Mean volatility regressions

Motivated by increment process modeling for two correlated random and non-random systems from a discrete-time asset pricing with both risk free asset and risky security, we propose a class of semiparametric regressions for a combination of a non-random and a random system. Unlike classical regressions, mean regression functions in the new model contain variance components and the model variables are related to latent variables, for which certain economic interpretation can be made. The motivating example explains why the GARCH-M of which the mean function contains a variance component cannot cover the newly proposed models. Further, we show that statistical inference for the increment process cannot be simply dealt with by a two-step procedure working separately on the two involved systems although the increment process is a weighted sum of the two systems. We further investigate the asymptotic behaviors of estimation by using sophisticated nonparametric smoothing. Monte Carlo simulations are conducted to examine finite-sample performance, and a real dataset published in Almanac of China's Finance and Banking (2004 and 2005) is analyzed for illustration about the increment process of wealth in financial market of China from 2003 to 2004.