1. Threshold Autoregressive (TAR) Models
• Movements between regimes governed by an observed variable.
• TAR model:
• Where st-k
is the state determining variable.
• The integer k determines with how many lags does the state-
determining variable influences the regime in time t.
• When st-k
= yt-k
we have a self-exciting TAR (SETAR) model:
• There are many possible variations of this simple model.



≥++
<++
=
−−
−−
rsifuy
rsifuy
y
kttt
kttt
t
2122
1111
φµ
φµ



≥++
<++
=
−−
−−
ryifuy
ryifuy
y
kttt
kttt
t
2122
1111
φµ
φµ

2. Threshold Autoregressive (TAR) Models
• Example: when st-k
= yt-k
we have a self-exciting TAR (SETAR) model:
• Consider k = 1. Parameters to be estimated:
µ1, µ2,σ1,σ2,
– r
• Estimation method: least squares with r estimated by a grid search.
• There are many possible variations of this simple model:
• Switching in only some of the parameters
• More than 2 regimes
• Different threshold variables
• Alternative dynamic specifications
Can use AIC or other information criteria to select models



≥++
<++
=
−−
−−
ryifuy
ryifuy
y
kttt
kttt
t
2122
1111
φµ
φµ

3. EXAMPLE: Threshold error correction (cointegration) model
2
4
6
8
10
12
14
16
18
1960 1965 1970 1975 1980 1985 1990
R3 R120

4. EXAMPLE: Threshold error correction (cointegration) model
-3
-2
-1
0
1
2
3
4
5
1960 1965 1970 1975 1980 1985 1990
SPREAD

5. EXAMPLE: Threshold error correction (cointegration) model
EVIEWS program:
series y = d(r120)
series x = d(r3)
series spread = r120 - r3
scalar th = 3.22
series _d = ( spread(-1) < th )
equation tar.ls y c y(-1) y(-2) x(-1) x(-2) _d*spread(-1) (1-_d)*spread(-1)

6. EXAMPLE: Threshold error correction (cointegration) model

7. EXAMPLE: Threshold error correction (cointegration) model