Journal of Forecasting
J. Forecast. 28, 445–464 (2009)
Published online 7 January 2009 in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/for.1122
On a Robust Test for SETAR-Type
Nonlinearity in Time Series Analysis
KING CHI HUNG,1 SIU HUNG CHEUNG,1,2
WAI-SUM CHAN1* AND LI-XIN ZHANG3
1
2
3
Chinese University of Hong Kong, Hong Kong
National Cheng Kung University, Taiwan
Zhejiang University, PR China
ABSTRACT
There has been growing interest in exploiting potential forecast gains from the
nonlinear structure of self-exciting threshold autoregressive (SETAR) models.
Statistical tests have been proposed in the literature to help analysts check for
the presence of SETAR-type nonlinearities in observed time series. However,
previous studies show that classical nonlinearity tests are not robust to additive
outliers. In practice, time series outliers are not uncommonly encountered. It
is important to develop a more robust test for SETAR-type nonlinearity in time
series analysis and forecasting. In this paper we propose a new robust nonlinearity test and the asymptotic null distribution of the test statistic is derived.
A Monte Carlo experiment is carried out to compare the power of the proposed
test with other existing tests under the influence of time series outliers. The
effects of additive outliers on nonlinearity tests with misspecification of the
autoregressive order are also studied. The results indicate that the proposed
method is preferable to the classical tests when the observations are contaminated with outliers. Finally, we provide illustrative examples by applying the
statistical tests to three real datasets. Copyright © 2009 John Wiley & Sons,
Ltd.
key words additive outliers; GM estimation; nonlinearity tests; robustness;
threshold time series
INTRODUCTION
Owing to the deficiency of linear models in capturing some commonly observed features of time
series data, many nonlinear time series models have been proposed in the literature. One popular
class is the self-exciting threshold autoregressive model (SETAR). It was first proposed by Tong
(1978) and is discussed in detail in Tong (1983, 1990).
If a time series Yt follows the SETAR(k; p; d) model, then
* Correspondence to: Wai-Sum Chan, Department of Finance, The Chinese University of Hong Kong, Shatin, Hong Kong.
E-mail: chanws@cuhk.edu.hk
Copyright © 2009 John Wiley & Sons, Ltd.
446
K. C. Hung et al.
p
Yt = φ0( j) + ∑ φi( j)Yt − i + ε t
if rj −1 ≤ Yt − d < rj
(1)
i =1
i.i.d.
N (0,s2); and s2 < ∞.
where j = 1,2, . . . , k, t = p + 1, . . . , n, d ≤ p are both positive integers, et ∼
Parameters p and d denote the autoregressive (AR) order and the delay parameter, respectively.
Furthermore, k is the number of regimes, rjs are the threshold parameters such that −∞ = r0 < r1
< . . . < rk = ∞, and n is the number of observations. In this way, different regimes possess different
(s)
AR(p) models. If f (j)
i = f i for all i = 0,1, . . . , p and j ≠ s = 1, 2, . . . , k, then the model reduces to
a linear AR(p) process. A more generalized version of the SETAR model allows different orders of
AR models inside different regimes.
With the introduction of threshold parameters, rjs, the process switches among different linear
autoregressive models. This class of piecewise linear models can effectively capture jump phenomena, amplitude-dependent frequency, and limit cycles. In application, testing problems for the
SETAR model against a linear model has aroused considerable interest and many tests have been
proposed in the literature (e.g., De Gooijer and Kumar, 1992; Petruccelli, 1990). Their power and
size properties are shown to be satisfactory.
However, realizations from the SETAR model can have an asymmetric distribution of
observations among regimes. Thus, it may look like a linear time series contaminated with outliers.
Owing to this similarity, standard testing procedures might not be able to distinguish between
a threshold-type nonlinear time series and a linear time series with an outlier contamination
structure.
This conjecture is supported in previous research. Moeanaddin and Tong (1988) report that some
nonlinearity tests are sensitive to outliers. An analysis of a real dataset by Chan and Cheung (1994)
further confirms this possibility. They apply Tsay’s (1989) F test for SETAR-type nonlinearity to
the monthly average wholesale prices of regular leaded gasoline in the United States between 1973
and 1987. It is found that outliers can induce aberrant F statistics. Furthermore, the simulation results
of Chan and Ng (2004) also stress the need for a more robust test for SETAR-type nonlinearity in
time series analysis and forecasting.
Regarding this insufficiency, Van Dijk et al. (1999) suggest a robust Lagrange multiplier
(LM) test for the smooth transition autoregressive (STAR) model. The class of STAR models contains the class of SETAR processes as a special case. The generalized-M estimate is used to ensure
the robustness of the standard LM test (Luukkonen et al., 1988) for STAR models. Although simulations indicate that the robustified LM test is often erroneously oversized, it is a first attempt to construct a robust nonlinearity test in time series analysis. In this paper, we propose a new robust
nonlinearity test for SETAR-type models. It does not have the oversized problem as does the robustified LM test, and it offers reasonable power in detecting SETAR-type nonlinearity in the presence
of outliers.
The paper proceeds as follows. In the next section we propose a new robust test for SETAR-type
nonlinearity. The asymptotic null distribution of the test statistic is derived. The third section
compares the power of the proposed test with other existing tests under the influence of time
series outliers via Monte Carlo experiments. The effects of additive outliers on nonlinearity
tests with misspecification of the autoregressive order are also studied. The fourth section
applies the tests to three real examples. A summary of the findings is given in the final
section.
Copyright © 2009 John Wiley & Sons, Ltd.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
A Robust Test for SETAR-Type Nonlinearity
447
TESTS FOR SETAR-TYPE NONLINEARITY
Various statistical tests for linearity (against SETAR-type nonlinearity) are available in the literature.
Petruccelli and Davies (1986) propose a CUSUM test for threshold nonlinearity, which is based on
the concept of arranged autoregression. The CUSUM test is modified by Petruccelli (1990) to achieve
a higher power. Tsay (1989) derives an F test, which is based on one-step-ahead forecast errors from
the arranged autoregression. Luukkonen et al. (1988) consider LM tests. Chan and Tong (1990)
employ the likelihood ratio approach for testing SETAR-type nonlinearity. Chan and Ng (2004)
conduct a small-scale simulation study on the outlier robustness of these tests. They conclude that
no single test is able to resist satisfactorily large outlying values.
In this section we first briefly describe the arranged autoregression approach of Tsay’s (1989) F
test. Tsay’s test is commonly used owing to its simple and regression-type structure. A robustified
version of Tsay’s F test is then proposed and its asymptotic null distribution is derived.
Tsay’s F test
Under Tsay’s arranged regression approach, the linear AR(p) model is regarded as the null model
against the alternative SETAR model. We consider an observed time series {Yt, t = 1, 2, . . . , n}.
Let h = max(1, p + 1 − d ) and denote pi as the index of the ith smallest value among {Yt : t = h,
h + 1, . . . , n − d}. The observations {Yt : t = h, h + 1, . . . , n − d} are sorted in ascending order.
Both the dependent vector Y and the corresponding lagged design matrix X are arranged according
to the threshold parameter Yt−d. Under the null hypothesis of linearity, the arranged autoregression
is written in the form Y = XΦ + e, where Φ = {f0, f1, . . . , fp} is a vector of the AR parameters and
e is a vector of noise, i.e.,
Yπ1 + d −1
 Yπ1 + d
 1

 Yπ 2 + d

Yπ 2 + d −1
1

 =

 
 Y
  1 Y
π n− d −h +1 + d −1
π n− d −h +1 + d
Yπ1 + d − p   φ0   ε π1 + d 
Yπ 2 + d − p   φ1   ε π 2 + d 
  +

  









Yπ n−d −h+1 + d − p φ p
επ n−d −h+1 + d 
(2)
Stepwise autoregression of the first j rows of Y on the first j rows of X is performed successively
for j = m, m+ 1, . . . , n − d − h, where m > p + 1 is the startup value. Let xj+1 be the ( j + 1)th row
of the X matrix and Xj be the submatrix containing the first j row of X. The corresponding one-stepahead prediction residuals ê pj+1+ d are obtained successively:
εˆ π j+1 + d =
Yπ j +1 + d − Yˆπ j +1 + d
1+ x
−1
( X X j ) x j +1
T
j +1
T
j
(3)
Tsay’s F test is developed based on the orthogonality property between the predictive residuals
given in (3) and the regressors {Ypj+1+ d−vv = 1, . . . , p, j = 1, . . . , n − d − h − m} under the null
hypothesis of linearity. This property will be destroyed if observations that are lying in other regimes
are involved, which in turn indicates nonlinearity.
Copyright © 2009 John Wiley & Sons, Ltd.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
448
K. C. Hung et al.
The orthogonality (thus the SETAR-type nonlinearity) can be tested by considering the regression
model
p
ε̂π j+1 + d = ω 0 + ∑ ωυ Yπ j+1 + d −υ + eπ j+1 + d
(4)
υ =1
for j = m, m + 1, . . . , n − d − h, i.e.,
 εˆ π m+1 + d   1 Yπ m+1 + d −1
 εˆ π m+2 + d   1 Yπ m+2 + d −1

 =

 
 εˆ
  1 Y
π n− d −h +1 + d −1
π n− d −h +1 + d
Yπ m+1 + d − p   ω 0   eπ m+1 + d 
Yπ m+2 + d − p   ω1   eπ m+2 + d 
  +

  

Yπ n−d −h+1 + d − p   ω p   eπ n−d −h+1 + d 
(5)
The usual F statistic of the regression in (5) is
F=
MSS ( p + 1)
RSS ( n − d − h − m − p)
(6)
where RSS = Σêt2 and MSS = Σε̂ t2 − RSS. Under the null hypothesis, the test statistic is asymptotically distributed as Fp+1,n−d−h−m−p.
The proposed test
To enhance the discriminative power of the F test in the presence of additive outliers, the Schweppe
type of generalized-M (GM) estimator is considered (Handschin et al., 1975). For simplicity, we
shall explain the concept of this type of GM estimation procedure for a simple linear AR(1) model
without intercept. However, the method can be easily generalized to any ordinary AR(p) models
(see, for example, Maronna et al., 2006, Ch. 8).
Following Franses and Van Dijk (2000, p. 67), we consider a simple AR(1) process:
Yt = φYt −1 + ε t
(7)
Given the observations {Yt : t = 1, . . . , n}, the ordinary least squares (OLS) estimate of f can be
obtained by minimizing the sum of the squared residuals under the first-order condition:
∑Y
t −1
( Yt − φYt −1) = 0
(8)
If an outlying observation appears at t = s, it creates two outliers in the regression. The point (Ys−1,
Ys) becomes a vertical outlier and (Ys, Ys+1) becomes a bad leverage point. In this situation, the OLS
estimator will be biased towards zero (Rousseeuw and Van Zomeren, 1990). The Schweppe-type
regression (with GM estimator) in computing the estimate of f has the advantage of utilizing the
information that is given by good leverage points while reducing the weights of the vertical outliers
and bad leverage points (Hampel et al., 1986). Under the Schweppe-type regression with a given
y-function (e.g., Maronna et al., 2006, p. 149), the first-order condition for the simple AR(1) model
becomes
Copyright © 2009 John Wiley & Sons, Ltd.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
A Robust Test for SETAR-Type Nonlinearity
∑ω
y
(Yt −1 ) Yt −1ψ ( zt ) = 0
449
(9)
where
zt =
εt
σ ε ω y (Yt −1 )
(10)
is the standardized residual series,
ε t = Yt − φYt −1
(11)
is the tth residual, and se is a robust measure of the scale of the residuals et. The function wy(·) in
(9) and (10) downweights the outlying observations in the regressor Yt−1. As proposed by Van Dijk
et al. (1999), the weight function for the regressor is chosen as
ψ ([δ ( Yt −1)]
2
ω y ( Yt −1) =
[δ ( Yt −1)]
)
2
(12)
where
δ ( Yt −1) =
Yt −1 − m y
σy
(13)
and my and sy are the robust measures of location and the scale of Yt−1, respectively. The y-function
must be specified under some regularity conditions such that the corresponding GM estimator is
asymptotically consistent (Hampel et al., 1986). We will discuss the choices of the y-function in
the next subsection.
Define the weight function for the standardized residuals as
 ψ ( zt )
 zt
ω z ( zt ) = 
 1

for zt ≠ 0
(14)
otherwise
Substituting (14) into the first-order condition (9), we have
∑ω
y
(Yt −1 ) Yt −1 ⋅ω z ( zt ) Z t = 0
(15)
Finally, combining (10), (11) and (15), we obtain an equivalent form of the first-order condition,
which is nonlinear in f, as
∑ ω ( z )Y
z
Copyright © 2009 John Wiley & Sons, Ltd.
t
t −1
( Yt − φYt −1) = 0
(16)
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
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K. C. Hung et al.
It should be noted that once a y-function has been chosen, the weight function for the regressor
wy(·) and the weight function for the standardized residuals wz(·) are fixed by equations (12) and
(14), respectively. Given an initial estimate of f, denoted by f̂ (0), the corresponding fitted residuals
ê t(0) can be computed via equation (11). In this paper we employ the median absolute deviation
(MAD) measure for estimating the se in (10), i.e.,
σˆ ε(0) = 1.483 median ( εˆ t(0) − mˆ ε(0)
)
(17)
with
mˆ ε(0) = median ( εˆ t(0) )
(18)
for t = 2, . . . , n. When et is normally distributed, the constant 1.483 is used to adjust the MAD
measure to become a consistent estimator of the scale of the residuals. Analogous to (17) and (18),
we employ the estimators ŝ y = 1.483 median (Yt−1 − m̂y) and m̂y = median(Yt−1) for the estimation
of sy and my in (13). Finally, the corresponding standardized residual series ẑ (0)
t can be computed.
Equation (16) indicates that the GM estimate of f can be obtained using the iterative weighted
least squares (IWLS) algorithm
n
∑ ω ( zˆ ( ) )Y
i
z
φˆ (i +1) =
t
t =2
n
Y
t −1 t
(19)
∑ ω ( zˆ )Y
(i )
t
z
2
t −1
t =2
where ẑ t(i)s are the fitted standardized residuals that are computed using the estimate f(i) at the ith
iteration. For the starting value f̂ (0), we suggest using the least median of squares estimator that is
proposed by Rousseeuw (1984).
For a general pth-order autoregression with intercept, as in (2) and (5), we need a multivariate
version of the IWLS algorithm in (19). In particular, we replace the d (Yt−1) value in equation (13)
by
δ ( Yt(−p1) ) =
(Yt(−p1) − M y( p) )
T
Σ y( p) (Yt(−p1) − M y( p) )
(20)
p)
= (Yt−1, Yt−2, . . . , Yt−p)T with the multivariate
which is the Mahalanobis distance for regressors Ỹ (t−1
p)
location My(p) and scatter matrix Σy(p). The value of d (Y (t−1
) in (20) can be efficiently computed by
using the minimum volume ellipsoid (MVE) estimator that is discussed in Rousseeuw and Van
Zomeren (1990). The corresponding FORTRAN subroutine MINVOL for computing MVE estimators is available at Statlib (www.stat.cmu.edu).
For our proposed test, we suggest replacing the OLS estimates with the GM estimates in the
arranged regressions in (2) and (4). Given p and d, we first compute the GM estimates and predictive
residual ê pj+1+ d successively for j = m, m+1, . . . , n − d − h. A set of n − d − h − m + 1 predictive
residuals are obtained. The Schweppe-type regression (4) is performed between the predictive residuals ê pj+1+ d and the last n − d − h − m + 1 rows of the design matrix X.
The proposed FGM statistic is formulated as
FGM =
Copyright © 2009 John Wiley & Sons, Ltd.
MSS ( p + 1)
RSS ( n − d − h − m − p)
(21)
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
A Robust Test for SETAR-Type Nonlinearity
451
where
ˆ T WX ( X T WWX )−1 X T W Ψ
ˆ;
MSS = Ψ
ˆ TΨ
ˆ − MSS;
RSS = Ψ
ˆ = (ψ ( zˆπ + d ) ,…,ψ ( zˆπ
Ψ
))T ;
m +1
n − d − h +1 + d
X = (Yπ(mp+)1 + d ,…, Yπ(np−)d −h+1 + d ) ;
T
W = diag (ω y (Yπ(mp+)1 + d ) ,…, ω y (Yπ(np−)d −h+1 + d )) ;
Yπ( jp+)1 + d = (1, Yπ( jp+)1 + d −1 ,…, Yπ( jp+)1 + d − p ) ;
T
and the standardized residual is defined as
zˆπ j +1 + d = εˆ π j +1 + d σˆ ε
(22)
for j = m, m + 1, . . . , n − d − h.
Theorem 1. Let Yt be a linear stationary AR(p) process. That is, Yt follows the model in (1) with
k = 1. Then, for large n, the statistic FGM that is defined in (21) follows approximately an F distribution with degrees of freedom (p + 1) and (n − d − h − m − p).
Proof. This theorem can be proved by using a similar argument to that used by Tsay (1986, p. 463).
For t = pm+1 + d, pm+2 + d, . . . , pn−d−h+1 + d, we denote
ˆ − Φ ) and Yt( p) = Yt(−p1) Φ + ε t
εˆ t = ε t + Yt(−p1) (Φ
It can be shown that Φ̂ and ŝe are n -consistent estimates of Φ and se, respectively. With
g = (y(epm+1+ d/se), . . . , (yepn−d−h+1+ d /se))T taking the place of Ψ̂, the asymptotic distribution of FGM in
(21) does not change. Now, X T W ϒ n = ∑ t Yt(−p1)ω y (Yt(−p1) ) ψ (ε t σ ε ) n satisfies a central limit
theorem (Billingsley, 1961) and converges in distribution to a (p + 1)-dimensional normal random
vector with a mean 0 and covariance matrix k1, with
κ 1 = E [ψ 2 (ε t σ ε )] E ω y2 (Yt(−p1) ) Yt(−p1) 
T
Also, X T WWX n = ∑ t ω y2 (Yt(−p1) ) Yt(−p1) n converges in probability to k2 with
T
κ 2 = E ω y2 (Yt(−p1) ) Yt(−p1) 
T
2
× E[y2(et /se)]. Finally, the large-sample F distribuThus, the numerator of (21) converges to c p+1
tion of the testing statistic FGM in (21) follows from an argument which is similar to that of the usual
analysis of variance.
The choice of the y-function
The outlier robust properties of the proposed FGM test depend on the choice of y-function. Table I
summarizes several commonly used y-functions (Hoaglin et al., 2000; Van Dijk et al., 1999). The
efficiencies for these y-functions for robust estimation of the location parameter u depend on their
tuning constants. They are often chosen with reference to the asymptotic efficiency of the Gaussian
Copyright © 2009 John Wiley & Sons, Ltd.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
452
K. C. Hung et al.
Table I. Some commonly used y-functions
Estimator with tuning constant(s)
Least squares (LS)
Huber1 (k)
k>0
Tukey’s biweight (c)
c>0
Polynomial2 (c1, c2)
c2 > c1 > 0
y-Function y(u)
Range of u
u
u
ksgn(u)
u[1 − (u/c)2]2
0
u
0
g(u)
(−∞; ∞)
u ≤ k
u > k
u ≤ c
u > c
u ≤ c1
u > c2
otherwise
Huber reduces to the LS (i.e., mean) estimator when k → ∞.
y(u) = u[1 − H(u − c1)] + H(u − c1)[1 − H(u − c2)]g(u); where H(.) is
the Heaviside function. H(z) = 1 if z > 0, and H(z) = 0 if z ≤ 0. g(u) is a fifthorder polynomial such that y(u) is twice continuously differentiable.
1
2
Table II. Tuning constant(s) of selected y-functions
Estimator
Tuning constant(s)
1.345 or 3.291
4.685 or 15
2.576, 3.291
Huber (k)
Tukey’s biweight (c)
Polynomial (c1, c2)
model and the square root of the quantiles of the c2 distribution with one degree of freedom (Spath,
1991, p. 195). At 95% and 99.95% efficiency of the Gaussian model, the tuning constants of the
Huber y-function are 1.345 and 3, respectively, while those of Tukey’s biweight y-function are
4.685 and 15, respectively. We also have χ12 (0.99) = 2.576 and χ12 (0.999) = 3.291. The tuning
constants for the y-functions that are considered in this paper are listed in Table II. Their impact on
the power of the proposed FGM test will be studied via simulation experiments in the next section.
MONTE CARLO RESULTS
In the first experiment, we consider a simple SETAR (2;1,1) model as the underlying outlier-free
data-generating process (DGP):
(1)
(1)
φ0 + φ1 Yt −1 + ε t
Yt =  (2)
( 2)
φ0 + φ1 Yt −1 + ε t
if Yt −1 ≤ r
if Yt −1 > r
i.i.d.
where et ∼
N(0, 1). A straightforward generalization of the definition of the additive outlier (AO)
in a linear time series by Denby and Martin (1979), in which both the number and the location of
the outliers are determined by a random mechanism, is given as follows:
Z t = Yt + ηt
Copyright © 2009 John Wiley & Sons, Ltd.
(23)
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
A Robust Test for SETAR-Type Nonlinearity
453
where Zts are the observations, hts are i.i.d. with density (1 − b)∆0(·) + bp, p follows N(0, w2), ∆0(·)
represents a degenerate density at 0, w is the size parameter of the contamination, and b is the percentage of contamination (0 ≤ b ≤ 1).
Chan and Ng (2004) extend the definition of an innovational outlier (IO) by Denby and Martin
(1979) to the SETAR models, and find that the existence of IOs seems to cause no harm to most
SETAR-type nonlinearity tests. Furthermore, Chan (1992) and Wei and Wei (1998) suggest that the
existence of innovational outliers is not an obstacle in model specification and estimation for linear
time series processes. Therefore, we do not consider IO in this section.
Time series data {Z1, . . . , Zn} are generated from (23). We consider sample sizes of n = 100 and
n = 200. The startup value, Y0, is set to zero. It is important to discard a sufficiently large number
of observations to remove transient effects in generating a SETAR time series. Therefore, the first
1500 observations are omitted in each replication.
The proposed FGM test with various choices of y-functions is considered. When the choice of yfunction is LS (see Table II), it is actually the ordinary least squares method with equal weighting,
and the FGM test reduces to the original Tsay (1989) F test.
Following Denby and Martin (1979), we set the size of contamination w = 0, 3, 6, and 10. The
percentage of contamination is fixed at b = 5%. We consider the level of significance a = 1%, 5%
and 10%. In each test, p = 1 and d = 1 are used. The startup value for the arranged regression is set
at m = n/10 + p.
Table III gives the empirical relative frequencies of rejecting the linear null hypothesis based on
(1)
(2)
1000 replications. We consider seven parameter combinations (r, f(1)
0 , f1 , f0 ) = (1.0, 0.5, 0.5, 0.5),
(2)
and f1 varies from −0.8 to + 0.8. Our experience has shown that the results are not sensitive to the
choice of AR(1) coefficient in regime 1; hence we only report the case of f 1(1) = 0.5 here. When f1(2)
= 0.5, the model reduces to the linear AR(1) process (i.e., the null model). Some of the parameter
combinations are taken from Tsay (1989). Other combinations are chosen such that there are adequate observations in both regimes for efficient parameter estimation and nonlinearity testing. Table
IV shows results that correspond to Table III but with a threshold parameter r = 0 and both intercept
(2)
terms f (1)
0 and f 0 set to zero. Chan and Ng (2004) find that the intercept term is related to the level
of a process and it has a significant impact on the power of SETAR-type nonlinearity testing.
First, we examine the effect of the size of the outlier contamination (w). Under the null model of
linearity (f1(2) = 0.5), then the empirical sizes of the tests are indicated by the relative rejection frequencies in the tables. If there are no outliers (w = 0), the empirical sizes of the tests are consistently
smaller than (but all satisfactorily close to) the nominal levels (1%, 5% and 10%). However, when
w increases, the empirical sizes of the LS test (i.e., the original Tsay F test) become significantly
larger than the corresponding nominal levels. Unfortunately, the adverse effects of the outliers on
the LS test cannot be diluted by increasing the sample size. On the other hand, the proposed FGM
tests can keep the sizes steady at around the nominal level without being affected by the outliers.
For example, in Table III (sample size n = 100 with a = 5%), under the null model (f1(2) = 0.5), the
empirical sizes for the LS test are 0.031 (w = 0), 0.176 (w = 3), and 0.312 (w = 6). For the FGM test
(with k = 3.291 Huber y-function) the empirical sizes are 0.025 (w = 0), 0.056 (w = 3), and 0.057
(w = 6). The results show that the original F test is sensitive to outlying observations. The test suffers
the erroneous oversized problem when the data are generated from the null model with outliers.
However, the proposed FGM tests are robust with respect to outliers in the null model.
As usual, the robust test encounters some power loss when there is no outlier in the data, especially
when the sample size is not large (see Tables III and IV for the case w = 0). When the degree of
SETAR-type nonlinearity gets stronger (e.g., f1(2) moves from 0.5 to −0.8), the powers of the proposed
Copyright © 2009 John Wiley & Sons, Ltd.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
Copyright © 2009 John Wiley & Sons, Ltd.
0.5
0.5
0.5
1.0
1.0
1.0
0.5
0.5
1.0
1.0
0.5
1.0
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.8
0.5
0.2
0.0
−0.2
−0.5
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
f 1(1) f (2)
f 1(2) w a =
f (1)
0
0
(a) Sample size n = 100
1.0 0.5 0.5 0.5 −0.8
r
Parameters
0.969
0.685
0.336
0.166
0.810
0.442
0.204
0.122
0.414
0.205
0.123
0.090
0.181
0.108
0.087
0.076
0.047
0.059
0.066
0.058
0.006
0.052
0.132
0.116
0.014
0.195
0.479
0.498
1%
0.996
0.836
0.494
0.275
0.927
0.679
0.350
0.209
0.648
0.410
0.248
0.162
0.398
0.271
0.204
0.142
0.152
0.154
0.145
0.116
0.034
0.191
0.307
0.257
0.066
0.349
0.689
0.722
1.000
0.879
0.572
0.341
0.964
0.767
0.449
0.268
0.767
0.530
0.344
0.222
0.535
0.382
0.289
0.206
0.224
0.250
0.220
0.179
0.079
0.283
0.428
0.371
0.138
0.448
0.783
0.801
0.721
0.609
0.601
0.586
0.518
0.438
0.419
0.406
0.232
0.190
0.176
0.178
0.090
0.064
0.062
0.059
0.023
0.019
0.019
0.015
0.002
0.005
0.003
0.003
0.008
0.016
0.015
0.016
5% 10% 1%
LS
0.870
0.817
0.807
0.808
0.754
0.661
0.646
0.648
0.469
0.396
0.383
0.369
0.257
0.193
0.199
0.198
0.079
0.061
0.072
0.067
0.027
0.027
0.024
0.022
0.045
0.058
0.083
0.070
0.934
0.881
0.873
0.877
0.838
0.772
0.766
0.757
0.600
0.509
0.506
0.492
0.391
0.312
0.299
0.293
0.146
0.133
0.123
0.124
0.042
0.050
0.046
0.042
0.087
0.104
0.151
0.135
0.948
0.784
0.700
0.643
0.767
0.548
0.448
0.417
0.371
0.237
0.196
0.173
0.171
0.094
0.083
0.067
0.046
0.032
0.025
0.024
0.005
0.007
0.008
0.014
0.015
0.048
0.073
0.070
0.990
0.911
0.850
0.813
0.911
0.783
0.684
0.648
0.633
0.450
0.385
0.368
0.380
0.269
0.224
0.215
0.132
0.101
0.087
0.088
0.035
0.044
0.044
0.045
0.068
0.154
0.214
0.203
0.998
0.948
0.906
0.886
0.952
0.864
0.793
0.751
0.746
0.585
0.519
0.495
0.520
0.392
0.342
0.327
0.210
0.167
0.161
0.157
0.073
0.099
0.105
0.087
0.125
0.240
0.339
0.304
0.501
0.439
0.461
0.467
0.322
0.298
0.295
0.298
0.140
0.134
0.135
0.138
0.065
0.051
0.051
0.055
0.014
0.014
0.013
0.010
0.003
0.006
0.003
0.006
0.019
0.016
0.012
0.015
0.717
0.662
0.677
0.687
0.542
0.509
0.508
0.530
0.309
0.285
0.279
0.294
0.162
0.139
0.145
0.145
0.058
0.054
0.060
0.058
0.024
0.024
0.021
0.021
0.046
0.053
0.056
0.051
0.786
0.753
0.763
0.778
0.650
0.619
0.618
0.647
0.443
0.398
0.410
0.410
0.266
0.221
0.241
0.240
0.113
0.097
0.105
0.096
0.056
0.043
0.048
0.056
0.084
0.092
0.094
0.092
0.948
0.771
0.670
0.649
0.756
0.551
0.452
0.444
0.368
0.238
0.200
0.191
0.163
0.108
0.090
0.093
0.039
0.032
0.026
0.022
0.003
0.008
0.014
0.012
0.015
0.044
0.037
0.033
5% 10% 1%
c = 4.685
Tukey
5% 10% 1%
k = 3.291
Huber
5% 10% 1%
k = 1.345
Huber
The FGM test with various y-functions
0.990
0.909
0.847
0.835
0.908
0.764
0.701
0.674
0.621
0.455
0.398
0.397
0.382
0.274
0.231
0.225
0.132
0.098
0.077
0.087
0.032
0.035
0.038
0.038
0.065
0.118
0.116
0.111
0.996
0.952
0.917
0.897
0.950
0.863
0.803
0.785
0.738
0.577
0.508
0.526
0.523
0.400
0.343
0.332
0.204
0.175
0.150
0.161
0.077
0.077
0.087
0.068
0.120
0.185
0.194
0.174
0.625
0.527
0.550
0.559
0.414
0.340
0.350
0.339
0.175
0.143
0.148
0.136
0.063
0.051
0.052
0.044
0.014
0.012
0.014
0.012
0.003
0.005
0.003
0.003
0.006
0.012
0.012
0.016
0.798
0.737
0.752
0.764
0.638
0.573
0.568
0.574
0.355
0.297
0.311
0.313
0.179
0.152
0.158
0.162
0.063
0.059
0.068
0.056
0.024
0.026
0.021
0.025
0.044
0.045
0.054
0.050
0.857
0.811
0.826
0.848
0.757
0.678
0.687
0.694
0.476
0.407
0.420
0.432
0.286
0.236
0.249
0.258
0.122
0.104
0.114
0.108
0.056
0.052
0.052
0.056
0.082
0.091
0.111
0.095
5% 10%
c1 = 2.576
c2 = 3.291
Poly
5% 10% 1%
c = 15
Tukey
Table III. The empirical relative frequencies of rejecting a linear model based on 1000 replications of a SETAR (2;1,1) with intercept model
454
K. C. Hung et al.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
0.5
0.5
0.5
0.5
0.5
0.5
1.0 0.5
1.0 0.5
Copyright © 2009 John Wiley & Sons, Ltd.
1.0 0.5
1.0 0.5
1.0 0.5
1.0 0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.8
0.5
0.2
0.0
−0.2
−0.5
(b) Sample size n = 200
1.0 0.5 0.5 0.5
−0.8
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
1.000
0.942
0.502
0.184
0.998
0.819
0.346
0.135
0.854
0.489
0.179
0.091
0.485
0.275
0.143
0.093
0.123
0.141
0.146
0.098
0.007
0.154
0.392
0.298
0.043
0.432
0.899
0.918
1.000
0.977
0.680
0.293
0.998
0.912
0.526
0.230
0.941
0.706
0.357
0.167
0.708
0.495
0.301
0.186
0.310
0.318
0.342
0.215
0.037
0.350
0.621
0.531
0.145
0.621
0.949
0.982
1.000
0.984
0.766
0.377
1.000
0.941
0.631
0.309
0.966
0.795
0.471
0.249
0.801
0.617
0.421
0.268
0.435
0.441
0.448
0.323
0.089
0.477
0.733
0.651
0.239
0.694
0.973
0.986
0.974
0.944
0.948
0.946
0.895
0.841
0.831
0.830
0.622
0.524
0.503
0.492
0.289
0.212
0.213
0.204
0.062
0.043
0.039
0.033
0.001
0.001
0.003
0.001
0.015
0.019
0.023
0.025
0.994
0.987
0.982
0.979
0.974
0.943
0.931
0.940
0.827
0.741
0.742
0.725
0.515
0.424
0.428
0.417
0.179
0.145
0.144
0.140
0.007
0.016
0.017
0.010
0.059
0.075
0.101
0.097
0.996
0.993
0.995
0.990
0.987
0.973
0.973
0.973
0.889
0.830
0.824
0.820
0.651
0.563
0.564
0.538
0.277
0.234
0.231
0.223
0.037
0.040
0.044
0.031
0.099
0.140
0.177
0.184
1.000
0.989
0.976
0.957
0.991
0.927
0.864
0.830
0.822
0.596
0.495
0.451
0.452
0.272
0.230
0.218
0.106
0.072
0.054
0.044
0.005
0.015
0.014
0.016
0.029
0.088
0.153
0.136
1.000
0.999
0.995
0.988
0.997
0.970
0.952
0.937
0.926
0.788
0.724
0.684
0.683
0.487
0.433
0.405
0.283
0.194
0.180
0.160
0.032
0.062
0.067
0.050
0.108
0.239
0.362
0.351
1.000
0.999
0.997
0.995
0.999
0.985
0.970
0.965
0.955
0.865
0.824
0.790
0.778
0.602
0.549
0.526
0.415
0.302
0.263
0.253
0.071
0.097
0.129
0.104
0.184
0.347
0.494
0.494
0.833
0.819
0.824
0.836
0.684
0.641
0.660
0.670
0.410
0.358
0.360
0.358
0.162
0.123
0.133
0.129
0.041
0.029
0.027
0.020
0.003
0.001
0.001
0.001
0.011
0.013
0.010
0.006
0.921
0.921
0.924
0.930
0.828
0.820
0.832
0.837
0.635
0.597
0.604
0.604
0.370
0.309
0.304
0.312
0.124
0.097
0.100
0.107
0.019
0.014
0.014
0.020
0.052
0.047
0.045
0.041
0.954
0.952
0.957
0.958
0.888
0.876
0.893
0.904
0.743
0.689
0.700
0.708
0.488
0.422
0.440
0.447
0.192
0.165
0.164
0.165
0.049
0.041
0.043
0.045
0.077
0.081
0.088
0.088
0.999
0.986
0.971
0.961
0.991
0.927
0.867
0.861
0.820
0.589
0.506
0.502
0.439
0.284
0.244
0.228
0.110
0.070
0.055
0.054
0.006
0.014
0.009
0.006
0.030
0.053
0.046
0.042
1.000
0.999
0.995
0.989
0.998
0.967
0.947
0.950
0.923
0.801
0.716
0.714
0.675
0.493
0.439
0.430
0.278
0.186
0.157
0.157
0.036
0.044
0.034
0.026
0.108
0.158
0.153
0.129
1.000
1.000
0.996
0.996
0.999
0.984
0.969
0.974
0.954
0.866
0.815
0.799
0.779
0.618
0.558
0.550
0.402
0.287
0.249
0.246
0.068
0.081
0.068
0.067
0.178
0.243
0.238
0.192
0.908
0.872
0.886
0.888
0.781
0.707
0.723
0.730
0.490
0.390
0.396
0.417
0.193
0.147
0.158
0.149
0.037
0.022
0.028
0.025
0.005
0.003
0.002
0.004
0.012
0.013
0.014
0.011
0.969
0.958
0.956
0.969
0.908
0.871
0.879
0.890
0.708
0.623
0.632
0.627
0.391
0.323
0.319
0.337
0.138
0.106
0.101
0.106
0.018
0.020
0.019
0.023
0.040
0.051
0.062
0.050
0.980
0.972
0.976
0.985
0.942
0.927
0.931
0.940
0.787
0.725
0.739
0.743
0.518
0.448
0.457
0.452
0.202
0.171
0.166
0.177
0.046
0.047
0.046
0.044
0.080
0.091
0.107
0.101
A Robust Test for SETAR-Type Nonlinearity
455
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
Copyright © 2009 John Wiley & Sons, Ltd.
0.5
0.5
1.0 0.5
1.0 0.5
0.5
1.0 0.5
0.5
0.5
1.0 0.5
1.0 0.5
0.5
1.0 0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.8
0.5
0.2
0.0
−0.2
−0.5
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0.560
0.307
0.160
0.109
0.330
0.190
0.115
0.082
0.137
0.105
0.099
0.082
0.062
0.076
0.095
0.100
0.021
0.063
0.091
0.077
0.006
0.052
0.132
0.116
0.029
0.161
0.319
0.302
f (1)
f 1(1) f (2)
f 1(2) w a = 1%
0
0
(a) Sample size n = 100
1.0 0.5 0.5 0.5
−0.8
r
Parameters
0.765
0.539
0.315
0.202
0.587
0.400
0.247
0.164
0.339
0.265
0.217
0.160
0.206
0.211
0.208
0.178
0.100
0.176
0.201
0.167
0.034
0.191
0.307
0.257
0.112
0.303
0.529
0.501
0.850
0.636
0.401
0.266
0.692
0.514
0.343
0.237
0.459
0.374
0.325
0.230
0.333
0.333
0.331
0.255
0.160
0.270
0.312
0.256
0.079
0.283
0.428
0.371
0.194
0.398
0.638
0.631
0.194
0.171
0.164
0.160
0.103
0.093
0.087
0.084
0.045
0.041
0.043
0.043
0.022
0.020
0.023
0.015
0.009
0.012
0.011
0.011
0.002
0.005
0.003
0.003
0.023
0.022
0.018
0.015
5% 10% 1%
LS
0.415
0.352
0.347
0.343
0.262
0.236
0.227
0.224
0.148
0.134
0.130
0.122
0.092
0.075
0.069
0.079
0.044
0.052
0.044
0.048
0.027
0.027
0.024
0.022
0.081
0.076
0.071
0.067
0.533
0.474
0.456
0.456
0.398
0.340
0.333
0.332
0.233
0.222
0.221
0.221
0.175
0.151
0.146
0.150
0.086
0.090
0.091
0.085
0.042
0.050
0.046
0.042
0.140
0.138
0.136
0.140
0.441
0.298
0.248
0.226
0.248
0.168
0.137
0.119
0.103
0.068
0.066
0.062
0.047
0.031
0.040
0.040
0.018
0.022
0.018
0.012
0.005
0.007
0.008
0.014
0.030
0.054
0.057
0.038
0.662
0.515
0.443
0.412
0.511
0.367
0.325
0.317
0.277
0.216
0.191
0.160
0.176
0.128
0.114
0.113
0.075
0.084
0.075
0.071
0.035
0.044
0.044
0.045
0.115
0.155
0.158
0.132
0.765
0.619
0.550
0.529
0.616
0.492
0.443
0.415
0.396
0.310
0.293
0.253
0.285
0.219
0.205
0.191
0.145
0.156
0.146
0.132
0.073
0.099
0.105
0.087
0.184
0.242
0.270
0.232
0.109
0.096
0.093
0.097
0.062
0.059
0.057
0.057
0.036
0.029
0.033
0.029
0.023
0.017
0.016
0.018
0.010
0.002
0.009
0.010
0.003
0.006
0.003
0.006
0.023
0.023
0.020
0.018
0.240
0.212
0.219
0.230
0.164
0.150
0.163
0.156
0.098
0.093
0.105
0.095
0.069
0.058
0.052
0.063
0.042
0.035
0.036
0.035
0.024
0.025
0.021
0.021
0.074
0.070
0.067
0.063
0.347
0.322
0.327
0.346
0.238
0.240
0.238
0.244
0.153
0.168
0.176
0.180
0.122
0.100
0.103
0.113
0.078
0.078
0.068
0.067
0.056
0.043
0.048
0.056
0.111
0.113
0.103
0.113
0.445
0.311
0.244
0.240
0.251
0.167
0.134
0.127
0.105
0.067
0.066
0.068
0.048
0.037
0.040
0.041
0.016
0.022
0.017
0.022
0.003
0.008
0.014
0.012
0.028
0.048
0.028
0.022
5% 10% 1%
c = 4.685
Tukey
5% 10% 1%
k = 3.291
Huber
5% 10% 1%
k = 1.345
Huber
The FGM test with various y-functions
0.662
0.514
0.439
0.442
0.505
0.374
0.318
0.303
0.274
0.212
0.187
0.180
0.177
0.127
0.112
0.120
0.080
0.081
0.065
0.071
0.032
0.035
0.038
0.039
0.113
0.131
0.105
0.087
0.762
0.625
0.555
0.548
0.621
0.493
0.442
0.420
0.399
0.314
0.273
0.272
0.279
0.218
0.190
0.191
0.134
0.136
0.138
0.114
0.077
0.077
0.087
0.068
0.184
0.213
0.171
0.153
0.154
0.117
0.119
0.127
0.080
0.068
0.072
0.069
0.036
0.034
0.038
0.029
0.018
0.019
0.015
0.015
0.006
0.005
0.009
0.007
0.003
0.005
0.003
0.003
0.014
0.012
0.012
0.014
0.314
0.267
0.277
0.284
0.195
0.170
0.168
0.169
0.112
0.107
0.121
0.110
0.070
0.057
0.067
0.065
0.033
0.033
0.038
0.038
0.024
0.026
0.021
0.025
0.056
0.061
0.062
0.057
0.412
0.373
0.371
0.388
0.315
0.256
0.269
0.267
0.192
0.178
0.189
0.189
0.133
0.100
0.111
0.118
0.080
0.070
0.074
0.072
0.056
0.052
0.052
0.056
0.105
0.108
0.107
0.105
5% 10%
c1 = 2.576
c2 = 3.291
Poly
5% 10% 1%
c = 15
Tukey
Table IV. The empirical relative frequencies of rejecting a linear model based on 1000 replications of a SETAR (2;1,1) without intercept model
456
K. C. Hung et al.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
0.5
0.5
0.5
0.5
0.5
0.5
1.0 0.5
1.0 0.5
Copyright © 2009 John Wiley & Sons, Ltd.
1.0 0.5
1.0 0.5
1.0 0.5
1.0 0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.8
0.5
0.2
0.0
−0.2
−0.5
(b) Sample size n = 200
1.0 0.5 0.5 0.5
−0.8
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0
3
6
10
0.926
0.673
0.288
0.127
0.761
0.471
0.230
0.130
0.412
0.295
0.199
0.111
0.203
0.205
0.197
0.128
0.067
0.164
0.219
0.144
0.007
0.154
0.392
0.298
0.096
0.391
0.722
0.693
0.976
0.836
0.501
0.242
0.897
0.706
0.418
0.227
0.650
0.523
0.394
0.237
0.420
0.423
0.413
0.286
0.196
0.358
0.456
0.330
0.037
0.350
0.621
0.531
0.244
0.553
0.884
0.876
0.991
0.908
0.593
0.334
0.943
0.808
0.554
0.318
0.752
0.644
0.531
0.333
0.532
0.553
0.534
0.387
0.287
0.476
0.601
0.460
0.089
0.477
0.733
0.651
0.357
0.649
0.925
0.926
0.506
0.464
0.448
0.450
0.329
0.261
0.261
0.265
0.144
0.117
0.119
0.111
0.063
0.058
0.058
0.051
0.018
0.019
0.021
0.020
0.001
0.001
0.003
0.001
0.044
0.032
0.027
0.025
0.715
0.675
0.680
0.680
0.550
0.495
0.501
0.496
0.317
0.298
0.289
0.284
0.199
0.194
0.187
0.179
0.069
0.079
0.073
0.063
0.007
0.016
0.017
0.010
0.120
0.115
0.120
0.103
0.814
0.783
0.779
0.777
0.659
0.621
0.605
0.605
0.458
0.429
0.414
0.413
0.294
0.281
0.285
0.266
0.139
0.139
0.125
0.127
0.037
0.040
0.044
0.031
0.198
0.199
0.197
0.186
0.842
0.686
0.599
0.569
0.618
0.427
0.366
0.350
0.314
0.231
0.181
0.161
0.149
0.105
0.081
0.071
0.046
0.049
0.043
0.038
0.005
0.015
0.014
0.016
0.087
0.108
0.114
0.089
0.932
0.856
0.803
0.772
0.828
0.665
0.601
0.570
0.570
0.439
0.384
0.363
0.356
0.249
0.251
0.223
0.150
0.134
0.110
0.114
0.032
0.062
0.067
0.050
0.210
0.265
0.293
0.253
0.973
0.909
0.876
0.847
0.899
0.780
0.728
0.689
0.697
0.555
0.512
0.488
0.451
0.371
0.362
0.335
0.240
0.230
0.207
0.189
0.071
0.097
0.129
0.104
0.338
0.382
0.421
0.377
0.258
0.241
0.249
0.271
0.164
0.136
0.147
0.156
0.078
0.082
0.070
0.066
0.032
0.037
0.030
0.031
0.016
0.013
0.014
0.013
0.003
0.001
0.001
0.001
0.025
0.016
0.016
0.024
0.476
0.445
0.462
0.464
0.351
0.314
0.326
0.334
0.200
0.194
0.185
0.186
0.129
0.134
0.132
0.123
0.063
0.059
0.044
0.045
0.019
0.014
0.014
0.020
0.072
0.066
0.064
0.072
0.575
0.566
0.580
0.576
0.449
0.431
0.435
0.447
0.296
0.277
0.271
0.269
0.206
0.207
0.201
0.202
0.102
0.098
0.093
0.097
0.049
0.041
0.043
0.045
0.121
0.111
0.118
0.122
0.841
0.693
0.600
0.590
0.619
0.432
0.376
0.368
0.329
0.225
0.171
0.156
0.150
0.098
0.077
0.077
0.050
0.036
0.028
0.029
0.006
0.014
0.009
0.006
0.085
0.067
0.056
0.048
0.938
0.855
0.798
0.787
0.838
0.664
0.610
0.604
0.576
0.435
0.367
0.360
0.350
0.253
0.218
0.206
0.152
0.122
0.105
0.088
0.036
0.044
0.034
0.026
0.211
0.198
0.146
0.133
0.971
0.900
0.872
0.855
0.900
0.776
0.734
0.725
0.688
0.545
0.488
0.487
0.452
0.360
0.330
0.309
0.247
0.194
0.166
0.144
0.068
0.081
0.068
0.067
0.322
0.284
0.232
0.204
0.364
0.294
0.318
0.334
0.215
0.172
0.185
0.190
0.089
0.095
0.082
0.078
0.045
0.049
0.049
0.039
0.012
0.008
0.012
0.011
0.005
0.003
0.002
0.004
0.021
0.013
0.020
0.018
0.573
0.526
0.552
0.547
0.419
0.376
0.370
0.378
0.219
0.204
0.199
0.216
0.146
0.148
0.149
0.138
0.067
0.050
0.052
0.050
0.018
0.020
0.019
0.023
0.075
0.060
0.061
0.066
0.673
0.642
0.660
0.656
0.511
0.498
0.482
0.493
0.325
0.311
0.298
0.313
0.249
0.234
0.223
0.218
0.123
0.107
0.103
0.099
0.046
0.047
0.046
0.044
0.135
0.130
0.120
0.119
A Robust Test for SETAR-Type Nonlinearity
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tests increase. In general, an increase in (w) will slightly lower the power of the proposed FGM tests,
but the effects can often be diluted by increasing the sample size. For example, in Table III (n = 100
with a = 5%), when f1(2) = −0.8, the powers for the FGM test (with c = 15 Tukey y-function) are
0.988 (w = 0), 0.899 (w = 3), and 0.860 (w = 6). For n = 200, the powers are 1.000 (w = 0), 0.996
(w = 3), and 0.990 (w = 6). Nevertheless, our simulations show that the proposed FGM tests can retain
a reasonable level of power in the presence of outliers.
In Tables III and IV, when f 1(2) = 0.8, the proposed tests have low power. This is not surprising because the nonlinearity is not strong. On the other hand, the LS test seems to have some
power, even if one uses the empirical critical values. We perform a simple simulation to study the
‘spurious’ power of the LS test in that situation. The DGP is a contaminated AR(1) model: Yt =
0.5 + fYt−1 + et, and the contamination mechanism is according to equation (23). Table V shows the
empirical rejection rates using the LS test with a = 5%, b = 5%, w = 0, 3, 6, 10 and various values
of f. The results indicate that the rejection rates of the LS test will normally be inflated when w
increases. The degree of inflation becomes quite extreme when f is close to 1 and w is large. The
‘spurious’ power of the LS test when f 1(2) = 0.8 in Tables III and IV might be due to this kind of
erroneous inflation of rejection rates.
As for the choice of the y-functions and tuning constants for the FGM tests, the simulation results
show that the Huber function (with k = 3.291) and Tukey’s biweight function (with c = 15) often
have higher powers than the other tests. For practical applications, we recommend these two tests
because they are robust with respect to outlying observations and do not result in large Type I errors
in the presence of outliers.
The results in Tables III and IV assume that the model is known. It might be interesting to study
the effects of additive outliers on SETAR-type nonlinearity tests, if the AR order is misspecified.
Another issue of interest is the effects of a patch of additive outliers (Justel et al., 2001). External
events may affect the underlying time series in a finite number of consecutive time periods. This
situation is less relevant in linear regression, but is important in time series analysis. Therefore, it
is important to examine the robustness of the proposed FGM test with respect to a patch of additive
outliers.
In order to examine the above two issues, we conduct another simulation experiment. The first
outlier-free DGP is a linear AR(3) model:
Yt = 2.1Yt −1 − 1.46Yt − 2 + 0.336Yt − 3 + ε t
(24)
Table V. The empirical relative frequencies of rejecting a linear
model based on 1000 replications of an AR(1) process, the LS test with
n = 100
w
0
3
6
10
f
0.95
0.8
0.5
0.2
0.038
0.263
0.668
0.853
0.032
0.313
0.676
0.742
0.034
0.191
0.307
0.257
0.043
0.074
0.079
0.079
Copyright © 2009 John Wiley & Sons, Ltd.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
A Robust Test for SETAR-Type Nonlinearity
459
with s2a = 1. The roots of the autoregressive polynomial are 0.6, 0.7 and 0.8, so that the model is
stationary. This model has been considered by Justel et al. (2001). The second outlier-free DGP is
a nonlinear SETAR(2;3,1) process:
Yt =
{
2.1Yt −1 − 1.46Yt − 2 + 0.336Yt − 3 + ε t
1.2Yt −1 − 0.74Yt − 2 + 0.060Yt − 3 + ε t
if Yt −1 ≤ −1.5
if Yt −1 > −1.5
(25)
with AR roots (0.6, 0.7, 0.8) in regime 1, and (0.3, 0.4, 0.5) in regime 2. The threshold parameter
r = −1.5 is chosen such that there are adequate observations in both regimes for efficient parameter
estimation and nonlinearity testing.
Time series data (with sample size n = 100) are generated from the two DGPs. We consider two
outlier contamination situations: (i) no outliers, and (ii) a patch of four consecutive additive outliers,
from t = 68 to t = 71, with sizes (11, 10, 9, 10). The proposed FGM tests with a = 5% and various
AR orders (p = 1 to 5) are applied to the simulated series. It should be noted that the correct AR
specification is p = 3. Table VI reports the empirical relative frequencies of rejecting the linear null
hypothesis based on 1000 replications.
When the underlying DGP is a linear AR(3) process, the rejection rates in Table VI are just the
empirical sizes of the tests. The problem of AR order misspecification does not seem to create
adverse effects on the sizes of the tests when there are no outliers. As can be seen from the
first panel of results in Table VI, all the figures are reasonably close to the nominal 5% level.
However, when a patch of additive outliers is introduced, the empirical sizes of the LS test become
significantly larger than the nominal level, except for the case of p = 1. We will attempt to explain
this exception later. On the other hand, the proposed FGM tests can keep the sizes steady at around
the nominal level without being affected by the patch of outliers (see the second panel of results in
Table VI).
When the underlying DGP is a nonlinear SETAR(2;3,1) process, the rejection rates in Table VI
(the third and the fourth panels) indicate the empirical powers of the tests. In general, the powers of
the proposed FGM tests are robust with respect to outliers, as well as to the specification of the AR
order. Analogous to the linearity case, the rejection rates of the LS test are significantly inflated due
to the adverse effects of the patch of outliers. On the other hand, when the under-specification of
the AR order is significant, the LS test might create a sudden drop in rejection rates. For example,
in the third panel of Table VI, the rejection rates for the LS test are 0.428 (p = 3), 0.457 (p = 2) and
0.013 (p = 1). This phenomenon can provide an explanation for the exception mentioned above,
where the empirical size of the LS test is not inflated by the effect of additive outliers for the case
of p = 1. The inflation effect by the outliers might be neutralized by the pushdown effect of the
under-specification of the AR order.
REAL EXAMPLES
In this section, we apply the new robust tests to analyze the linearity behavior (against the SETARtype nonlinearity alternative) of three well-known real datasets. In addition to applying FGM tests on
the data, we would like to investigate the consistency of the results with the commonly used SETARCopyright © 2009 John Wiley & Sons, Ltd.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
460
K. C. Hung et al.
Table VI. The empirical relative frequencies of rejecting a linear model based on 1000 replications for the FGM
test (a = 5%) with various specifications of the AR order p and a patch of additive outliers, n = 100
The FGM test with various y-functions
AR specification
k = 1.345
Huber
k = 3.291
Huber
c = 4.685
Tukey
c = 15
Tukey
c1 = 2.576
c2 = 3.291
Poly
0.010
0.065
0.046
0.045
0.037
0.014
0.073
0.039
0.039
0.033
0.009
0.089
0.067
0.049
0.056
0.010
0.075
0.039
0.041
0.032
0.002
0.081
0.053
0.047
0.034
(ii) With a patch of AO’s from t = 68 to 71, with sizes (11, 10, 9, 10)
1
0.016
0.014
0.018
0.014
2
0.410
0.057
0.061
0.095
3
0.399
0.039
0.038
0.065
4
0.390
0.027
0.031
0.047
5
0.543
0.028
0.013
0.050
0.017
0.044
0.019
0.015
0.007
0.010
0.092
0.056
0.041
0.039
(b) DGP: Nonlinear SETAR(2;3,1) process
(i) No outliers
1
0.013
0.004
2
0.457
0.367
3
0.428
0.299
4
0.375
0.217
5
0.340
0.193
0.002
0.217
0.167
0.133
0.096
0.005
0.456
0.381
0.280
0.206
0.008
0.234
0.178
0.128
0.106
(ii) With a patch of AO’s from t = 68 to 71, with sizes (11, 10, 9, 10)
1
0.091
0.016
0.046
0.002
2
0.746
0.318
0.406
0.188
3
0.665
0.241
0.302
0.167
4
0.630
0.166
0.207
0.106
5
0.801
0.159
0.130
0.099
0.028
0.196
0.137
0.068
0.028
0.006
0.191
0.141
0.085
0.068
p
(a) DGP: Linear AR(3)
(i) No outliers
1
2
3
4
5
LS
process
0.012
0.057
0.038
0.030
0.033
0.012
0.517
0.446
0.338
0.281
type nonlinearity tests, which include Tsay’s F test (Tsay, 1989); the CUSUM test (Petruccelli and
Davies, 1986); the RC test (Petruccelli, 1990); the bootstrap version of the likelihood ratio LR–BS
test (Hansen, 1996), and the Lagrange multiplier LM test (Luukkonen et al., 1988). Chan and Ng
(2004) provide a brief review of these five tests.
We first consider the Australian blowfly data (BLOWFLY). These are the bi-daily population
sizes of blowflies obtained by Nicholson (1957). The dataset, together with a detailed description
and nonlinear time series analysis, are given in Tong (1990). We shall only use the first 206 observations, because many researchers found SETAR-type nonlinearity and/or aberrant observations in
this portion of the data (e.g., Tsay, 1988; Tong, 1990; Wong and Kohn, 1996). Following Tsay
(1988), we employ Yt = Zt/1000 transformation, where Zt is the original series.
Copyright © 2009 John Wiley & Sons, Ltd.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
A Robust Test for SETAR-Type Nonlinearity
461
Table VII. SETAR-type nonlinearity test results for the real datasetsa
Test
Datasets
(a) Non-robust tests
F
CUSUM
RC
LR–BS
LM
(b) Robust FGM tests
Huber (k = 1.345)
Huber (k = 3.291)
Tukey (c = 4.685)
Tukey (c = 15)
Poly (c1 = 2.576, c2 = 3.291)
a
BLOWFLY
GAS
UORDER
NL (0.000)
L (0.317)
NL (0.030)
NL (0.022)
NL (0.000)
NL (0.000)
NL (0.000)
NL (0.003)
NL (0.000)
NL (0.000)
L (0.222)
L (0.133)
L (0.085)
L (0.424)
L (0.085)
NL (0.000)
NL (0.000)
NL (0.007)
NL (0.000)
L (0.060)
L (0.320)
L (0.116)
L (0.361)
L (0.120)
L (0.543)
L (0.235)
L (0.332)
L (0.604)
L (0.294)
L (0.907)
L, linear; NL, nonlinear; p-values are in parentheses.
For the second dataset, we consider the monthly average wholesale prices of regular leaded gasoline in the United States between January 1973 and December 1978 (GAS). The data are listed in
Liu (1991). The studies by Chan and Cheung (1994) and Chen (1997) conclude that nonlinear and/or
outlier models should be employed for this time series. We shall use the first differenced series of
the logarithmic transformed data in our analysis.
The third real dataset is the monthly US industry-unfilled orders for radio and TV, in millions of
dollars, from January 1958 to October 1980 (UORDER). We use the logged data and focus on the
seasonally adjusted series, where the seasonal component was removed by the well-known X11ARIMA procedure. The original data are listed in Zellner (1983). Justel et al. (2001) detect a patch
of additive outliers in the seasonally adjusted series.
For the practical application of the tests, we need to determine p and d. Following Tsay (1989)
and Sarantis (2001), we estimate the model order p by the cutoff pattern of the sample partial autocorrelation function (SPACF) of the observed series. The value of d is then searched among
{1, 2, . . . , p} such that its corresponding p-value of the nonlinearity test is the smallest.
Table VII provides the test results at the 5% level. Both the existing non-robust tests and the proposed FGM tests indicate SETAR-type nonlinearity for the BLOWFLY dataset in most cases. After
downweighting the aberrant observations in the dataset, the p-values of the proposed FGM tests are
in general less than those of the non-robust tests. The results for the GAS dataset are interesting. All
the traditional non-robust tests identify the data as NL (nonlinear), while all of the FGM tests support
the linearity (L) conclusion. The robust and non-robust tests give completely different conclusions
in this dataset. Finally, all tests favor linearity conclusion of the UORDER dataset. The p-values for
the non-robust tests are much less than those of the proposed FGM tests, which might be due to the
potential effect of the patch of additive outliers in the data.
CONCLUSION
We have investigated the robustness properties of the classical Tsay (1989) F test for SETAR-type
nonlinearity. The test is not robust to extreme additive outliers, and becomes very unreliable when
Copyright © 2009 John Wiley & Sons, Ltd.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
462
K. C. Hung et al.
the data contain large outlying values. In practice, time series outliers are not encountered very often,
so it is important to develop a more robust test for SETAR-type nonlinearity in time series analysis
and forecasting. In this paper we propose a new robust nonlinearity test and the asymptotic null
distribution of the test statistic is derived. A Monte Carlo experiment is carried out to compare the
power of the proposed test and other existing tests under the influence of time series outliers. The
effects of additive outliers on nonlinearity tests with misspecification of the autoregressive order are
also studied. The results indicate that the proposed method is preferable to the classical tests when
the observations are contaminated with outliers. In the empirical study we applied the tests to three
well-known datasets, which have been judged to be outlier-contaminated by many time series analysts. The proposed FGM tests give consistent conclusions for each dataset under different choices of
y-functions and they do not seem to be adversely affected by the outliers. The overall performance
of the proposed tests is supported by the real examples.
ACKNOWLEDGEMENTS
The authors thank the departmental editor and an anonymous referee for their valuable suggestions,
which helped to shape the final version of this paper. This work was supported by a grant from the
Research Grants Council of the Hong Kong Special Administrative Region (Competitive Earmarked
Research Grant Project No. CUHK7413/05H).
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Authors’ biographies:
King Chi Hung gained her MPhil degree in Statistics from the Chinese University of Hong Kong in 2008. Her
research interests include robust statistics and nonlinear time series analysis.
Siu Hung Cheung is a professor in the department of Statistics at the Chinese University of Hong Kong and an
adjunct professor in the Department of Statistics at National Cheng Kung University. He received a PhD in Statistics from Temple University, USA. His current research interests include multiple comparisons, adaptive designs
and nonlinear time series analysis.
Wai-Sum Chan is a professor in the Department of Finance at the Chinese University of Hong Kong. He received
a PhD in Statistics from Temple University, USA. His current research interests include nonlinear time series
analysis, clinical statistics and actuarial modeling.
Li-Xin Zhang is a Professor in the Institute of Statistics and Department of Mathematics at Zhejiang University.
He received a PhD in Statistics from Fudan University, PR China. His current research interests include clinical
statistics, limit theorems, stochastic processes and time series analysis.
Copyright © 2009 John Wiley & Sons, Ltd.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for
464
K. C. Hung et al.
Authors’ addresses:
King Chi Hung, Department of Statistics, the Chinese University of Hong Kong, Shatin, Hong Kong.
Siu Hung Cheung, Department of Statistics, The Chinese University of Hong Kong, Shatin, Hong Kong; Department of Statistics, National Cheng Kung University, Tainan, Taiwan.
Wai-Sum Chan, Department of Finance, the Chinese University of Hong Kong, Shatin, Hong Kong.
Li-Xin Zhang, Department of Mathematics, Zhejiang University, Zheda Road No. 38, Hangzhou 310027, PR
China.
Copyright © 2009 John Wiley & Sons, Ltd.
J. Forecast. 28, 445–464 (2009)
DOI: 10.1002/for