Smooth Transition Autoregressive Model

Smooth Transition
Autoregressive Model
Eni Sumarminingsih

Eni Sumarminingsih, SSi, MM

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

• 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
Eni Sumarminingsih, SSi, MM

• 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

• 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.
Eni Sumarminingsih, SSi, MM

• 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

• As Yt-1 moves further from c, 
approach 1  behavior of Yt 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}
Step 2 : Estimate the auxiliary
equation

Eni Sumarminingsih, SSi, MM

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

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

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.


JKG R 
F

JKGU  /  kU  k R 
~ F kU  k R ,n  kU 
JKGU /  n  kU 

JKGR: JK galat restricted model
JKGU: JK galat unrestricted model
kU: jumlah peubah eksogen (termasuk konstanta)
pada unrestricted model
kR: jumlah peubah eksogen (termasuk konstanta)
pada restricted model