Nonparametric Analysis for the Reality of the CPI when Ignoring
Some Financial Asset - Evidence from Taiwan
Chien-Chung Nieh*
Department and Graduate Institute of Banking and Finance, Tamkang University,
Taiwan
Chu-Chun Wang
Graduate Institute of Banking and Finance, Tamkang University, Taiwan
Fang-Wen Yeh
Graduate Institute of Banking and Finance, Tamkang University, Taiwan
Abstract
This research attempts to judge the reality of the consumer price index (CPI)
without incorporating the prices of stock and real estate over the period of 1986Q1 to
2004Q4 in Taiwan by employing both conventional Johansen maximum likelihood
and Bierens nonparametric cointegration techniques to investigate the long-run
relationships among CPI, stock index and real estate index. The conventional
Johansen cointegration test has mixed findings, especially in the two variables system.
However, the confirmation of the nonlinearity conducts us to believe more in the
Bierens nonparametric cointegration, which shows a consistent result that long-run
equilibrium relationship exists with one cointegration rank no matter for two or three
variables considered. The findings of long-run equilibrium relationships among CPI
and each of stock and real estate index provides us a perceptual information that the
price index without incorporating the prices of stock and real estate might be spurious.
This ‘spurious price index’ contains an important policy implication in constructing
consumer price index in Taiwan.
Key Words: Nonparametric Cointegration, Nonlinear, Consumer Price Index,
Stock Index, Real Estate Index
JEL classification: C22; E31; G12
Corresponding author: Chairman and Director of the Department and Graduate Institute of Banking
and Finance, Tamkang University, Tamsui, Taipei, 251,Taiwan. TEL: 886-2-2621-5656 ext.2591.
E-Mail: niehcc@mail.tku.edu.tw
*
I. Introduction
Without considering the asset prices of stock and real estate, the consumer price
index (CPI) in Taiwan seems not an appropriate one to reflect the real aggregate price
level. There is probably another type of latent inflation if the asset prices of stock
and real estate are incorporated into the price index. For instance, Taiwan had
experienced a dramatic increase in the stock and real estate markets during years of
1986 to 1990. The stock index of Taiwan boomed from 900 in 1986 to 10756 in
1990, twelvefold within four years. The real estate index also fivefold jumped from
24 in 1986 to 124 in 1990. CPI, however, increasing smoothly and only 8% during
the same period, seems not being able to catch the speed of the increases of the stock
index and the real estate index. Figure 1 to Figure 3 depict the trends of these three
variables.
<Insert Figure 1 to Figure 3 about here>
The interrelationships between stock index and price index are generally
acknowledged strong. For example, Flannery and Protopapadakis (2002) and
Nikkinen and Sahlstrom (2004) argued that price index is a crucial
macro-fundamental, which shows strong impact on the stock movement. The other
way around, Amihud (1996) and Luintel (2002) found that stock price is one of the
key determinants of the price level.
Moreover, the examination of the
interrelationships between real estate index and price index can be found in Chatrath
and Liang (1998), Chaudhry, Myer and Webb (1999) and Deng, Gabriel and Nothaft
(2003), among others. Most of these literature found that not only short-run
dynamic, but long-run equilibrium relationships exist between real estate index and
price index.1
The main objective of this paper is to investigate the long-run equilibrium
relationships between CPI and each of asset prices of stock price and real estate price,
respectively. The existence of the long-run equilibrium relationship between CPI
and asset prices considered can be described as that the current CPI is not an
appropriate one to reflect the real aggregate price level and it should contain a policy
implication of incorporating the stock price and real estate price when estimating the
CPI.
The conventional Johansen (1988, 1994) multivariate likelihood cointegration
1
The interrelationships between asset prices of real estate and stock are also numerously investigated,
which can found in Liu et al. (1990), Gyourko and Keim (1992), Eichholtz (1997), Okunev and
Wilson(1997), Goldstein and Nelling (1999), He (2000) and Fu and Ng (2001).
1
test, was once believed the most powerful techniques among others,2 has been widely
employed in the field of time series for testing for the long-run equilibrium
relationships among variables. However, it is based on the linear autoregressive (AR)
model and, as such, assumes that the underlying dynamics are in a linear form. A
numbers of empirical evidence having flatly rejected such a linear paradigm has been
largely reported recently. From a theoretical perspective, there is no sound reason to
assume that economic systems are intrinsically linear (see, Barnett and Serletis, 2000).
Numerous studies have empirically demonstrated that financial time series, such as
stock prices, exhibit nonlinear dependencies (see, Hsieh, 1991; Abhyankar et al.,
1997). Besides, substantive evidence from the Monte Carlo simulations in Bierens
(1997) indicated that the conventional Johansen cointegration framework is
misspecified when the true nature of the adjustment process is nonlinear and the speed
of adjustment varies with the magnitude of the disequilibrium. The work of Balke
and Fomby (1997) as well as that of Enders and Granger (1998) has also pointed out a
potential loss of power in conventional cointegration tests under the threshold
autoregressive (TAR) data generating process (DGP).
Motivated by the above argument, this paper investigates the long-run
equilibrium relationships between CPI and each of asset prices of stock price and real
estate price, respectively, by employing the more persuadable nonparametric
cointegration test developed by Bierens (1997) and, for comparison, the conventional
Johansen cointegration test as well. The findings of this study are used to judge the
reality of the price index without incorporating the prices of stock and real estate.
We perceive that the results from the former (i.e., the Bierens nonparametric
cointegration test) are expected to confirm the long-run equilibrium relationships
between CPI and indexes of stock and real estate and it contains a policy implication
of incorporating the stock price and real estate price when estimating the CPI.
The remainder of this paper is organized as follows: Section II describes the data.
Section III introduces the methodologies and presents the empirical results. Section
IV concludes this paper.
II. Data
The data sets consist of quarterly time series on stock price index, real estate price
index and consumer price index covering the period of 1986Q1 to 2004Q4, a total of
76 observations for each variable. Stock price index is obtained from the AREMOS
2
See Gonzalo’s (1994) comparison of five cointegration tests.
2
database of the Ministry of Education of Taiwan; consumer price index is downloaded
from Directorate-General of Budget, Accounting and Statistics, Executive Yuan of
ROC;3 real estate price index is collected and constructed by Hsin-Yi Real Estate Inc.
Examination of the individual data series makes it clear that the logarithmic
transformations were required to achieve stationarity in variance; therefore, all the
data series were transformed to logarithmic form. Figure 1 to Figure 3 exhibit the
plots of the three variables considered in this paper.
III. Methodologies and Empirical Results
3.1 Stationary test
In this study we apply several conventional unit root tests, such as ADF, PP, KPSS,
DF-GLS, ERS, and NP, to test for the ‘stationarity’ for each of three variables
considered.4 The appropriate lag length for each test is selected based on Modified
Akaike information criterion (MAIC) suggested by Ng and Perron (2001). Empirical
results shown in Table 1 indicate that, with the trivial exception of an I(0) series of the
real estate price (LR) by KPSS, all three variables, consumer price index (LP), stock
price (LS), and real estate price, are found integrated of order one whatever
techniques are used.
<Insert Table 1 about here>
Recently, a general consensus has been emerging in support of the likelihood that
financial data exhibit nonlinearity and that such conventional tests (as shown in
previous paragraph) for stationarity have too low the power to detect the
mean-reverting tendency of a series. It follows that stationarity tests must be applied
in a nonlinear framework. Therefore, in this study, we use the nonlinear unit root
test advanced by Kapetanios et al. (2003) (henceforth, KSS test) for the stationary
test.
The KSS test is based on detecting the presence of non-stationarity against a
nonlinear but globally stationary exponential smooth transition autoregressive
(ESTAR) process. The model is given by
Yt  Yt 1{1  exp( Yt 21 )}   t ,
3
(1)
http://www.dgbas.gov.tw/
Various unit root tests of ADF, DF-GLS, ERS, PP and NP, are developed by Dickey and Fuller(1981),
Elliot, Rothenberg, and Stock(1996), Elliot, Rothenberg, and Stock(1996), Philips and Perron (1988)
and Ng and Perron(2001), respectively, for testing for the null of a unit root against the alternative of
stationarity, whereas, KPSS test developed by Kwiatkowski et al. (1992) is for testing for the null of
4
3
where Yt is the data series of interest, vt is an i.i.d. error with a zero mean and
constant variance, and   0 is the transition parameter of the ESTAR model and
governs the speed of transition. Under the null hypothesis Yt follows a linear unit
root process, but under the alternative, Yt follows a nonlinear stationary ESTAR
process. One shortcoming of this framework is that the parameter  is not identified
under the null hypothesis. Kapetanios et al. (2003) utilized a first-order Taylor series
approximation for { 1  exp( Yt 21 ) } under the null hypothesis   0 and have then
approximated equation (1) by using the following auxiliary regression:
k
Yt    Yt 31   bi Yt i  t ,
t = 1, 2,…., T
(2)
i 1
Under this framework, the null hypothesis and the alternative hypotheses are
expressed as   0 (non-stationarity) against   0 (non-linear ESTAR stationarity).
The results of KSS nonlinear stationary test, as shown in Table 2, clearly indicate that
all three variables considered in this study are integrated of order one, as noted as the
I(1) series.
<Insert Table 2 about here>
3.2 Johansen Cointegration Test
Since all the variables considered in this study are found to be I(1) series by the
various unit root test techniques and further reinforced by the more persuadable
nonlinear KSS stationary test, we consequently go for our test for the reality of the
CPI by employing both the conventional Johansen cointegration test and the Bierens
(1997) nonparametric cointegration approaches.
The five VECM models elaborated by Johansen (1988, 1990, 1994) are
presented as the following forms:5
H0(r) : X t  1 X t 1  ...  k 1 X t k 1   X t 1  Dt  t
(1988) (3)
H1*(r): X t  1 X t 1  ...  k 1 X t k 1   (  ,  0 )( X t1 ,1)  Dt  t
(1990) (4)
H1(r) : X t  1 X t 1  ...  k 1 X t k 1   X t 1   0  Dt  t
(1990) (5)
H2*(r): X t  1 X t 1  ...  k 1 X t k 1   (  ,  1 )( X t1 , t ) 0  Dt  t (1994) (6)
stationarity against the alternative of a unit root.
5
The 1990 eqution form is indeed from Johansen and Juselius (1990).
4
H2(r) : X t  1 X t 1  ...  k 1 X t k 1   X t 1   0  1t  Dt  t
(1994) (7)
To analyze the deterministic term, Johansen decomposed the parameters 0 and
1 in the directions of α and α as i =αβi +α i, thus we have βi = (α'α)-1
α'i and i = (α'α)-1α'i. The nested sub-models of the general model of null
hypothesis  = αβ' are, therefore, defined as:
H0(r) :
H1*(r):
Y=0
Y = αβ0
H1(r) :
H2*(r):
H2(r) :
Y = αβ0 +α0
Y = αβ0 +α0 + αβ1t
Y = αβ0 +α0 + (αβ1+α1)t
Johansen (1994) emphasized the role of the deterministic term, Y = 0 + 1t,
which includes constant and linear terms in the Gaussian VAR. Following Nieh and
Lee (2001), a decision procedure among the hypotheses H(r) and H*(r) for five
different models is presented in the following procedure:6
H0(0)  H1*(0)  H1(0)  H2*(0)  H2(0)  H0(1)
 H1*(1)  H1(1)  H2*(1)  H2(1)
 ...  ...  H0(p-1)  H1*(p-1) H1(p-1)  H2*(p-1)  H2(p-1)
To elaborate on our long-run equilibrium relationship test, we step our tests first
by two variables system and then three variables system. The empirical findings of
our cointegration tests with the consideration of a linear trend and a quadratic trend
are summarized in Table 3. We first found that there is no cointegration relationship
between CPI and stock index (Panel A of Table 3); whereas, the relationship between
CPI and real estate are shown to be stationary in the original data since the
cointegration ranks are larger than one (Panel B of Table 3). When both stock index
and real estate index are incorporated into the three variables system, we found that
the long-run equilibrium relationship exists with two cointegration ranks for our three
variables system (Panel C of Table 3). That is, stock index and real estate index
together with CPI share common trends in the long-run. This finding offers a policy
implication that, when estimating the CPI, the index level of stock and real estate
should be incorporated to reflect the real aggregate price level.
<Insert Table 3 about here>
3.3 Nonlinear Test of the Error-Correction Term
As mentioned earlier, the evidence from the Monte Carlo simulations in Bierens
Nieh and Lee’s (2001) decision procedure for the Johansen models is to diagnose from model of
lower number to model of higher number and then from lower rank to higher rank until the model that
cannot be rejected for the null.
6
5
(1997) indicates that the conventional Johansen cointegration framework has a
misspecification problem when the true nature of the adjustment process is nonlinear
and the speed of adjustment varies with the magnitude of the disequilibrium.
Bearing this in mind, we follow Granger and Teräsvirta (1993) and Teräsvirta (1994)
by employing a nonlinear test on our error-correction term. The STECM is formulated
as following form:
y t   0   1/Wt  ( 0   1/Wt ) F ( z t  d ; r , c)   t
(8)
where Wt  (Z t 1 , yt 1 ,..., yt  p , X t/ ,..., X t/ p ) / and  t ~ NID (0,  2 ) . The coefficient vector
 1 has a dimension of m  1 and m  ( p  1)( k  1) . The function F ( z t d ; r , c) is a
continuous transition function with the transition variable z t  d and parameter (r, c)
that provides a variety of nonlinear models, e.g., logistic, exponential or quadratic
logistic functions.7 Van Dijk, Teräsvirta and Franses (2002) has a good survey for the
recent developments of smooth transition autogressive (STAR) models and some
good applications of STECM can be found in Huang, Lin and Cheng (2001) and
Milas and Otero (2002), among others.
Table 4 presents the results for the different delay parameters (up to six); these
demonstrate that the true nature of the adjustment process is nonlinear and that the
speed of adjustment varies with the magnitude of the disequilibrium.
<Insert Table 4 about here>
3.4 Bierens (1997) Nonparametric Cointegration Test
As pointed out by Bierens (1997), one of the major advantages of his
nonparametric method lies in its superiority to detect cointegration when the error
correction mechanism is nonlinear.
The Bierens nonparametric cointegration test considers the general framework to
be:
z t   0   1t  yt
(9)
where  0 (qx1) and  1 (qx1) are the terms for the optimal mean and trend vectors,
respectively, and yt is a zero-mean unobservable process such that yt is
stationary and ergodic.
7
Apart from these conditions of regularity, the method does
The detailed procedures are not presented here due to space consideration.
6
not require further specifications of the DGP for z t , and in this sense, it is completely
nonparametric.
The Bierens method is based on the generalized eigenvalues of the
matrices Am and ( Bm  cT 2 Am1 ) , where Am and B m are defined in the following
matrices:
Am 
8 2
T
m
k 2 (
k 1
m
Bm  2T  (
k 1
1 T
1 T
cos(
2
k

(
t

0
.
5
)
/
T
)
z
)(

 cos(2k (t  0.5) / T ) zt )
t
T t 1
T t 1
1 T
1 T
cos(
2
k

(
t

0
.
5
)
/
T
)

z
)(

 cos(2k (t  0.5) / T )zt )
t
T t 1
T t 1
(10)
(11)
which are computed as the sums of the outer-products of the weighted means of
z t and z t , and where T is the sample size. To ensure invariance in the test statistics
to drift terms, we recommend using the weighted functions of cos( 2k (t  0.5) / T ) .
Very much like the properties in the Johansen likelihood ratio method are the ordered
generalized eigenvalues that we obtain from this nonparametric method. These
serve as the solution to the problem det[ PT  QT ]  0 when we define the pair of
random matrices PT  Am and QT  ( Bm  cT 2 Am1 ) . Thus, we can use this to test
the hypothesis for the cointegration rank r. To estimate r, Bierens (1997) proposed
two statistics tests. One is the  min test which corresponds to Johansen’s
maximum likelihood procedure, and it tests hypothesis H 0 (r ) against
hypothesis H1 (r  1) . The second statistic is the gˆ m (r0 ) test, which is computed
from the Bierens’s generalized eigenvalues:
 n ˆ 1
( k ,m ) , if ....r0  0
 k 1
n
 nr
ˆg m (r0 )  ( ˆk ,m ) 1 (T 2 r  ˆk ,m ), if ...r0  1,...., n  1
k  n  r 1
 k 1
n
 2n
T  ˆk ,m , if ....r0  n
k 1

(12)
This statistic employs the tabulated optimal values (see Bierens, 1997, Table 1) for
m when n  r0 , provided that we select m  n for n  r0 . This verifies that
gˆ m (r0 )  O p (1) for r  r0 , and in terms of probability, it converges to infinity if
r  r0 . Hence, a consistent estimate of r is given by rˆm  arg min r0  n {gˆ m (r0 )} . This
7
statistic is an invaluable tool when double-checking the determination of r. Table 5
presents the results of both the  min test and the gˆ m (r0 ) test. The results of
 min test strongly suggest that there are long-run relationships between CPI and each
of stock index and real estate index. The three variables are also shown to be
cointegrated when they are examined as a whole. These findings are further
supported by the gˆ m (r0 ) statistics (Table 5) that all the results shown to be
cointegrated with cointegrating rank of r  1 . These results reveal that, without
considering the asset prices of stock and real estate, the consumer price index (CPI) in
Taiwan seems not an appropriate one to reflect the real aggregate price level during
the 1986Q1 to 2004Q4 period. On account of the nonparametric method’s
superiority in detecting cointegration when the error-correction mechanism is
nonlinear, we firmly believe that these results are considerably more reliable than
those derived from the conventional Johansen approach.
<Insert Table 5 about here>
3.5 Generalized Impulse Response Function (G-IRF)
The more recent researches have largely applied the impulse response functions
(IRF) to conquer the difficulty of interpreting the estimated coefficients of a VAR
model. An IRF traces the response of one of the innovations on current and future
values of the endogenous variables to a one standard deviation shock. This shock to a
variable directly affects itself, and is also transmitted to all of the endogenous
variables through the dynamic structure of the VAR. In this study, we further employ
the impulse response function to investigate the dynamic influence among our three
variables, especially the effects of the shocks of real estate price and the stock price
on CPI.
The traditional orthogonalized-IRF (O-IRF) based on Cholesky decomposition,
however, is criticized. Cooley and LeRoy (1985) found that O-IRF is not meaningful.
King et al. (1991) and Zhou (1996) further pointed out that as there is more than one
common trend in a model, different ordering of variables may significantly affect the
results of O-IRF if the common trends are not absolutely uncorrelated. The newly
developed Generalized-IRF (G-IRF) model by Pesaran and Shin (1998) overwhelm
the shortcoming of O-IRF since G-IRF holds the advantage of being invariant to the
ordering of the variables. Dekker et al. (2001) compared traditional VAR with G-VAR
in estimating linkages of the Asia Pacific equity markets. It found that the generalized
approach significantly gives more realistic results, particularly for those markets with
closest geographical and economic links. Moreover, study by Naka and Tufte (1997)
argued that IRF of the two models of VAR in level and VECM are similar at short
8
horizon, but different at long horizon. At short horizons, VECM estimates are known
to perform poorly relative to those from a VAR in level in terms of forecast errors.
Therefore, in our study, the IRF are constructed using a VAR in level for the analysis.
The mutual impacts of shocks among three variables are exhibited in Figure
4.1-4.9. These diagrams of G-IRF show that real estate price and stock price have
significant impacts on the CPI, especially in the long run. However, in the short run
dynamic, this influence is moderate.8
<Insert Figure 4 about here>
IV. Conclusion
This research attempts to judge the reality of the consumer price index (CPI)
without incorporating the prices of stock and real estate over the period of 1986Q1 to
2004Q4 in Taiwan by employing both conventional Johansen maximum likelihood
and Bierens nonparametric cointegration techniques to investigate the long-run
relationships among CPI, stock index and real estate index. The conventional
Johansen cointegration test has mixed findings, especially in the two variables system.
However, the confirmation of the nonlinearity conducts us to believe more in the
Bierens nonparametric cointegration, which shows a consistent result that long-run
equilibrium relationship exists with one cointegration rank no matter for two or three
variables considered. The findings of long-run equilibrium relationships among CPI
and each of stock and real estate index provides us a perceptual information that the
price index without incorporating the prices of stock and real estate might be spurious.
This ‘spurious price index’ contains an important policy implication in constructing
consumer price index in Taiwan.
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12
Table 1. The results of various unit root test based on MAIC(NP, 2001)
Level
Difference
Level
DF-GLS Difference
Level
ERS
Difference
Level
PP
Difference
Level
NP
Difference
Level
KPSS Difference
ADF
LP
-2.436(2)
-8.730(0) ***
-0.284(5)
-8.772(0) ***
851.42(2)
0.703(0) ***
-2.353(2)
-8.748(2) ***
-1.110(2)
-4.548(0) ***
1.109(6) ***
0.694 *
LS
-3.209(0) *
-9.800(0) ***
-1.871(0)
-9.861(0) ***
21.450(0)
2.484(0) ***
-3.127(4)
-9.859(3) ***
-1.671(0)
-4.306(0) ***
0.136(6) *
0.078(6)
LR
-3.468(1) *
-5.013(0) ***
-1.287(1)
-4.825(0) ***
89.317(1)
3.511(0) ***
-2.896(4)
-5.093(4) ***
-1.118(1)
-3.694(0) ***
0.216(6) ***
0.148(5) **
Notes: 1. LP, LS, and LR are the symbols for the logarithm of consumer price index, stock index,
and real estate index, respectively.
2. ***, **, and * denote significant at the 1%, 5%, and 10% levels, respectively.
3. The critical values for 1%, 5% and 10% levels of ADF, DF-GLS, ERS, PP, NP and KPSS
are (-4.101, -3.478 and -3.167), (-3.717, -3.145 and -2.848), (4.236, 5.668 and 6.778),
(-4.097, -3.476 and -3.166), (-3.42, -2.91 and -2.62), and (0.216, 0.146 and 0.119),
respectively.
4. The test statistic for NP test is MZt.
5. The number in the parentheses are the appropriate lag lengths based on MAIC (Modified
Akaike information criterion) suggested by Ng and Perron (2001).
6. The null of KPSS test is testing for I(0), the null of the rest five tests are testing for I(1).
7. Based on the decision procedure suggested by Dolado, Jenkinson, and Sosvilla-Rivero
(1990), the appropriate models for the level and the first difference are both with trend
and intercept and model with intercept, respectively.
Table 2. KSS’s Nonlinear Unit Root Test Results
Variable
KSS statistic
LP
Level
1.869(2)
Difference
6.169(0)***
LS
Level
0.935(0)
Difference
5.527(0)***
LR
Level
2.660(1)**
Difference
4.518(0)***
Notes: 1. Simulated critical values are from Table 1 in Kapetanios et al. (2003).
2. The number in parentheses indicates the selected lag order based on SBC.
3. ***, **, and * denote significant at the 1%, 5%, and 10% levels, respectively.
4. The critical values for 1%, 5% and 10% levels are 2.82, 2.22, and 1.92, respectively.
13
Table 3. Johansen Cointegration Test in the Presence of a LT and a QT
Panel A. CPI and Stock Index
Rank
T0(r)
C0(5%)
T1*(r)
C1*(5%)
T1(r)
C1(5%)
T2*(r)
C2*(5%)
T2(r)
C2(5%)
r=0
35.13
12.32
49.65
20.26
30.86
15.49
32.64
25.87
17.52
18.40
r1
12.47
4.13
16.84
9.16
13.42
3.84
15.03
12.52
0.00
3.84
SBC
2
2
2
2
2
Panel B. CPI and Real Estate Index
Rank
T0(r)
r=0
r1
SBC
C0(5%)
45.61
12.32
10.94
4.13
2
T1*(r)
C1*(5%)
69.18
20.26
21.75
9.16
2
T1(r)
C1(5%)
51.50
15.49
18.28
3.84
2
T2*(r)
C2*(5%)
51.77
25.87
18.53
12.52
2
T2(r)
C2(5%)
35.04
18.40
3.85
3.84
2
Panel C. CPI, Stock Index and Real Estate Index
Rank
T0(r)
C0(5%)
T1*(r)
C1*(5%)
T1(r)
C1(5%)
T2*(r)
C2*(5%)
T2(r)
C2(5%)
r=0
59.32
24.28
83.72
35.19
65.64
29.80
68.97
42.92
51.58
35.01
r1
23.48
12.32
34.36
20.26
30.82
15.49
34.11
25.87
19.21
18.40
8.28
4.13
12.54
9.16
12.45
3.84
15.44
12.52
3.528
3.84
r2
SBC
2
2
2
2
2
notes: 1. T0(r), T1*(r), T1(r), T2*(r), and T2(r) denote the LR test statistics for all the null of H(r) versus the
alternative of H(p) of Johansen’s five models.
2. C0(5%), C1*(5%), C1(5%), C2*(5%) and C2(5%) are 5% LR critical values for Johansen’s five models,
which are extracted from MacKinnon, Haug and Michelis (1999).
3. LT and QT are abbreviations of Linear Trend and Quadratic Trend
4. The model selection follows Nieh and Lee’s (2001) decision procedure, diagnosing models one by one
until the model that cannot be rejected for the null.
5. The bold number with underline indicates the selection of the rank in the presence of linear trend and
quadratic trend.
6. VAR length is 2 for all the models, which is selected based on the smallest number of SBC.
Table 4. Smooth Transition Error Correction Model Tests for Nonlinearity
Null
d=1
d=2
d=3
d=4
d=5
d=6
H0
1.446(0.149)
1.830(0.046)
1.861(0.041)
1.660(0.079)
1.024(0.458)
1.682(0.402)
H 03
1.886(0.095)
0.850(0.552)
0.772(0.614) 1.989(0.079)* 0.984(0.456)
1.033(0.423)
H 02
1.277(0.280)
3.727(0.002)
3.740(0.002)
1.460(0.203)
0.971(0.463)
1.039(0.417)
H 01
0.879(0.528)
0.760(0.623)
0.922(0.496)
1.105(0.372)
1.128(0.359)
1.158(0.342)
Note: 1. The numbers out and in the parentheses are the F-statistics and the p-value, respectively.
2. d indicates the delay variable.
3. Bold number indicate the findings of nonlinearity at the 5% level.
Table 5. Bierens Nonparametric Cointegration Test Results
14
Panel A. CPI and Stock Index
min Teat (5%)
0.0029**
H0 : r  0
(0,0.017)
H :r 1
min Teat (10%)
0.0021*
(0,0.005)
r0  0
gˆ m (r0 )
1183.79
0.4022
(0,0.111)
r0  1
29.37
r0  2
26728.31
r0  0
gˆ m (r0 )
926.79
r0  1
5.92
r0  2
34140.10
r0  0
gˆ m (r0 )
198436.12
a
H0 : r  1
Ha : r  2
0.4022
(0,0.054)
Panel B. CPI and Real Estate Index
min Teat (5%)
min Teat (10%)
0.0018**
0.0011*
H0 : r  0
(0,0.017)
(0,0.005)
H :r 1
a
H0 : r  1
Ha : r  2
1.1022
(0,0.054)
1.1022
(0,0.111)
Panel C. CPI, Stock Index and Real Estate Index
min Teat (5%)
min Teat (10%)
0.00009**
0.00009*
H0 : r  0
(0,0.008)
(0,0.017)
H :r 1
a
H0 : r  1
Ha : r  2
H0 : r  2
Ha : r  3
0.07458
(0,0.017)
0.07458
(0,0.034)
r0  1
3.30
1.24258
(0,0.111)
1.24258
(0,0.187)
r0  2
103.30
r0  3
896906.11
Notes: 1. * and ** denote significance at the 10% and 5% level, respectively.
2. Both the results of the min test and the gˆ m (r0 ) test indicate one cointegration rank
15
4.7
4.6
4.5
4.4
4.3
4.2
86
88
90
92
94
96
98
00
02
04
LP
Figure 1. Logarithm of Consumer Price Index (1986Q1:2004Q4)
9.5
9.0
8.5
8.0
7.5
7.0
6.5
86
88
90
92
94
96
98
00
02
04
02
04
LS
Figure 2. Logarithm of Stock Index (1986Q1:2004Q4)
5.0
4.5
4.0
3.5
3.0
86
88
90
92
94
96
98
00
LR
Figure 3. Logarithm of Real Estate Index (1986Q1:2004Q4)
16
Figue 4.1- 4.9 : Response to Generalized One S.D. Innovations ± 2 S.E.
Figure 4.1 : Response of LP to LP
Figure 4.2 : Response of LP to LRE
Figure 4.3 : Response of LP to LST
.012
.012
.012
.008
.008
.008
.004
.004
.004
.000
.000
.000
-.004
-.004
1
2
3
4
5
6
7
8
9
10
-.004
1
Figure 4.4 : Response of LRE to LP
2
3
4
5
6
7
8
9
10
1
Figure 4.5 : Response of LRE to LRE
.10
.10
.08
.08
.08
.06
.06
.06
.04
.04
.04
.02
.02
.02
.00
.00
.00
-.02
-.02
-.02
-.04
1
2
3
4
5
6
7
8
9
10
Figure 4.7 : Response of LST to LP
2
3
4
5
6
7
8
9
10
1
Figure 4.8 : Response of LST to LRE
.3
.2
.2
.2
.1
.1
.1
.0
.0
.0
-.1
-.1
-.1
-.2
2
3
4
5
6
7
8
9
10
5
6
7
8
9
10
2
3
4
5
6
7
8
9
10
Figure 4.9 : Response of LST to LST
.3
1
4
-.04
1
.3
-.2
3
Figure 4.6 : Response of LRE to LST
.10
-.04
2
-.2
1
2
3
4
5
6
7
8
Figure 4. Generalized-Impulse Response Function
17
9
10
1
2
3
4
5
6
7
8
9
10