270 G
. Kapetanios Economics Letters 69 2000 267 –276
since f . 0, f
. 0 Experiments A5–A7, B5–B7 and C5–C7. Finally, within each subset of
1,1 2,1
experiments we vary both the magnitude of the difference of the autoregressive coefficient between the two regimes as well as the absolute magnitude of the autoregressive coefficient.
The threshold parameter, r, is estimated by grid search. The grid points are obtained using the quantiles of the sample under investigation as suggested by Tsay 1989 and Tong and Lim 1980, In
particular 21 equally spaced quantiles are used starting from the 10 quantile and ending at the 90 quantile, i.e. we use the 10, 14, 18 . . . 86, 90 quantiles of the sample. The sample size, T,
takes the values 100, 200 and 400. The error terms are constructed to be zero mean normal variates. For each replication a sample of size T 1 200 is initially generated. The first 200 observations of each
2
sample are discarded to minimise the effect of initial conditions . For each of the DGPs and for each T, 1000 replications are carried out. The bias and the mean square error of the estimated parameters
over the 1000 replications for all the experiments are given in Tables 2–4. The coefficients of the SETAR models do not exhibit significant bias and the bias that appears is
reduced significantly when the sample size is increased. The mean square error of the estimators behaves similarly. The major exception to this finding is the behaviour of the estimators of the
constant terms which exhibit bigger biases than the other coefficients and large mean square errors. Nevertheless, even these estimators improve significantly when the sample size is increased.
The estimator of the threshold parameter exhibits large biases in a number of cases, especially when the constant terms in the model are different from zero. Examples include experiments B1, B5, B6
and their counterparts in the set of experiments C. The biases tend to be reduced for larger sample sizes but not uniformly over the experiments. Additionally, in a number of experiments the MSEs of
the estimator remain large even for 400 observations in contrast to the MSE of the other parameter estimators. It is worthwhile to note that whereas the MSE of the threshold parameter estimator is
comparable to the MSEs of the other parameter estimators in small sample sizes e.g. T 5 100 it is always larger than them for T 5 400, sometimes by a factor of ten. It is clear that the performance of
the estimator improves much more as the sample size increases when the autoregressive parameters in the two regimes differ significantly. Further, when the autoregressive parameters are large in absolute
values then the performance of the threshold parameter estimator deteriorates always in terms of bias and in most cases in terms of MSE.
Overall, we conclude that despite the superconsistency result mentioned earlier, the threshold parameter estimator behaves poorly in small samples, relatively to the other parameter estimators.
4. The conditional sum of squares function
In order to investigate further this result we analyse the conditional sum of squares CSS of the models as a function of the threshold parameters. The CSS function is defined as
m
CSSr , . . . ,r ;
f 5
O
e r , . . . ,r ;
f9e r , . . . ,r ;
f
1 m 21
j 1
m 21 j
1 m 21
j 51
2
The starting values are set to zero.
G . Kapetanios Economics Letters 69 2000 267 –276
271 Table 2
Experiments A1–A10
a
Exp. Par. Obs. Exp. Par.
Obs. 100
200 400
100 200
400
b c
Bias MSE
Bias MSE
Bias MSE
Bias MSE Bias
MSE Bias
MSE f
20.08 1.77
20.04 0.69
20.04 0.34
f 20.37
1.90 20.19
0.60 20.13
0.27
1,0 1,0
f 20.05
0.59 20.02
0.23 20.02
0.11 f
20.19 0.31
20.09 0.09
20.06 0.04
1,1 1,1
f 0.03
1.20 20.07
0.48 20.03
0.23 f
0.13 0.93
0.07 0.47
0.03 0.24
2,0 2,0
A1 f
20.02 0.47
0.04 0.19
0.01 0.09
A6 f
20.12 0.25
20.06 0.10
20.03 0.05
2,1 2,1
2 2
s 20.09
0.07 20.04
0.03 20.02
0.02 s
20.11 0.07
20.05 0.03
20.02 0.01
1 1
2 2
s 20.09
0.06 20.05
0.03 20.02
0.02 s
20.09 0.06
20.04 0.03
20.02 0.02
2 2
r 20.07
0.64 20.04
0.62 20.10
0.65 r
20.21 1.29
20.07 1.23
20.13 1.22
P 0.52
0.52 0.52
P 0.54
0.55 0.54
f 20.19
1.55 20.17
0.44 20.10
0.17 f
20.23 1.21
20.19 0.43
20.11 0.15
1,0 1,0
f 20.11
0.44 20.10
0.14 20.06
0.05 f
20.13 0.28
20.11 0.11
20.06 0.04
1,1 1,1
f 20.14
0.85 20.07
0.42 20.08
0.14 f
20.07 0.78
20.06 0.29
20.07 0.14
2,0 2,0
A2 f
0.10 0.42
0.04 0.19
0.05 0.07
A7 f
0.03 0.38
0.04 0.14
0.04 0.07
2,1 2,1
2 2
s 20.10
0.07 20.04
0.03 20.02
0.01 s
20.08 0.06
20.05 0.03
20.02 0.01
1 1
2 2
s 20.09
0.06 20.05
0.03 20.02
0.01 s
20.10 0.07
20.04 0.03
20.03 0.02
2 2
r 20.18
0.60 20.12
0.55 20.09
0.45 r
20.23 0.71
20.20 0.59
20.12 0.48
P 0.58
0.58 0.58
P 0.61
0.61 0.61
f 20.21
1.02 20.14
0.19 20.07
0.05 f
0.06 1.59
0.03 0.58
0.03 0.26
1,0 1,0
f 20.12
0.25 20.08
0.06 20.04
0.02 f
0.02 0.55
0.02 0.20
0.02 0.09
1,1 1,1
f 20.11
0.59 20.09
0.20 20.08
0.07 f
0.03 1.18
0.05 0.59
20.00 0.28
2,0 2,0
A3 f
0.09 0.38
0.07 0.13
0.06 0.05
A8 f
20.02 0.44
20.04 0.20
20.00 0.09
2,1 2,1
2 2
s 20.09
0.05 20.04
0.02 20.01
0.01 s
20.12 0.07
20.05 0.03
20.02 0.02
1 1
2 2
s 20.09
0.06 20.05
0.03 20.03
0.02 s
20.08 0.05
20.04 0.03
20.03 0.02
2 2
r 20.25
0.58 20.13
0.36 20.09
0.24 r
20.01 0.62
20.01 0.61
0.07 0.65
P 0.65
0.65 0.65
P 0.49
0.49 0.49
f 20.26
0.52 20.12
0.10 20.05
0.02 f
0.17 1.86
0.12 0.70
0.05 0.29
1,0 1,0
f 20.13
0.10 20.07
0.03 20.03
0.01 f
0.08 0.35
0.06 0.13
0.03 0.05
1,1 1,1
f 20.11
0.52 20.11
0.17 20.07
0.07 f
20.01 1.34
0.01 0.51
20.00 0.22
2,0 2,0
A4 f
0.09 0.43
0.09 0.15
0.06 0.06
A9 f
0.01 0.26
0.01 0.09
0.00 0.04
2,1 2,1
2 2
s 20.06
0.04 20.02
0.02 20.01
0.01 s
20.10 0.07
20.04 0.04
20.03 0.02
1 1
2 2
s 20.10
0.07 20.06
0.04 20.03
0.02 s
20.10 0.06
20.05 0.03
20.03 0.02
2 2
r 20.22
0.49 20.13
0.28 20.06
0.13 r
0.00 1.24
0.01 1.23
0.01 1.19
P 0.73
0.73 0.73
P 0.50
0.49 0.49
f 20.10
2.02 20.07
0.65 20.01
0.29 f
0.14 1.69
0.10 0.43
0.09 0.14
1,0 1,0
f 20.08
0.66 20.05
0.21 20.01
0.09 f
0.08 0.67
0.07 0.18
0.06 0.06
1,1 1,1
f 20.04
1.01 20.05
0.48 20.02
0.25 f
0.11 1.18
0.11 0.46
0.08 0.15
2,0 2,0
A5 f
0.02 0.40
0.03 0.18
0.01 0.09
A10 f 20.06
0.35 20.05
0.13 20.04
0.05
2,1 2,1
2 2
s 20.11
0.07 20.05
0.03 20.02
0.02 s
20.10 0.07
20.04 0.03
20.02 0.02
1 1
2 2
s 20.10
0.06 20.05
0.03 20.03
0.02 s
20.10 0.05
20.05 0.03
20.02 0.01
2 2
r 20.08
0.64 20.07
0.60 20.03
0.63 r
0.09 0.64
0.13 0.58
0.12 0.48
P 0.52
0.52 0.52
P 0.44
0.44 0.44
a
Exp.: Experiment; Par.: Parameters; Obs.: Sample size; P: Proportion of observations in lower regime; MSE: Mean square error.
b N
n n
ˆ ˆ
The bias is given by 1 N o a 2 a where a is the relevant parameter estimate for replication n and a is the true
n 51
value of the relevant parameter and N 5 1000.
c N
n 2
ˆ The mean square error is estimated by 1 N o
a 2 a .
n 51
272 G
. Kapetanios Economics Letters 69 2000 267 –276 Table 3
Experiments B1–B10
a
Exp. Par. Obs. Exp. Par.
Obs. 100
200 400
100 200
400
b c
Bias MSE
Bias MSE
Bias MSE
Bias MSE Bias
MSE Bias
MSE f
20.05 1.20
20.06 0.33
20.02 0.15
f 20.14
0.70 20.08
0.24 20.04
0.09
1,0 1,0
f 20.03
0.53 20.05
0.17 20.01
0.08 f
20.18 0.28
20.09 0.10
20.05 0.04
1,1 1,1
f 0.01
1.72 20.04
0.73 20.05
0.29 f
0.24 2.33
0.09 0.97
0.00 0.40
2,0 2,0
B1 f
20.02 0.47
0.01 0.20
0.02 0.08
B6 f
20.11 0.24
20.05 0.09
20.01 0.04
2,1 2,1
2 2
s 20.10
0.07 20.05
0.03 20.02
0.02 s
20.10 0.07
20.05 0.03
20.03 0.02
1 1
2 2
s 20.07
0.07 20.05
0.03 20.02
0.02 s
20.08 0.06
20.05 0.03
20.02 0.01
2 2
r 0.30
0.74 0.29
0.64 0.26
0.57 r
0.85 2.09
0.83 1.95
0.80 1.75
P 0.39
0.39 0.39
P 0.25
0.25 0.26
f 20.14
0.98 20.09
0.30 20.07
0.09 f
20.23 0.86
20.12 0.23
20.05 0.06
1,0 1,0
f 20.09
0.38 20.06
0.13 20.05
0.04 f
20.16 0.31
20.08 0.09
20.04 0.03
1,1 1,1
f 20.06
1.14 20.09
0.47 20.08
0.16 f
20.07 1.39
20.05 0.41
20.08 0.14
2,0 2,0
B2 f
0.02 0.39
0.05 0.16
0.04 0.06
B7 f
0.02 0.40
0.02 0.12
0.04 0.05
2,1 2,1
2 2
s 20.11
0.06 20.04
0.03 20.02
0.01 s
20.09 0.06
20.04 0.03
20.02 0.01
1 1
2 2
s 20.08
0.06 20.04
0.03 20.02
0.02 s
20.09 0.06
20.04 0.03
20.02 0.01
2 2
r 0.13
0.56 0.19
0.52 0.21
0.41 r
0.13 0.65
0.18 0.52
0.24 0.42
P 0.46
0.46 0.46
P 0.45
0.45 0.45
f 20.22
0.59 20.10
0.14 20.05
0.04 f
20.02 1.19
0.02 0.38
20.00 0.19
1,0 1,0
f 20.14
0.22 20.07
0.06 20.04
0.02 f
20.02 0.61
0.01 0.21
0.00 0.10
1,1 1,1
f 20.18
0.85 20.11
0.30 20.07
0.07 f
0.04 1.37
20.03 0.60
0.04 0.29
2,0 2,0
B3 f
0.10 0.36
0.06 0.12
0.04 0.04
B8 f
20.03 0.37
0.01 0.16
20.02 0.07
2,1 2,1
2 2
s 20.05
0.05 20.04
0.02 20.02
0.01 s
20.09 0.07
20.04 0.03
20.02 0.02
1 1
2 2
s 20.08
0.06 20.05
0.03 20.02
0.01 s
20.08 0.05
20.05 0.03
20.02 0.02
2 2
r 0.06
0.47 0.11
0.34 0.13
0.20 r
0.26 0.66
0.28 0.61
0.23 0.58
P 0.53
0.53 0.53
P 0.37
0.37 0.37
f 20.20
0.40 20.09
0.07 20.04
0.02 f
0.18 1.57
0.05 0.50
0.02 0.18
1,0 1,0
f 20.13
0.13 20.07
0.03 20.03
0.01 f
0.10 0.40
0.03 0.14
0.01 0.05
1,1 1,1
f 20.16
0.57 20.12
0.17 20.06
0.06 f
0.04 1.27
0.01 0.61
0.03 0.23
2,0 2,0
B4 f
0.11 0.31
0.08 0.10
0.04 0.04
B9 f
20.01 0.24
0.00 0.11
20.01 0.04
2,1 2,1
2 2
s 20.07
0.04 20.02
0.02 20.01
0.01 s
20.10 0.07
20.05 0.03
20.03 0.02
1 1
2 2
s 20.09
0.07 20.04
0.03 20.02
0.02 s
20.09 0.06
20.05 0.03
20.03 0.01
2 2
r 0.01
0.34 0.11
0.22 0.11
0.12 r
0.15 1.03
0.16 0.96
0.11 0.85
P 0.61
0.61 0.61
P 0.44
0.44 0.43
f 20.03
0.92 20.05
0.37 20.02
0.13 f
0.10 1.28
0.07 0.37
0.07 0.15
1,0 1,0
f 20.04
0.46 20.04
0.19 20.01
0.07 f
0.08 0.79
0.06 0.23
0.06 0.10
1,1 1,1
f 20.03
1.55 20.03
0.66 20.01
0.35 f
0.15 1.37
0.06 0.58
0.04 0.14
2,0 2,0
B5 f
0.00 0.41
0.01 0.16
20.00 0.09
B10 f 20.07
0.32 20.04
0.14 20.02
0.04
2,1 2,1
2 2
s 20.11
0.07 20.04
0.03 20.02
0.01 s
20.10 0.07
20.04 0.03
20.03 0.02
1 1
2 2
s 20.08
0.06 20.05
0.03 20.02
0.02 s
20.09 0.05
20.05 0.02
20.01 0.01
2 2
r 0.28
0.68 0.31
0.68 0.33
0.63 r
0.30 0.68
0.21 0.57
0.09 0.42
P 0.38
0.38 0.38
P 0.34
0.34 0.34
a
Exp.: Experiment; Par.: Parameters; Obs.: Sample size; P: Proportion of observations in lower regime; MSE: Mean square error.
b N
n n
ˆ ˆ
The bias is given by 1 N o a 2 a where a is the relevant parameter estimate for replication n and a is the true
n 51
value of the relevant parameter and N 5 1000.
c N
n 2
ˆ The mean square error is estimated by 1 N o
a 2 a .
n 51
G . Kapetanios Economics Letters 69 2000 267 –276
273 Table 4
Experiments C1–C10
a
Exp. Par. Obs. Exp. Par.
Obs. 100
200 400
100 200
400
b c
Bias MSE
Bias MSE
Bias MSE
Bias MSE Bias
MSE Bias
MSE f
20.01 1.20
20.04 0.57
20.02 0.23
f 20.21
1.07 20.16
0.45 20.11
0.18
1,0 1,0
f 20.02
0.49 20.04
0.24 20.01
0.09 f
20.17 0.35
20.12 0.13
20.08 0.05
1,1 1,1
f 20.02
1.41 20.04
0.54 20.05
0.30 f
0.16 1.72
0.08 0.78
0.05 0.42
2,0 2,0
C1 f
20.00 0.46
0.02 0.16
0.02 0.09
C6 f
20.09 0.24
20.05 0.11
20.03 0.06
2,1 2,1
2 2
s 20.10
0.07 20.05
0.03 20.03
0.02 s
20.10 0.07
20.05 0.04
20.02 0.02
1 1
2 2
s 20.09
0.06 20.04
0.03 20.02
0.02 s
20.09 0.06
20.05 0.03
20.03 0.02
2 2
r 0.18
0.69 0.10
0.63 0.12
0.65 r
0.42 1.41
0.40 1.36
0.46 1.41
P 0.44
0.44 0.44
P 0.36
0.36 0.36
f 20.18
1.38 20.13
0.49 20.11
0.16 f
20.15 1.51
20.14 0.34
20.09 0.12
1,0 1,0
f 20.10
0.45 20.08
0.17 20.07
0.06 f
20.11 0.45
20.09 0.11
20.06 0.04
1,1 1,1
f 20.16
1.03 20.08
0.38 20.10
0.20 f
20.07 1.17
20.11 0.37
20.08 0.12
2,0 2,0
C2 f
0.08 0.38
0.04 0.15
0.06 0.08
C7 f
0.02 0.42
0.06 0.13
0.04 0.05
2,1 2,1
2 2
s 20.08
0.06 20.05
0.03 20.02
0.01 s
20.11 0.06
20.04 0.03
20.02 0.01
1 1
2 2
s 20.09
0.06 20.05
0.03 20.02
0.01 s
20.09 0.06
20.05 0.03
20.02 0.01
2 2
r 20.00
0.57 20.02
0.51 0.04
0.46 r
0.02 0.65
20.01 0.54
0.03 0.47
P 0.50
0.50 0.50
P 0.50
0.50 0.50
f 20.25
0.98 20.13
0.23 20.07
0.06 f
0.04 1.35
20.01 0.61
0.03 0.21
1,0 1,0
f 20.15
0.30 20.08
0.08 20.05
0.03 f
0.02 0.60
20.01 0.26
0.01 0.10
1,1 1,1
f 20.17
0.71 20.12
0.19 20.09
0.06 f
20.00 1.53
0.02 0.64
0.02 0.30
2,0 2,0
C3 f
0.11 0.36
0.08 0.10
0.06 0.04
C8 f
20.00 0.45
20.02 0.19
20.01 0.08
2,1 2,1
2 2
s 20.09
0.06 20.04
0.02 20.02
0.01 s
20.10 0.06
20.07 0.03
20.02 0.02
1 1
2 2
s 20.08
0.06 20.05
0.03 20.01
0.01 s
20.09 0.06
20.05 0.03
20.02 0.02
2 2
r 20.09
0.50 20.05
0.34 20.04
0.22 r
0.21 0.69
0.19 0.65
0.18 0.63
P 0.57
0.57 0.57
P 0.42
0.41 0.41
f 20.22
0.44 20.11
0.08 20.06
0.03 f
0.12 1.63
0.10 0.55
0.06 0.24
1,0 1,0
f 20.13
0.13 20.07
0.03 20.04
0.01 f
0.07 0.36
0.06 0.12
0.03 0.05
1,1 1,1
f 20.16
0.48 20.09
0.14 20.06
0.05 f
0.04 1.54
20.03 0.57
20.01 0.24
2,0 2,0
C4 f
0.09 0.29
0.06 0.10
0.04 0.04
C9 f
20.00 0.26
0.02 0.09
0.01 0.04
2,1 2,1
2 2
s 20.06
0.04 20.03
0.02 20.01
0.01 s
20.09 0.07
20.05 0.03
20.03 0.02
1 1
2 2
s 20.09
0.07 20.05
0.03 20.02
0.02 s
20.10 0.06
20.05 0.03
20.02 0.02
2 2
r 20.10
0.37 20.06
0.22 20.02
0.12 r
0.09 1.23
0.13 1.18
0.07 1.15
P 0.64
0.64 0.64
P 0.46
0.46 0.46
f 20.03
1.30 20.04
0.44 20.03
0.25 f
0.09 1.07
0.08 0.36
0.06 0.15
1,0 1,0
f 20.04
0.55 20.02
0.19 20.02
0.11 f
0.05 0.53
0.04 0.19
0.04 0.08
1,1 1,1
f 0.03
1.58 20.06
0.67 20.01
0.31 f
0.13 1.27
0.10 0.39
0.10 0.16
2,0 2,0
C5 f
20.02 0.47
0.02 0.20
0.01 0.09
C10 f 20.07
0.33 20.05
0.10 20.05
0.04
2,1 2,1
2 2
s 20.09
0.06 20.05
0.03 20.03
0.02 s
20.10 0.07
20.05 0.03
20.02 0.02
1 1
2 2
s 20.10
0.07 20.05
0.03 20.02
0.01 s
20.08 0.06
20.05 0.03
20.02 0.01
2 2
r 0.19
0.67 0.19
0.67 0.16
0.67 r
0.25 0.68
0.16 0.55
0.17 0.54
P 0.43
0.43 0.43
P 0.38
0.38 0.38
a
Exp.: Experiment; Par.: Parameters; Obs.: Sample size; P: Proportion of observations in lower regime; MSE: Mean square error.
b N
n n
ˆ ˆ
The bias is given by 1 N o a 2 a where a is the relevant parameter estimate for replication n and a is the true
n 51
value of the relevant parameter and N 5 1000.
c N
n 2
ˆ The mean square error is estimated by 1 N o
a 2 a .
n 51
274 G
. Kapetanios Economics Letters 69 2000 267 –276
where e r , . . . ,r
; f 5 y 2 X f is the error vector for regime j implicitly defined in 2 and
j 1
m 21 j
j j
9 9
f 5 f , . . . ,f 9. Note that we modify slightly our notation to explicitly state the dependence of
1 m
the error vectors on the threshold parameters and the fact that we take the coefficients, f, as given. To
simplify notation we supress the dependence of the above quantities on the sample size, T. We would like to obtain the expected value of CSSr;
f for the SETAR2,1,1 model we are investigating over a grid of possible values of r. We resort to simulation techniques to obtain an estimate of that
expectation. The simulation estimator is given by
N 2
2 1
n n
] CSSr;
f 5
O O
e r; f9e r; f 3
j j
N
n 51 j 51 n
n n
n n
where e r; f 5 y 2 X f . y and X , j 5 1,2 are obtained from simulated samples constructed
j j
j j
j j
using the DGPs given in Table 1. N is the number of replications. The parameter values, f, used to
2 2
p
evaluate CSSr; f are the true parameter values from Table 1. It is clear that CSSr; f
→ ECSSr;
f as N →
`. We set N 5 2000, T 5 100 and use a grid of equally spaced values for r which starts at 2
21 and ends at 1. The distance between the points is 0.02. We plot 2CSSr; f over r for all
experiments in Fig. 1. For comparative purposes we additionally define the CSS as a function of one
Fig. 1. Opposite of CSS as a function of r for all experiments.
G . Kapetanios Economics Letters 69 2000 267 –276
275
of the coefficients which we choose to be f . In the notation we introduced above this is denoted as
1,1
CSSf ; r, f
, f , f
. We estimate the expectation of CSSf ; r, f
, f , f
using a
1,1 0,1
2,0 2,1
1,1 0,1
2,0 2,1
similar estimator to the one given in 3. Once again the parameters used to generate the simulated samples are taken from Table 1 and N 5 2000, T 5 100. Denoting the true value of f
for
1,1
experiment i by f , the grid used for f
and experiment i is given by G 5 h f 2 0.5,
i 1,1
1,1 i
i 1,1
f 2 0.48, . . . ,
f 2 0.02,
f ,
f 1 0.02, . . . ,
f 1 0.48,
f 1 0.5j. We
plot
i 1,1
i 1,1
i 1,1
i 1,1
i 1,1
i 1,1
2 2CSSf ; r, f , f , f
over G for all experiments in Fig. 2.
1,1 0,1
2,0 2,1
i
The plots reveal the reason for the poor performance of the threshold parameter estimator. Whereas 2
2 CSSf
; r, f , f
, f is a smooth quadratic function of f
, CSSr; f is less well behaved.
1,1 0,1
2,0 2,1
1,1
2 In general, when the autoregressive parameters do not differ significantly between regimes CSSr;
f is very flat exhibiting variation less than 0.001 throughout the grid of r. When the autoregressive
parameters differ significantly the variation increases dramatically. When there is a nonzero constant 2
term which differs between regimes, CSSr; f exhibits a kink at the true threshold parameter value.
The appearance of the kink seems to depend more on the fact that the constant is different between 2
regimes and less on the fact that it is non-zero. Finally, in a number of cases, CSSr; f is asymmetric
Fig. 2. Opposite of CSS as a function of f for all experiments.
1,1
276 G
. Kapetanios Economics Letters 69 2000 267 –276
around the true threshold parameter value. The above results do not change when we consider larger
3
sample sizes . Overall, it is safe to conclude that the nonstandard features of the CSS function with respect to the
threshold parameter lie behind the poor performance of the threshold parameter estimator in small samples.
5. Conclusions
