268 G
. Kapetanios Economics Letters 69 2000 267 –276
2. Theory
We consider the following self exciting threshold autoregressive SETAR model y 5 f
1 f y
1 ? ? ? 1 f y
1 s e ,
j 5 1, . . . ,m, t 5 p, . . . ,T,
s . 0 1
t j,0
j,1 t 21
j, p t 2p
j t j
The model has m regimes. The process is in regime j if r y
, r where d is an integer valued
j 21 t 2d
j
delay parameter. r 5 2 ` and r 5 `. hr . . . r j is a strictly increasing sequence of parameters to
m 1
m 21
be estimated. e is an i.i.d. zero mean process with unit variance. For simplicity, throughout this paper,
t
we will assume that e has a standard normal distribution. This model will be denoted as SETARm, p,
t 1
d . The number of regimes, m, and the delay parameter, d, are assumed known in our setup . We note that p is the true lag order for all the m regimes. Estimation is carried out by constructing a grid of
possible values for r , j 5 1, . . . ,m 2 1 and running the regressions
j
y 5 X f 1 e ,
j 5 1, . . . ,m 2
j j
j j
for each point in the threshold parameter grid, where y and X are a vector and matrix, respectively,
j j
containing the observations for regime j. f and e are the coefficient and error vectors for regime j. In
j j
matrix notation, y 5 y , y , . . . , y 9, X 5 x , . . . ,x 9, x 5 y , y
, . . . , y 9,
f 5
j j
j j
j j
j j
j 21 j 22
j 2p j
1 2
T 1
T i
i i
i
j j
f , . . . ,f 9 e 5 e , . . . ,e 9 and h j , j , . . . , j j are the time indices of the observations
j,1 j, p
j j
j 1
2 T
1 T
j
j
belonging to regime j, j 5 1, . . . ,m. The grid point which minimises the sum of squared residuals from the m regressions is adopted as the estimate for the threshold parameters. Chan 1993 proves
that under geometric ergodicity and some other regularity conditions the threshold parameters are consistent, tend to their true value at rate T, and suitably normalised follow asymptotically a
21 2
compound Poisson process. The other parameters of the model are T consistent and are
asymptotically normally distributed. To the best of our knowledge investigation of the small samples properties of the parameter estimates and particularly the threshold parameter estimates has not been
undertaken. A Monte Carlo investigation is carried out in the next section.
3. Monte Carlo investigation
We aim to investigate the properties of the conditional least squares estimator for a variety of parametric setups so as to minimise the risk of drawing invalid conclusions due to idiosyncracies of a
particular model design. Preliminary investigation suggested that the results obtained for SETAR models of two regimes readily extend to models with three regimes. SETAR models with more
regimes have not been considered due to excessive computational cost. The same conclusion was reached for models with more than one lags. We therefore provide the results of an extensive
investigation on SETAR2,1,1 models. It is likely that the performance of the conditional least squares estimator will depend on the magnitude and signs of the coefficients f , j 5 1, 2, i 5 0, 1.
j,i
The set of DGPs we therefore consider is given in Table 1.
1
The number of regimes is usually dictated by theory or preliminary examination of the data. The assumption that d is known may be dropped without affecting the asymptotic properties of the estimators see Chan, 1993.
G . Kapetanios Economics Letters 69 2000 267 –276
269 Table 1
DGPs for Monte Carlo simulations for SETAR models Experiments
f f
f f
1,0 1,1
2,0 2,1
A1 0.05
20.05 A2
0.25 20.25
A3 0.45
20.45 A4
0.65 20.65
A5 0.15
0.05 A6
0.75 0.65
A7 0.55
0.05 A8
20.15 20.05
A9 20.75
20.65 A10
20.55 20.05
B1 0.2
0.05 0.4
20.05 B2
0.2 0.25
0.4 20.25
B3 0.2
0.45 0.4
20.45 B4
0.2 0.65
0.4 20.65
B5 0.2
0.15 0.4
0.05 B6
0.2 0.75
0.4 0.65
B7 0.2
0.55 0.4
0.05 B8
0.2 20.15
0.4 20.05
B9 0.2
20.75 0.4
20.65 B10
0.2 20.55
0.4 20.05
C1 0.2
0.05 0.2
20.05 C2
0.2 0.25
0.2 20.25
C3 0.2
0.45 0.2
20.45 C4
0.2 0.65
0.2 20.65
C5 0.2
0.15 0.2
0.05 C6
0.2 0.75
0.2 0.65
C7 0.2
0.55 0.2
0.05 C8
0.2 20.15
0.2 20.05
C9 0.2
20.75 0.2
20.65 C10
0.2 20.55
0.2 20.05
We fix the threshold parameter, r ; r , to 0 and s , j 5 1, 2, to 1. A number of comments of the
1 j
choice of the DGPs is in order. The set of experiments A generates zero mean processes. Set B considers processes with non zero mean where the constant switches between regimes. Finally, set C
considers processes with non zero mean but where the constant does not switch between regimes. Within each set of experiments we consider three subsets. In the first subset experiments A1–A4,
B1–B4 and C1–C4 the autoregressive coefficient has different signs in the two regimes suggesting that the process is more likely to be in one regime than the other. In our case the most common
regimes is the lower one. To see this note that if y
, 0 then f y
, 0 and if y . 0 then
t 21 1,1
t 21 t 21
f y
, 0 as well. The other two subsets consider processes which when at some period are in one
2,1 t 21
regime then they are either more likely to be in the opposite regime in the next period since f , 0,
1,1
f , 0 Experiments A8–A10, B8–B10 and C8–C10 or are more likely to be in the same regime
2,1
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
Kategori