Document Type : Research Article


Faculty of science, Urmia university of technology, Urmia,Iran


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.‎


[1] Barndorff-Nielsen OE., Shephard, N., 2001, Non-Gaussian Ornstein-Uhlenbeck-
based models and some of their uses in  nancial economics, J R Stat Soc B 63,167{241.
[2] Chaiyapo, N., Phewchean, N., 2017, An application of Ornstein-Uhlenbeck pro-
cess to commodity pricing in Thailand, Advances in Difference Equations, 179,1-10.
[3] Delgado, M.A., Robinson, P.M., 1992, Nonparametric and semiparametric methods for economic research, J. Eco. Sur., 6, 201{249.
[4] Farnoosh, R., Mortazavi, S.J., 2011, A Semiparametric Method for Estimating nonlinear autoregressive model with dependent errors, Journal of NonlinearAnalysis, 74(17), 6358{6370.
[5] Hajrajabi, A., Fallah, A., 2017, Nonlinear semiparametric AR(1) model with skewsymmetric Innovations, Communications in Statistics Simulation and Computation, Taylor and Francis, 1-10.
[6] Hidalgo, F.J., 1992, Adaptive semiparametric estimation in the presence of autocorrelation of unknown form, J. Time Series Anal., 13, 47{78.
[7] Hjort, N.L., Jones, M.C., 1996, Locally parametric nonparametric density estimation, Ann. Statist., 24, 1619{1647.
[8] Tjostheim, D., 1994, Nonlinear time series: A selective review, J. Stat.
[9] Tsay, R. S., 2013, An Introduction to Analysis of Financial Data with R, NewJersey: John Wiley and Sons.
[10] Valdivieso, L., Schoutens, W., Tuerlinckx, F., 2009, Maximum likelihood estimation in processes of Ornstein Uhlenbeck type, Stat Infer Stoch Process, 1{19.
[11] Zhuoxi, Y. Dehui, W. and Ningzhoneg, S., 2009, Semiparametric estimation of regression function in autoregressive models, Journal of Statistics and Probability Letters, 79(2), 165-172.