Smooth Transition Autoregressive Model

1. Smooth Transition Autoregressive Model EniSumarminingsih Eni Sumarminingsih, SSi, MM

2. Smooth Transition Autoregressive Model • For some process, it may not seem reasonable to assume that the threshold is sharp • Smooth Transition Autoregressive (STAR) Model allow the autoregressive parameters to change slowly. Eni Sumarminingsih, SSi, MM

3. Consider the special NLAR model given by • If f() is a smooth continuous function, the autoregressive coefficient (α1 + β1) will change smoothly along with the value of Yt-1 • There are two particularly useful forms of the STAR model : the Logistic STAR and the Exponential STAR Eni Sumarminingsih, SSi, MM

4. The LSTAR Model generalizes the standard AR model such that the AR coefficient is a logistic function : • where •  is called the smoothness parameter • In the limit, as  --> 0 or ∞, LSTAR become an AR(p) model since the value of  is constant. Eni Sumarminingsih, SSi, MM

5. For intermediate value of , the degree of autoregressive decay depends on the value of Yt-1 • As Yt-1 -,   0 so that the behavior of Ytis given by • As Yt-1 +,   1 so that the behavior of Ytis given by • Thus the intercept and the AR coefficient smoothly change between these two extremes as the value of Yt-1 changes. EniSumarminingsih, SSi, MM

6. The ESTAR model uses ,  > 0 • As  approach zero or infinity, the model becomes an AR(p) model since  is constant • Otherwise, the model display nonlinear behavior • As Yt-1 approach c,  approach 0  behavior of Yt is given by Eni Sumarminingsih, SSi, MM

7. As Yt-1 moves further from c,  approach 1  behavior of Yt is given by Eni Sumarminingsih, SSi, MM

8. Test for STAR Model Step 1 : Estimate the linear portion of the AR(p) model to determine the order and to obtain the residual {et} Step 2 : Estimate the auxiliary equation Eni Sumarminingsih, SSi, MM

9. Test the significance of the entire regression by comparing TR2 to the critical value of 2. If the calculated value of TR2 exceed the critical value from a 2 table, reject the null hypothesis of linearity and accept the alternative hypothesis of a smooth transition model. Alternatively, you can perform an F test Eni Sumarminingsih, SSi, MM

10. Step 3 : If you accept the alternative hypothesis (i.e., if the model is nonlinear), test tre restriction a31 = a32 = … = a3p = 0 using an F test. If you reject the a31 = a32 = … = a3p = 0, the model has LSTAR form. If you accept the restriction, conclude that the model has the ESTAR form Eni Sumarminingsih, SSi, MM

11. Uji F JKGR: JK galat restricted model JKGU: JK galat unrestricted model kU: jumlahpeubaheksogen (termasukkonstanta) pada unrestricted model kR: jumlahpeubaheksogen (termasukkonstanta) pada restricted model Hipotesisnol: restricted model valid Mendugarestricted model danunrestricted model Memperoleh JK Galatuntukrestricted modeldan JK Galatuntuk unrestricted model, danmenghitungstatistikuji F.