Empirical likelihood based nonparametric testing for CAPM 105On the basis of these assumptions we can derive a model that relates the expected
return of a risky asset with the risk-free rate and the return of market portfolio; in
the latter, all assets are held according to their value weights. We will denoteR ̃ja
random variable that describes the return of risky assetj.LetR ̃fbe the risk-free rate
andR ̃Mthe return on market portfolio. Under the assumptions above and assuming
that the following expectations exist, the theory of CAPM states that there exists the
following relation:
E[R ̃j]=E[R ̃f]+βj(
E[R ̃M]−E[R ̃f])
. (1)
The termβjin the CAPM equation (1) is the key to the whole model’s implications.
βjrepresents the risk assetjcontributes in the market portfolio, measured relative to
the market portfolio’s variance:
βj=Cov[R ̃j,R ̃M]
Va r [R ̃M]. (2)
βis a measure of systematic risk: since it is correlated with the market portfolio’s
variance and the market portfolio is efficient, an investor cannot possibly diversify
away from it. The theory predicts that each asset’s return depends linearly on its beta.
Notice that the CAPM equation is a one-period model; this means that this equation
should hold period by period. In order to estimate and test the CAPM equation date
by date, we need to make further assumptions in order to estimate the betas first.
2.2 Testing strategy
The beauty of the CAPM theory is that in order to predict assets’ return we only
need information about prices and no further expensive information is needed. The
tests conducted over the last 45 years have brought up different issues and contrasting
views and results. Whereas the first test found no empirical evidence for the theory of
equilibrium asset prices, a very famous test, conducted in 1973 by Fama and MacBeth
(see [7]), provided evidence in favour of the validity of the SLB-CAPM model. How-
ever, later studies (e.g., Fama and French [6]) have challenged the positive and linear
relationship between betas and returns (i.e., CAPM’s theory’s main conclusion) by
introducing other variables which proved to have a much greater explanatory power
but at some costs.
The main contribution of this paper is a nonparametric test about the linear specifi-
cation of the CAPM. Our nonparametric testing approach is based on the comparison
between the predicted returns obtained via the parametric linear model implied by the
CAPM and the returns predicted by a kernel estimator. This testing strategy implies
two steps: the first step (or “parametric step”) is to estimate the predicted returns
based on the CAPM equation; the second step (or nonparametric step) is to predict
the returns on the basis of a kernel regression. We describe the two steps in detail.