A Study On The Threshold Autoregressive Model With A Unit Root

Threshold autoregressive model is one of important nonlinear time series model infinance time series analysis and the econometrics,which is firstly put forward by DoctorH.Tong in 1978.however, threshold autoregressive model is established on the nature ofstable time and ergodicity.so Caner and Hansen originally studied Threshold autoregres-sive model with a unit root,This article based on their theory and study the asymptoticnature of statistical inference in three stages threshold model ,enrich and develope non-stationary theories of time series in nonlinear modelFirst,this article introduces the originality and development of threshold autoregressivemodel,and then studies Bi-Threshold autoregressive model on the basis of threshold au-toregressive model.Bi-Threshold autoregressive model displays obvious superiority in theprocess of solving financial time-series with periods:it can manifest the position which theeconomy locates, the growth (accelerates to grow, generally growth), declination (acceler-ates to decline, generally decline), or the time of shaking.Second,the article studies thresh-old autoregressive model with a uint root,under the null hypothesis H₀ :θ₁ =θ₂ =θ₃,weobtain the theorem 1.The mathematical proofs of the lemma 1,lemma 2 and lemma 3 arepresented in the Appendix.this paper focuses creativity on the following aspects:1, This paper systematically introduces the definition of Threshold autoregressivemodel and the application of Threshold autoregressive model in finance,astronomy, pop-ulation dynamics and so on;2, Explains realistic background of Bi-Threshold autoregressive model, thus gives itsdefinition, finally carries on the parameter estimation and recognition of the model;3, Study of the Threshold autoregressive model with a unit root, under the jointhypothesis H₀ :θ₁ =θ₂ =θ₃, the asymptotic distribution of the Wald statistic for H0 is4, Discussing two bootstrap approximations to the asymtotic distribution of W_T ,onebased on the unrestricted estimates,and the other enforcing the restriction of a unit root.These bootstrap approximation can be used to calculate critical values and p-values.