Statistical Inference On Semiparametric Models With Applications To Finance

In recent 20 years, much research has been focused on the semiparametric models. The advantage of such semiparametric models is that they reduce the risk of misspecification relative to a fully parametric model and avoid some serious drawbacks of fully nonparametric methods. Firstly we discuss the statistical inference for partially linear models. Using Huber's M-estimator, we establish the robust estimator for the parametric part. The resulting estimator is shown to be consistent and asymptotically normally distributed. Simulation results show that the M-estimator performs as well as least square estimator under normally distributed errors. Especially, when the random errors are distributed as Cauchy distribution, the M-estimator is superior to the least square estimator. Secondly we propose a new kind of semiparametric models, called partially nonlinear models, which not only nest the partially linear models as a special case, but also contain many nonlinear functions widely used in applications. Compared to general partially nonlinear models, our models are convenient to make statistical inferences. We develop the sieve method to estimate the nonparametric part, which was approximated by the piecewise linear functions and B-spline functions, and then get the least square estimator and the maximum likelihood estimator, respectively. Both the two resulting estimators are consistent and asymptotically normally distributed. The simulation results support our theory procedures. By introducing the acceleration stress model, we establish the partially nonlinear model to simulate the relation among the exchange rate of USD/EUR, USD/RMB and oil price. And then we study the inner relations of the global stock market, the model is applied to simulate the relationship among the Dow Jones Industrial index, Nikkei index and USD/JPY exchange rate. The partially nonlinear models are found to be an appropriate setting in such cases in finance. Finally, we construct a criterion function using copula function, which can select the best one among the methods on trend estimation in financial time series.