likelihood based nonparametric testing for CAPM 111periods we validate the linearity of the regression function, but this is probably a
constant.
As Case 2 we label the percentage of testing periods where the estimated linear
coefficients in the parametric step are jointly equal to zero at levelα=5%, and we
reject theH 0 hypothesis in the nonparametric stage at the same level. That is, no
linear relationship can be detected but there is some evidence of nonlinear structures.
As Case 3 we report the percentage of testing periods where the estimated linear
coefficients in the parametric step are jointly different from zero at levelα=5%, and
we do not reject theH 0 hypothesis in the nonparametric stage at the same level. The
case is in favour of the CAPM theory because here we are saying that the estimated
linear coefficients in the parametric step are jointly different from zero at levelα=
5%, and we do not reject the linearity hypothesis (H 0 ) in the nonparametric stage at
the same level. This means that when this situation occurs we are validating the idea
behind the CAPM, that is: historical information on prices can be useful to explain the
cross-section variations of assets’ returns. For this case the best occurs for modelB,
which is the one that uses the pair of regressors(β,σ). The statistical conclusion we
draw from case 3 is that whenw=22 for 26.6% of the testing periods we validate the
linear relation between assets’ returns,βs and the nonsystematic risk (σ). Moreover
this result does not depend on the rolling window (even thoughw=66 produces
slightly better results).
Finally, as Case 4 we label the percentage of testing periods where the esti-
mated linear coefficients in the parametric step are jointly different from zero at level
α=5%, and we reject theH 0 hypothesis in the nonparametric stage at the same
level. That is, a relationship is present but the nonparametric testing step supports the
evidence for nonlinear effects. It is worthwhile to observe that for all the cases con-
sidered, conclusions do not seem to be related to the particular choice of the rolling
window. They remain basically stable when moving across different choices. Despite
the limitations of the present analysis the open question remains whether the betas
are a determinant of the cross-section variations of assets’ return at all. From this
study we cannot conclude that modelBis validated. But certainly, this occurs in
approximately one quarter of the testing periods. This encourages us to investigate
other possibilities that could reveal stronger paths in the data. There are several issues
that would be worth investigating further: dependence structures, group structures in
the risk behaviour of assets, robustness issues and nonparametric specifications of the
first stage.
References
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Statistics. Springer, New York (1998) - Chen, S.X., Gao, J.: An adaptive empirical likelihood test for time series models. J. Econo-
metrics 141, 950–972 (2007)