Non-linear properties of conditional returns under scale mixtures

This article provides a framework for analyzing multifactor financial returns that violate the Gaussian distributional assumption. Analytical expressions are provided for the non-linear regression equation and its prediction error (heteroscedasticity) by modeling the returns of financial assets as scale mixtures of the multivariate normal distribution. The expressions involve conditional moments of the mixing variable. These conditional moments are explicitly derived when the mixing variable belongs to the generalized inverse Gaussian family, of which gamma, inverse gamma and the inverse Gaussian distributions are distinguished members. The derived expressions are non-linear in the parameters and involve the modified Bessel function of the third kind. The effects of the non-linear model, in terms of both the regression equation and heteroscedasticity against the corresponding values for the standard linear regression model, are captured through simulations for the gamma, inverse gamma and inverse Gaussian distributions. The proposed scale mixture models extend the well-known arbitrage pricing theory (APT) in financial modeling to non-Gaussian cases. The methodology is applied to analyze the intra-day log returns quarterly data for DELL and COKE regressed against S&P 500 for the years 1998–2000.