The Estimation Of A Semiparametric Model Based On The Risky And Riskless Assets

Semiparametric Regression Models, which integrate the advantages of parametric and nonparametric regression models, have been developed intensively since the 1980s. They have wider adaptability and stronger explanatory ability than parametric and nonparametric regression models in practice, so they are used in medicine science, biology, economics, finance and so on.Motivated by discrete-time asset pricing with both riskless assets and risk securities in mathematical finance, we introduce a new semipametric regression model. Because asset pricing model could be writen as a BSDE, the estimation of a BSDE and asset pricing model are equivalent. To estimate a BSDE model is actually to estimate the generator. The estimation of BSDE models has been developed rapidly in recent years. Yang,W.,Yang,L.(2006) studied the nonparametric estimation of FBSDEs; YuXia Su(2008) studied the estimation of FBSDEs when the generator is linear; LuLin (2008) developed a modeling method of FBSDEs and studied the estimation of FBSDEs when the generator is variable coefficient linear model.In this paper, we introduce a new semiparametric regression model based on discrete-time asset pricing with both riskless assets and risk securities. Its limit form is FBSDE. It is worth pointing out that, difference from the classical semiparametric regression, the linear part of the regression function contains an unknown standard deviation function. The new model iswhere aY(X(t)) + bZ(X(t)) is the parametric part, the parameters a, b are unknown time-independent, f is a unknown function about Y(X(t)) and it is the nonparametric part.△_t is time interval;ε(t) is a random vector with standard normal distribution. (?)(t) , Y(X(t)) and Z(X(t)) depend on an observable variable X(t) ,but Z(X(t)) is an unobservable random variable satisfying Var((?)(t)|X(t)) =Z~2(X(t)).When the normality condition onε(t) is false, the above model becomes the following modelIn this paper, we need to estimate Z(X(t)), parameter vector (a b)' and unknown function f. The estimation method is that, firstly, we can construct an estimator for Z~2(X(t)) using the second equation of the model by the kernelestimation; after plugging the estimator of Z~2(X(t)) into the mean regressionfunction, we use semiparametric estimation method to estimate (a b)' and f by kernel estimation and the least squares estimation; at last we prove the asymptotic properties of the estimators and give the simulation results.We have the following conclusions in given conditions:Conclusions:...