Smooth Transition Autoregressive Model

Smooth Transition

Autoregressive Model

Eni Sumarminingsih

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

• _{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 Y_t-1

• _{There are two particularly useful} forms of the STAR model : the

Logistic STAR and the Exponential STAR

• _{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}

• _{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

Yt is given by

• As Yt-1  +,   1 so that the behavior of

Yt is given by

• _{Thus the intercept and the AR coefficient}

smoothly change between these two

extremes as the value of Yt-1 changes.

• _{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 Y_t-1 moves further from c, 

approach 1  behavior of Y_t is given by

Eni Sumarminingsih, SSi, MM

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}

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

Step 3 : If you accept the alternative hypothesis (i.e., if the model is

Uji F

• _{Hipotesis nol: restricted model valid}

• _{Menduga restricted model dan unrestricted model}

• _{Memperoleh JK Galat untuk restricted model dan JK Galat}

untuk unrestricted model, dan menghitung statistik uji F.

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JKG_R: JK galat restricted model JKG_U: JK galat unrestricted model

k_U: jumlah peubah eksogen (termasuk konstanta) pada unrestricted model