Central Limit Theorems For Convolutional Double-smooth Estimators Of Diffusion Model

In the past decades,the estimation of drift and diffusion terms has become one of the most active research fields in high-frequency financial econometrics.Song et al.[23]proposed the convolutional double-smooth estimator by replacing the average with the weighted average as a compromise,and gave the asymptotic normality.In this paper,the convolutional double-smooth local linear estimator is obtained by combining the convolutional double-smooth estimator with the local linear estimator,but obtaining the asymptotic normality of the local linear estimator is a complex problem.The reason is that when the local linear estimator is used to estimate the coefficient of the diffusion process,the predictability structure of the estimator is destroyed,and the estimator no longer has the martingale structure.Song and Wang[24]used Jacod’s stable convergence theorem for the first time to obtain the asymptotic normality of the local linear threshold estimator for diffusion processes with jumps.With the help of Song and Wang’s result,this paper studies the asymptotic normality of local linear estimator based on the convolutional double-smooth kernel estimation method for the first time.The convolutional double-smooth local linear estimation method can effectively reduce the deviation and improve the boundary effect at the same time.In this paper,Monte Carlo simulation is used to verify the asymptotic normality of the convolutional double-smooth local linear estimator,and the comparison of the boundary effect,deviation and mean square error of the convolutional doublesmooth local linear estimator and the convolutional double-smooth kernel estimator for the stationary diffusion model and the non-stationary diffusion model also confirms this point.