Stergios Fotopoulos, Washington State University, Pullman


Non-Linear regression properties for log-returns
under scale mixtures


We discuss analytical properties of the regression and the
variance-covariance matrix representing conditional log-returns of a
set of assets given the returns of another set.  Expressions derived
for the regression equation show that they are not linear when the
returns are modeled by members of the scale mixture family.
Furthermore under some mild conditions, we show that the regression
equation is always finite a.s. for m>1, where m is the dimensionality
of the assets we condition upon. When m=1, it remains finite if and
 only if certain low moment conditions are satisfied by the mixing variable. 
Similar properties are shown to hold for the variance-covariance matrix
which is nondegenerate as in the Gaussian model. Explicit expressions
are given in special cases for which the simulations are also presented. 
Empirical evidence for the analysis is included extending the 
Arbitrage Pricing Theory for log returns.