  We discuss the foundations of factor or regression models in the light of the
self-consistency condition that the market portfolio (and more generally the
risk factors) is (are) constituted of the assets whose returns it is (they are)
supposed to explain. As already reported in several articles, self-consistency
implies correlations between the return disturbances. As a consequence, the
alpha's and beta's of the factor model are unobservable. Self-consistency leads
to renormalized beta's with zero effective alpha's, which are observable with
standard OLS regressions. Analytical derivations and numerical simulations show
that, for arbitrary choices of the proxy which are different from the true
market portfolio, a modified linear regression holds with a non-zero value
$\alpha_i$ at the origin between an asset $i$'s return and the proxy's return.
Self-consistency also introduces ``orthogonality'' and ``normality'' conditions
linking the beta's, alpha's (as well as the residuals) and the weights of the
proxy portfolio. Two diagnostics based on these orthogonality and normality
conditions are implemented on a basket of 323 assets which have been components
of the S&P500 in the period from Jan. 1990 to Feb. 2005. These two diagnostics
show interesting departures from dynamical self-consistency starting about 2
years before the end of the Internet bubble. Finally, the factor decomposition
with the self-consistency condition derives a risk-factor decomposition in the
multi-factor case which is identical to the principal components analysis
(PCA), thus providing a direct link between model-driven and data-driven
constructions of risk factors.
