This paper studies a special class of vector smooth-transition autoregressive (VS- TAR) models that contains common nonlinear features (CNFs), for which we proposed a triangular representation and developed a procedure of testing CNFs in a VSTAR model.
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Working papers in transport, tourism, information technology and microdata analysis
Testing Common Nonlinear Features in Nonlinear Vector
Autoregressive Models
Dao Li
Changli He
Editor: Hasan Fleyeh
Working papers in transport, tourism, information technology and microdata analysis
First version: September 2010, Second version: September 2011,
October 24, 2012
Abstract
This paper studies a special class of vector smooth-transition autoregressive (VS-
TAR) models that contains common nonlinear features (CNFs), for which we proposed a
triangular representation and developed a procedure of testing CNFs in a VSTAR model.
We rst test a unit root against a stable STAR process for each individual time series
and then examine whether CNFs exist in the system by Lagrange Multiplier (LM) test if
unit root is rejected in the rst step. The LM test has standard Chi-squared asymptotic
distribution. The critical values of our unit root tests and small-sample properties of
the F form of our LM test are studied by Monte Carlo simulations. We illustrate how
to test and model CNFs using the monthly growth of consumption and income data of
United States (1985:1 to 2011:11).
Keywords: Common features, Lagrange Multiplier test, Vector STAR models.
1 . Introduction
A family of STAR models has been comprehensively studied for modelling nonlin-
ear structures such as data break and regime switching in time series data since
Teräsvirta (1994) introduced many theoretical issues and applications of the STAR
models. The extensions of the univariate STAR models have been developed for
exploring nonlinearity of time series data. Escribano (1987) and Granger and
Swanson (1996) introduced smooth-transition error-correction model for nonlin-
ear adjustment. Johansen (1995) incorporated the smooth-transition mechanism
∗Correspondence to: Dao Li, Statistics, School of Technology and Business Studies, DalarnaUniversity, 78170 Sweden, E-mail: [email protected], Tel: +46-(0)23-778509; Örebro University Schoolof Business, Sweden. The material from this paper has been presented at GRAPES workshop,Örebro, August 2010, Econometric Society Australasian Meeting, Melbourne, July 2012 and the66th European Meeting of the Econometric Society, Málaga, August 2012.
CNFs in VSTAR Models 2
in an error-correction model. Moreover, van Dijk and Franses (1999) contributed
that multiple regime STAR models could accommodate more than two regimes
in which it cannot be done by the STAR models in Teräsvirta (1994). Lin and
Teräsvirta (1994) and He and Sandberg (2006) studied time-varying STAR models
which are characteristic for the time series of structural instability in time spans.
However, there has not been much research work undertaken for a multivariate
nonlinear system and modelling vector nonlinear time series is little considered in
the literature. There is a discussion about parameters constancy in a VSTAR
model in He et al. (2009).
This paper studies a special class of VSTAR models that contains CNFs. A
group of time series is said to have CNFs if each of the time series is nonlin-
ear but that their combination does not contain nonlinearity. Modelling vector
nonlinear time series can be moderately improved by imposing CNFs. Previous
studies of common features are mostly related to the generalized method of mo-
ments estimator and canonical correlation, for example, Engle and Kozicki (1993)
and Anderson and Vahid (1998). However, we propose an LM test based on a
triangular representation of the system that contains CNFs to investigate CNFs
in a VSTAR model. In the triangular representation, testing CNFs in a VSTAR
model is to test whether the population residual in the regression of CNF relations
is linear. The proposed method is convenient to perform.
The empirical application in this paper concerns the monthly growth of con-
sumption expenditures and disposable income of United States, dating from 1985:1
until 2011:11. The permanent-income hypothesis and the study in Campbell and
Mankiw (1990) imply that consumption and income are linearly cointegrated in
the levels (see Engle and Granger; 1987, for more discussions). But are there any
other common features in those two dynamic time series? Removing the possi-
bility of cointegration by looking at the growth, we investigate whether there are
CNFs between consumption and income when each of them presents nonlinearity.
The remaining paper is outlined as follows. Section 2 studies VSTAR models;
Section 3 discusses CNFs in a VSTAR model; The procedure of testing CNFs
3 CNFs in VSTAR Models
is provided in Section 4; The asymptotic distribution of our LM test is derived
in Section 5; Simulation studies are in Section 6; An empirical application is
illustrated in Section 7; and conclusions can be found in Section 8.
The probability shown at the head of the column is the area in the left-hand tail.
in which G (st; γ, c) = (1 + exp −3(yt−1 − 0.5))−1 and (v1t, v2t)′ ∼ n.i.d(0, 0.1I).
The CNFs in yt is expressed in terms of0.05 +(
0.12 0.18) y1t−1
y2t−1
G (st; γ, c) .
The estimation of parameter ψ is from (4.16) under the null of θ = 0 in which,
in this case, f(yt−s;ψ) has a form
f(yt−s;ψ) =µ+ δ1y1t−1 + δ2y1t−2 +3∑
s=−2
δ3sy2t−s
+2∑
s=−2
δ4sy2t−1−sG (st−s; γ, c) +2∑
s=−2
δ5sG (st−s; γ, c).
To calculate our FLM statistic, wt is given in details such that
wt =(y21t−1, y1t−1y2t−1, y22t−1, y
21t−2, y1t−2y2t−2, y
22t−2, y1t−1y1t−2, y1t−1y2t−2,
y2t−1y1t−2, y2t−1y2t−2)′.
CNFs in VSTAR Models 18
Table 3: The estimated size of FLM using critical values of F .given signicance levels
T 10% 5% 2.5% 1%
100 0.123 0.068 0.026 0.011
200 0.135 0.077 0.032 0.016
Table 4: Empirical power of FLM test using critical values of F .given signicance levels
T 10% 5% 2.5% 1%
100 0.139 0.076 0.041 0.018
200 0.153 0.081 0.044 0.016
1000 0.333 0.228 0.153 0.089
We compare the small-sample distribution of our FLM test to the F -distribution
and present the estimated size in Table 3. Our FLM test is oversize, but it is
acceptable because the number of the parameters is relatively large compared
to the small sample sizes in nonlinear least square estimation. The numerical
problem, due to that the residual vector may not be orthogonal to the gradient
matrix, is the potential reason of the size problem.
TheDGP under the alternative for power study follows a VSTAR model (with-
out CNFs) such that
y1t
y2t
=
0.4
0.2
+
0.3 0.05
0.2 −0.1
y1t−1
y2t−1
+
0.1
−0.16
+
0.24 −0.18
0.12 −0.36
y1t−1
y2t−1
G (st; γ, c) +
v1t
v2t
in which G (st; γ, c) = (1 + exp −3(yt−1 − 0.5))−1 and (v1t, v2t)
′ ∼ n.i.d(0, 0.1I).
The way of how to perform our FLM test is similar to it is in the previous size
study.
Table 4 gives the empirical power of our FLM statistic for both small samples
and large sample. It is obvious that power increases as sample size increases,
although power is low. Several tests in related area in the literature commonly
19 CNFs in VSTAR Models
1985 1990 1995 2000 2005 2010
−2
−1
0
1
2
1985 1990 1995 2000 2005 2010
−4
−2
0
2
4
Figure 4: Percentage change of personal consumption expenditures (upper) and disposableincome (lower), United States, billions of 2005 dollars, monthly, 1985:1-2011:11.
suer such power problems, for example, Eitrheim and Teräsvirta (1996). One
possible reason is that, in the simulations, it is dicult to make the nonlinear
estimators good in each replication when only one set of initial values of the
parameters is used in all replications as we mostly do. Sometimes, the iterations
cannot converge with inappropriate initial values.
7 . Application
7.1 Data
The data set, monthly personal consumption expenditures and personal dispos-
able income in United States dating from 1985:1 until 2011:11 (323 observations),
is download from U.S. bureau of economic analysis to analyse CNFs between con-
sumption and income. Both of them are seasonally adjusted at annual rates.
Natural logs of the raw data were taken and then multiplied by 100. Figure 4
presents the percentage growth (∆y1t and ∆y2t). We use the observations from
1985:l to 2006:l2 (264 observations) for modeling and save the remaining ones for
out-of-sample forecasting (future work).
CNFs in VSTAR Models 20
Table 5: The unit root tests.DF statistics critical values
time series st t F t F
∆y1t ∆2y1t−1 -2.7476 4.3017 -2.87 3.12
∆y1t−1 -2.6744 4.8862 -1.96 3.52
∆y2t ∆2y2t−1 -5.7069 18.029 -2.87 3.12
∆y2t−1 -5.6537 17.810 -1.96 3.52
The data in two gures have obvious uctuations in the late 1980s, the early
1990s and 2000s. A well-known credit crunch of U.S. savings and loan was during
1980s and led to the U.S. recession of 1990-1991 as a major factor. Another well-
known nancial crisis was the crash of the dot-com bubble in 2000-2001. In late
2008 and 2009, the subprime mortgage crisis and the bursting of other real estate
bubbles around the world has led to recession in the U.S. and a number of other
countries, and there is a now-deating United States housing bubble. Besides, the
international nancial crises, as many Latin American countries defaulted on their
debt in the early 1980s, may cause the big data break in the middle of 1980s. The
data behavior suggest that a nonlinear model could be a good consideration.
7.2 Testing procedure
Applying our procedure to consumption and income growth, our unit root tests
in Section 4.1 is imposed rst. Table 5 presents the results of Dickey-Fuller t and
F statistics for those unit root tests. From Table 5, both ∆y1t and ∆y2t likely
perform as nonlinearly stabilized processes.
In the following analysis, we are interested in whether ∆y1t and ∆y2t have
CNFs by the testing procedure in Section 4.2. The test regression follows (4.16)
choosing p = 1 and l = 2. Based on the nonlinear least square estimation, the
FLM test statistic in (4.23) is calculated. FLM = 1.7526 and the critical value is
2.11. The data is unable to reject the null of CNFs. It is reasonable to believe
that there are CNFs in the system of consumption and income.
21 CNFs in VSTAR Models
Table 6: Choose transition variable.data test st
x1,t−1 x1,t−2 x1,t−3 x1,t−4 x1,t−5 x1,t−6
x1t t -2.6744 -2.7522 -3.1715 -2.3702 -2.4773 -2.5426
F 4.8862 3.7963 8.7313 5.9177 6.1487 3.8504
x2t t -4.6066 -5.3846 -4.5765 -5.0576 -4.7381 -5.2491
F 19.504 14.806 15.725 15.762 15.604 14.271
data test stx1,t−7 x1,t−8 x1,t−9 x1,t−10 x1,t−11 x1,t−12
x1t t -3.1982 -2.5386 -2.4964 -2.0108 -3.4666 -2.9038
F 6.1462 3.7753 5.1453 6.4307 9.1321 6.0254
x2t t -5.2798 -4.2107 -5.3020 -5.3532 -5.3216 -4.4308
F 24.610 20.225 14.169 14.402 16.495 19.319
data test stx2,t−1 x2,t−2 x2,t−3 x2,t−4 x2,t−5 x2,t−6
x1t t -2.8004 -2.5715 -2.6882 -2.7418 -2.8424 -2.6694
F 4.1458 4.2530 3.7920 3.9519 4.0765 4.1619
x2t t -5.6537 -5.6306 -5.2734 -5.3828 -5.1479 -5.4606
F 17.810 16.894 14.196 14.506 15.249 15.322
data test stx2,t−7 x2,t−8 x2,t−9 x2,t−10 x2,t−11 x2,t−12
x1t t -2.8831 -2.7102 -2.6630 -2.4212 -2.5734 -2.7697
F 4.4869 3.7515 3.9776 4.8118 4.4453 3.9163
x2t t -5.6110 -4.5739 -5.0931 -5.4967 -5.2708 -5.4674
F 17.485 17.585 16.843 15.108 14.160 16.565
*Critical values of Dickey-Fuller t and F tests at 5% signicance level are -1.96 and3.92, respectively.
7.3 Modeling
Based on the conclusion above, we want to model the dierenced data ∆y1t and
∆y2t by a VSTAR model containing CNFs. Let xit = ∆yit denote the percentage
change of consumption (i = 1) and income (i = 2). Before we do estimation of
a VSTAR model containing CNFs, the model needs to be specied by concerning
how to choose the form of the ST function G(st; γ, c). We take the simplest one
with k = 1. Then to specify the transition variable st in an ST-type model,
a technique in Teräsvirta (1994) can be applied by using the result of testing
for nonlinearity such that the one which most signicantly presents nonlinearity
is the best choice. Following the principle, it is found from testing unit root
CNFs in VSTAR Models 22
against nonlinearity in Table 6 that x1,t−d is better than x2,t−d because st = x1,t−d
contributes more nonlinearity of xit than st = x2,t−d. Further, st = x1,t−7 is
recognized as a good consideration. st = ∆xi,t−d is not considered because ∆xi,t−d
is not so meaningful when xit is global stationary (however, it is reasonable to
include it as an option of st under the null of unit root in the tests).
The following estimated model is suggested: x1t
x2t
=
0.3682
0.3328
+
−0.3778 0.0291
0.0087 −0.3228
x1t−1
x2t−1
+
−0.0378
−0.0545
(0.0493 + 0.0100x1t−1 − 0.0850x2t−1)G(st; γ, c)
in whichG(st; γ, c) =(1 + exp
−47.5(x1t−7 − 0.98)/σx1t−7
)−1and σx1t−7 = 0.5146.
The Ljung-Box test is considered to examine the autocorrelation of the residual
from the estimated model above. It is approximately distributed as Chi-square
with degrees of freedom n2(s− k) (see Ljung and Box; 1978; Hosking; 1980). For
our data, n = 2, k = 1, and s = 60 (s is reasonably taken T/4 following the
literature, see for example Juselius (2007), Chapter 4.3.3), the degree of freedom
is 236. The residual cannot reject the null of no autocorrelation.
8 . Conclusions
In this paper we propose a testing procedure to examine the existence of CNFs in a
VSTAR model. A triangular representation to perform our LM test in a VSTAR
model containing CNFs is provided. The LM test under the null of existing CNFs
in a VSTAR model has an asymptotic Chi-squared distribution. We illustrate
our testing procedure by applying it to United States consumption and income
data, and observe that the growth of consumption and income contain CNFs.
Afterwards, modeling and estimation are briey discussed.
23 CNFs in VSTAR Models
Acknowledgments
Li is grateful to Sune Karlsson, Kenneth Carling and Johan Lyhagen for invaluable
advice and support. Li also thanks the Riksbankens Jubileumsfond foundation for
partial nancial support, Grant No. P2005-1247.
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The probability shown at the head of the column is the area in the left-hand tail.
−4 −2 0 2 4
0.0
0.2
0.4
0.6
0.8
1.0
(b)
s1t=−2 s1t=2s1t=−1
γ=0.01
γ=0.1
γ=1
γ=2γ=50
G
s1t
−2 −1 0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
(b)
s2t=1 s2t=2s2t=−1
γ=0.01
γ=0.1
γ=1
γ=2γ=50
G
s2t
Figure 5: Smooth-transition function G in s1t ∼ N(0, 1) in panel (a) and s2t ∼ N(0, 0.5) inpanel (b), when k = 2 with dierent both sjt and cj = −1, 2.
CNFs in VSTAR Models 30
−4 −2 0 2 4
0.0
0.2
0.4
0.6
0.8
1.0
(b)
s1t=2s1t=1
γ=0.01
γ=0.1
γ=1
γ=2
γ=50
G
s1t
−2 −1 0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
(b)
s2t=1s2t=0.5
γ=0.01
γ=0.1
γ=1
γ=2
γ=50
G
s2t
Figure 6: Smooth-transition function G in s1t ∼ N(0, 1) in panel (a) and s2t ∼ N(0, 0.5) inpanel (b), when k = 2 with dierent both sjt but same cj = 1.
−4 −2 0 2 4
0.0
0.2
0.4
0.6
0.8
1.0
(b)
s1t=1
G
s1t
−0.4 −0.2 0.0 0.2 0.4
0.0
0.2
0.4
0.6
0.8
1.0
(b)
s2t=0.1
G
s2t
Figure 7: Smooth-transition function G in s1t ∼ N(0, 1) in panel (a) and s2t ∼ N(0, 0.1) inpanel (b), when k = 2 with dierent both sjt but same cj = 1.
31 CNFs in VSTAR Models
−4 −2 0 2 4
0.0
0.2
0.4
0.6
0.8
1.0
(b)
s1t=1
G
s1t
−0.4 −0.2 0.0 0.2 0.4
0.0
0.2
0.4
0.6
0.8
1.0
(b)
s2t=−0.1
G
s2t
Figure 8: Smooth-transition function G in s1t ∼ N(0, 1) in panel (a) and s2t ∼ N(0, 0.1) inpanel (b), when k = 2 with dierent both sjt but same cj = −1.