WOLFRAM|DEMONSTRATIONS PROJECT

Two-Regime Threshold Autoregressive Model Simulation

​
threshold parameter
threshold k
0
parameters of first AR(1) process
α
1
0
β
1
-1.5
parameters of second AR(1) process
α
2
0
β
2
0.5
new random case
ListPlot[Table[If[x[i-1]<FE`tresh$$9671556467574808365456143473968549077299,x[i]=
FE`alfa1$$9671556467574808365456143473968549077299+
FE`beta1$$9671556467574808365456143473968549077299x[i-1]+random〚i〛,x[i]=
FE`alfa2$$9671556467574808365456143473968549077299+
FE`beta2$$9671556467574808365456143473968549077299x[i-1]+random〚i〛],{100,
2,500}],JoinedTrue,PlotStyle
,PlotRangeAll,FrameTrue,
FrameLabel{t,x(t),two-regime TAR(1) simulated time series,},ImageSize{380,
300},ImagePadding{{75,30},{40,20}}]

This Demonstration allows you to study realizations from a two-regime threshold autoregressive (TAR) process of the first order by changing its parameters. The two-regime TAR(1) model is represented by:
​
​
x
t
=
α
1
+β
1
x
t-1
+ϵ
t
x
t-1
<k
α
2
+β
2
x
t-1
+ϵ
t
x
t-1
≥k
​
Parameters are initially set to
β
1
=-1.5
,
β
1
=-1.5
, and
β
2
=0.5
to obtain the following two-regime TAR(1) process:
x
t
=
-1.5x
t-1
+ϵ
t
x
t-1
<0
0.5x
t-1
+ϵ
t
x
t-1
≥0
​
Note that the process is stationary and geometrically ergodic despite the coefficient -1.5 in the first regime. The series contains large upward jumps when it becomes negative (due to the -1.5 coefficient) and there are more positive than negative jumps. The model also contains no constant term, but
E(x
t
)
is not zero.