The first order nonlinear autoregressive model ‎ ‎with Ornstein Uhlenbeck processes driven by white ‎noise

This paper presents a nonlinear autoregressive model with Ornstein Uhlenbeck processes innovation driven with white noise. Notations and preliminaries are presented about the Ornstein Uhlenbeck processes that have important applications in finance. The parameter estimation for these processes is constructed from the time continuous likelihood function that leads to an explicit maximum likelihood estimator. A semiparametric method is proposed to estimate the nonlinear autoregressive function using the conditional least square method for parametric estimation and the nonparametric kernel approach by using the nonparametric factor that is derived by a local L2-fitting criterion for the regression adjustment estimation. Then the Monte Carlo numerical simulation studies are carried out to show the efficiency and accuracy of the present work. The mean square error (MSE) is a measure of the average squared deviation of the estimated function values from the actual ones. The values of MSE indicate that the innovation in noise structure is performed well in comparison with the existing noise in the nonlinear autoregressive models.