Chapter 4. Data and Methodology
4.3 VECM and Cointegration Test
國
立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
24
therefore, we can reject null hypothesis and have sufficient evidence to say the time
series data are stationary after 1st differenced.
Table 2. Result of Augmented Dickey-Fuller Test
Level 1st difference
t-statistic p-value. t-statistic p-value.
TAIALL(all sample) 1.7075 0.9995 -4.7185 0.0004
NEW 1.1289 0.9971 -5.2719 0.0001
PRESALE 1.5278 0.9991 -4.0508 0.0030
SINYI 1.1729 0.9975 -4.6439 0.0005
4.3 VECM and Cointegration Test
The methodology we employed is Vector Error Correction Model (VECM).
Traditional econometric such as OLS should align with traditional regression
assumptions which assumed the time series data of economic variable should be
stationary and residuals should be non-serial correlation. In general, most of economic
variables are non-stationary. If we find that the variables are not stationary after
unit-root test, there are two ways to resolve the problem. First, we can eliminate the
deterministic trend of the economic variable. Second, you can eliminate stochastic trend
by difference which is the most simplest and common way. However, data may lose
long term information after difference. Therefore, Engle and Granger (1987) proposed
‧
國立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
25
the cointegration concept to resolve this problem. The concept of cointegration applies
to a wide variety of economic models. Any equilibrium relationship among a set of
non-stationary variables implies that their stochastic trend must be linked. After all, the
equilibrium relationship means that the variables cannot move independently of each
other. This linkage among the stochastic trends necessitates that the variables be
cointegrated and there is a linear relationship among the variables. Since the
cointegrated variables are linked, there are two types of change when the economic
variables varying. One is temporal another is permanent. Permanent change is the
change of long trend which is low frequencies. And temporal change is that the dynamic
pahts of such variables must bear some relation to the current deviation from the
equilibrium relationship by some reasons but it will back to long term equilibrium
during the time passing. Engle and Granger (1987) provided famous Granger
Representation which stated two major concepts. First, an error correction model for I(1)
variables necessarily implies cointegration. Second, It can also be shown that
cointegration implies error correction. Therefore, when two variables exist cointegrated
relationship, they expressed as error correction model. To test cointrgration relationship,
I can use Engle and Granger two steps cointegration test processing. Step 1, If both of
‧
國立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
26
variables are unit-root and I(1) series, then we can use OLS to compute its residual. Step
2, To test the residuals whether it exists unit-root, if there is unit-root then we will say
both of two variable are not integrated. On the other hand, if the residual do not exist
unit-root then both of variables are cointegrated. I also can use the rank of π to
determine whether or not the variables in 𝑥𝑡 are integrated. To elaborate, consider the
simple case of a first order VAR:
𝑥𝑡 = 𝐴1𝑥𝑡−1+ 𝜀𝑡
Where 𝑥𝑡 is the (n x 1) vector (𝑥1𝑡, 𝑥2𝑡,… 𝑥𝑛𝑡). 𝜀𝑡 is the (n x 1) vector
(𝜀1𝑡, 𝜀2𝑡,… 𝜀𝑛𝑡). 𝐴1 is an (n x n) matrix of parameters. Subtracting from each side of
the equation, we get
∆𝑥𝑡 = −(𝐼 − 𝐴1)𝑥𝑡−1+ 𝜀𝑡= 𝜋𝑥𝑡−1+ 𝜀𝑡
If the rank of π matrix is zero, each element of it must equal zero. In this instance,
the equation is equivalent to an n-variable VAR in first differences:
∆𝑥𝑡 = 𝜀𝑡
All the {𝑥𝑡} sequences are unit-root processes and there is no linear combination
of the variable which is stationary. At the other extreme, suppose that π is of full rank
The long-run solution to dynamic system is given by the n independent equations:
‧
國立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
27
𝜋11𝑥1𝑡− 𝜋12𝑥2𝑡+ 𝜋13𝑥3𝑡+ ⋯ + 𝜋1𝑛𝑥𝑛𝑡 = 0 𝜋21𝑥1𝑡− 𝜋22𝑥2𝑡+ 𝜋23𝑥3𝑡+ ⋯ + 𝜋2𝑛𝑥𝑛𝑡 = 0 𝜋𝑛1𝑥1𝑡− 𝜋𝑛2𝑥2𝑡+ 𝜋𝑛3𝑥3𝑡+ ⋯ + 𝜋𝑛𝑛𝑥𝑛𝑡 = 0
In this case, each of the n variables contained in the vector must be stationary with
the long-run values given by solving the above system. In this paper, I use the Johansen
cointegration method to test the conintergration. We know from linear algebra that the
rank of a matrix is equal to the number of non-zero characteristic roots. Therefore, the
number of distinct co-integrating vectors can be obtained by checking the significance
of the characteristic roots of π . The following table is the Johansen test results. I
assumed that there is no intercept or trend in CE or test VAR. As the result represented,
the TAIALL and Sinyi index significantly exist at most 1 cointegration relationship
under the trace and maximum eigenvalue method. The presale house and Sinyi also
have at most 1 cointegration relationship under the trace and maximum eigenvalue
method. The conintegration test of new house and Sinyi represented a conintergration
pattern which p-value are 0.0005 under trace method and 0.0104 under maximum
eigenvalue method. And even presale and new house represented a significant
cointegration relationship. After cointegration test, we can conclude that all of the
variables exist cointegration. Therefore, I can employ VECM model to test the price
lead-lag relationship.
‧
Table 3. Johnansen Cointegration Test
Unrestricted Cointegration Rank Test (Trace) Unrestricted Cointegration Rank Test (Maximum Eigenvalue)
TAIALL and SINYI TAIALL and SINYI
Hypothesized
None 0.370929 25.39180 12.32090 0.0002 None 0.370929 19.46746 11.22480 0.0015
At most 1 0.131559 5.924341 4.129906 0.0177 At most 1 0.131559 5.924341 4.129906 0.0177
PRESALE and SINYI PRESALE and SINYI
Hypothesized
None 0.483328 32.92621 12.32090 0.0000 None 0.269193 27.73461 11.22480 0.0000
At most 1 0.116275 5.191601 4.129906 0.0270 At most 1 0.135553 5.191601 4.129906 0.0270
NEW HOUSE and SINYI NEW HOUSE and SINYI
Hypothesized
PRESALE and NEWHOUSE PRESALE and NEWHOUSE
Hypothesized
None 0.269193 19.28939 12.32090 0.0029 None 0.269193 13.17143 11.22480 0.0225
At most 1 0.135553 6.117955 4.129906 0.0159 At most 1 0.135553 6.117955 4.129906 0.0159
‧
For a VECM model, first, we considered a VAR(p) model:
∆𝑦𝑡= 𝑚 + 𝐴1𝑦𝑡−1+ 𝐴2𝑦𝑡−2+ ⋯ + 𝐴𝑝𝑦𝑡−𝑝+ 𝜀𝑡, 𝜀𝑡~𝑊𝑁(0, Ω)
According to Granger Representation theorem, VAR(p) model can be expressed to
vector error correlation model:
Δ𝑦𝑡 = 𝑚 + ∑ 𝐵𝑗∆𝑦𝑡−𝑗
To study the price lead-lag relationship between spot and futures, I applied the
VECM model as follow:
𝐸𝑞 1 ∶ 𝑒𝑐𝑚̂ = 𝛼𝑡−1 0+ 𝐹𝑡−1+ 𝛼1𝑆𝑡−1
𝐸𝑞 2 ∶ ∆𝐹𝑡 = 𝛽0+ 𝛽1𝑒𝑐𝑚̂ + 𝛽𝑡−1 2∆𝐹𝑡−1+ 𝛽3∆𝑆𝑡−1+ 𝛽4∆𝐹𝑡−2+ 𝛽5∆𝑆𝑡−2
+ 𝛽6𝑅𝑎𝑡𝑒𝑡+ 𝛽7ln (𝐺𝐷𝑃𝑝𝑒𝑟)𝑡+ 𝛽8ln (𝑆𝑡𝑜𝑐𝑘)𝑡
𝐸𝑞 3 ∶ ∆𝑆𝑡 = 𝛽0+ 𝛽1𝑒𝑐𝑚̂ + 𝛽𝑡−1 2∆𝐹𝑡−1+ 𝛽3∆𝑆𝑡−1+ 𝛽4∆𝐹𝑡−2+ 𝛽5∆𝑆𝑡−2
+ 𝛽6𝑅𝑎𝑡𝑒𝑡+ 𝛽7ln (𝐺𝐷𝑃𝑝𝑒𝑟)𝑡+ 𝛽8ln (𝑆𝑡𝑜𝑐𝑘)𝑡
‧
國立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
30
Here, Eq1 represented long term equilibrium equation. Eq2 and Eq3 showed short
term equilibrium equation. Also, We can transform the model into reduced form like:
∆𝐹𝑡 = (𝛽0+ 𝛼0𝛽1) + 𝛽1𝐹𝑡−1+ 𝛼1𝛽1𝑆𝑡−1+ 𝛽2∆𝐹𝑡−1+ 𝛽3∆𝑆𝑡−1+ 𝛽4∆𝐹𝑡−2
+ 𝛽5∆𝑆𝑡−2+ 𝛽6𝑅𝑎𝑡𝑒𝑡+ 𝛽7ln (𝐺𝐷𝑃𝑝𝑒𝑟)𝑡+ 𝛽8ln (𝑆𝑡𝑜𝑐𝑘)𝑡
∆𝑆𝑡 = (𝛽0 + 𝛼0𝛽1) + 𝛽1𝐹𝑡−1+ 𝛼1𝛽1𝑆𝑡−1+ 𝛽2∆𝐹𝑡−1+ 𝛽3∆𝑆𝑡−1+ 𝛽4∆𝐹𝑡−2
+ 𝛽5∆𝑆𝑡−2+ 𝛽6𝑅𝑎𝑡𝑒𝑡+ 𝛽7ln (𝐺𝐷𝑃𝑝𝑒𝑟)𝑡+ 𝛽8ln (𝑆𝑡𝑜𝑐𝑘)𝑡
In my studies, ∆F is 1st difference of the proxy futures price index at time t, ∆S is
1st difference of the proxy of spot price index at time t, ln(GDPPER) is exponential
logarithmic of GDP per capita, ln(STOCK) is exponential logarithmic of TAIEX, RATE
is residential mortgage borrowing rate. Sample period is 2002Q1 to 2012 Q4. Using this
equation, I test the price lead-lag relationship for my experimenta
‧
國立 政 治 大 學
‧
N a tio na
l C h engchi U ni ve rs it y
31