Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
This handout illustrates the steps to carry out Unit Root tests, Johansen cointegration test,
Granger Causality, variance decomposition and Impulse response functions.
This example analyzes the importance of the monetary policy and its transmission mechanism in
the fast-growing Malaysian economy. The monetary model is:
M3 = f(Y, R, P)
(1)
where M3 is money supply; Y is industrial production index; R is 3-month Treasury Bill
(Interest Rate); and P is Consumer Price Index (CPI).
The data file is MacroM3.xls; where the dataset is covering from 1980M1 to 2008:M12.
Year
1980M1
1980M2
1980M3
1980M4
1980M5
1980M6
1980M7
1980M8
1980M9
1980M10
1980M11
1980M12
1981M1
:
:
:
M3
26281.8
27301
27926
28721.9
29483
30130
30747.6
31157.6
30864
31272.7
31906.4
32687.6
34458.1
:
:
:
2008M9
2008M10
2008M11
2008M12
912780
900443
909231
931656
R
P
3.49
3.59
3.66
3.75
4.1
4.17
4.14
4.14
4.26
4.33
4.47
4.46
4.42
:
:
:
3.56
3.55
3.38
3.02
46.1
46.21
46.28
46.13
46.49
46.86
47.36
47.78
47.62
47.89
48.54
48.94
49.78
Y
12.54
12.61
13.22
13.56
13.72
12.93
14.06
13.29
13.95
13.83
13.56
13.39
13.06
:
:
:
:
:
:
114.73
114.23
112.93
111.83
105.04
104.35
100.38
95.92
1
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
1. Open Eview 6 – File – New -- Workfile
2. Choose the frequency as Monthly – from 1980 M1 to 2008 M12 and then Click “OK”
3. Click the button “Quick” and “Empty Group (Edit Series)”
2
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
4. Place your cursor to the left of the first row (obs)
5. Copy the original data from Excel file.
3
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
6. Paste the data to Eview worksheet
7. Transform the variables into logarithm from [Type the following generate (genr)
command]:
genr lm3 = log(m3)
genr lp = log(p)
genr lr = log(r)
genr ly = log(y)
4
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
STEP 1: Unit Root Tests
8. Checking for Unit Root – For example: lm3. Double click on “lm3”, click “View” and
choose the Unit Root Test.
9. We can choose Augmented Dickey Fuller (ADF) test and the optimal lag length is
selected by Akaike Information Criteria (large sample size).
i)
First, we perform the unit root test of “lm3”: level model with constant but
without trend model (let say the maximum lag is 16) .
5
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
ii)
Second, we perform the Unit root test again for the level model but now with
constant with trend model.
Eview Output for level Unit Root Test:
Constant without Trend Model:
Null Hypothesis: LM3 has a unit root
Exogenous: Constant
Lag Length: 16 (Automatic based on AIC, MAXLAG=16)
Augmented Dickey-Fuller test statistic
Test critical values:
1% level
5% level
10% level
t-Statistic
Prob.*
-1.166712
-3.449917
-2.870057
-2.571377
0.6897
t-Statistic
Prob.*
-1.559270
-3.985941
-3.423418
-3.134664
0.8069
*MacKinnon (1996) one-sided p-values.
Constant with Trend Model:
Null Hypothesis: LM3 has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 15 (Automatic based on AIC, MAXLAG=16)
Augmented Dickey-Fuller test statistic
Test critical values:
1% level
5% level
10% level
*MacKinnon (1996) one-sided p-values.
6
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
After that, we estimate the first difference with and without trend models (repeat the
same process but now select 1st difference).
Eview Output for First Different Unit Root Test:
Constant without Trend Model:
Null Hypothesis: D(LM3) has a unit root
Exogenous: Constant
Lag Length: 14 (Automatic based on AIC, MAXLAG=16)
t-Statistic
Prob.*
Augmented Dickey-Fuller test statistic
-2.865942
0.0505
Test critical values:
-3.449857
-2.870031
-2.571363
1% level
5% level
10% level
*MacKinnon (1996) one-sided p-values.
Constant with Trend Model:
Null Hypothesis: D(LM3) has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 15 (Automatic based on AIC, MAXLAG=16)
Augmented Dickey-Fuller test statistic
Test critical values:
1% level
5% level
10% level
*MacKinnon (1996) one-sided p-values.
7
t-Statistic
Prob.*
-3.051348
-3.986026
-3.423459
-3.134688
0.1200
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
10. We also can perform another unit root test namely Phillips-Perron (PP) test.
Eviews Output:
i.
Level, Constant Without Trend
Null Hypothesis: LM3 has a unit root
Exogenous: Constant
Bandwidth: 9 (Newey-West using Bartlett kernel)
Phillips-Perron test statistic
Test critical values:
1% level
5% level
10% level
Adj. t-Stat
Prob.*
-2.469495
-3.448998
-2.869653
-2.571161
0.1239
Adj. t-Stat
Prob.*
-1.041872
0.9355
*MacKinnon (1996) one-sided p-values.
ii.
Level, Constant With Trend
Null Hypothesis: LM3 has a unit root
Exogenous: Constant, Linear Trend
Bandwidth: 9 (Newey-West using Bartlett kernel)
Phillips-Perron test statistic
Test critical values:
1% level
5% level
10% level
-3.984726
-3.422828
-3.134315
*MacKinnon (1996) one-sided p-values.
iii.
First Difference, Constant Without Trend
Null Hypothesis: D(LM3) has a unit root
Exogenous: Constant
Bandwidth: 9 (Newey-West using Bartlett kernel)
Phillips-Perron test statistic
Test critical values:
1% level
5% level
10% level
Adj. t-Stat
Prob.*
-16.62462
0.0000
-3.449053
-2.869677
-2.571174
*MacKinnon (1996) one-sided p-values.
8
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
iv.
First Difference, Constant With Trend
Null Hypothesis: D(LM3) has a unit root
Exogenous: Constant, Linear Trend
Bandwidth: 8 (Newey-West using Bartlett kernel)
Phillips-Perron test statistic
Test critical values:
1% level
5% level
10% level
Adj. t-Stat
Prob.*
-16.73624
-3.984804
-3.422865
-3.134337
0.0000
*MacKinnon (1996) one-sided p-values.
Table 1 below presents the results of Unit Root Tests:
Variable
LM3
Table 1 Unit Root Tests
Augmented Dickey Fuller
Phillips Perron
(ADF)
(PP)
Level
Constant
Constant
Constant
Constant
Without Trend With Trend
Without Trend
With Trend
-1.1667
-1.5593
-2.4695
-1.0419
(16)
(15)
[9]
[9]
LP
LR
LY
LM3
-2.866**
(14)
First Difference
-3.0513
-16.6246***
(15)
[9]
-16.7362***
[8]
LP
LR
LY
Note: *** and ** denotes significant at 1%, and 5% significance level, respectively. The figure in
parenthesis (…) represents optimum lag length selected based on Akaike Info Critirion. The figure in
bracket […] represents the Bandwidth used in the KPSS test selected based on Newey-West
Bandwidth critirion.
Please complete the above results of Unit Root Tests
9
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
STEP 2: Estimate the Multivariate VECM Model
11. After testing the variables are stationary at first order or I(1), then the step is to estimate
the Vector Error-correction Model (VECM). Firstly, we need to select an optimum lag of
VECM model before performing the Johansen cointegration test.
(You should show all the four log variables).
Then click “Quick” – “Estimate VAR” – “Vector Error Correction”
10
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
12. From equation (1), the VECM model can be written as:
(2)
(3)
(4)
(5)
11
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
STEP 2.1: Select the Optimum Lag Length
(a) First, we estimate the VECM model with lag 1
Type in all variables with
lm3 first (dependent
variable)
LM3 LY LR LP
Change 2 to 1
(b) Make residuals for the VECM models, click “Proc” – “Make Residuals”
EViews will show 4 residuals in the EViews Workfile – resid01 (residual in Equation 1),
resid02 (residual in Equation 2), resid03 (residual in Equation 3), and resid04 (residual in
Equation 4).
12
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
(c) Now, the autocorrelation of the error terms in each regression is checked by using the
Ljung-Box Q-statistic.
We double click the “resid01” – “View” – “Correlogram…” – “OK”
13
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
EViews Output:
The p-value is less
than 0.05. This
implies that the
regression residuals
have autocorrelation
problem.
The Q-statistic shows that the error terms are statistically significant from lag 12 for
“resid01”. This indicates that the model with lag 1 has autocorrelation problem. Hence,
we need to re-estimate the VECM model by increasing one lag (repeat the same process
but now with lag 2).
This process will continue until each of the regression error terms is free from
autocorrelation problem (where the p-values of Q-statistic are greater than 0.05).
In this case, we repeat the same process and the optimum lag is 12.
14
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
13. The EViews output with 12 lag is as follows:
Long-run
Equation
Error
correction
terms
(ECT)
STEP 2.2: Johansen Cointegration Test
14. After obtaining the optimum lag, the next step is to estimate the Johansen Cointegration
Test. Click “View” – “Cointegration Test” – “OK”.
15
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
0.05 represent 5%
significance level.
EViews Output:
Sample (adjusted): 1981M02 2008M12
Included observations: 335 after adjustments
Trend assumption: Linear deterministic trend
Series: LM3 LP LR LY
Lags interval (in first differences): 1 to 12
Unrestricted Cointegration Rank Test (Trace)
Hypothesized
No. of CE(s)
Eigenvalue
Trace
Statistic
0.05
Critical Value
Prob.**
None *
At most 1
At most 2
At most 3
0.088734
0.043366
0.014884
0.002934
51.98841
20.86000
6.007864
0.984286
47.85613
29.79707
15.49471
3.841466
0.0194
0.3664
0.6946
0.3211
Trace test indicates 1 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values
Unrestricted Cointegration Rank Test (Maximum Eigenvalue)
Hypothesized
No. of CE(s)
Eigenvalue
Max-Eigen
Statistic
0.05
Critical Value
Prob.**
None *
At most 1
At most 2
At most 3
0.088734
0.043366
0.014884
0.002934
31.12841
14.85214
5.023578
0.984286
27.58434
21.13162
14.26460
3.841466
0.0168
0.2994
0.7388
0.3211
16
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 level
* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values
Table 2 presents the Johansen-Juselius Cointegration test. The result shows that both
Trace test and Max-Eigen test are statistically significant to reject the null hypothesis of r
= 0 at 5% significance level. Therefore, only one long run cointegration relationship
between M3 and it determinants.
Table 2: Johansen-Juselius Cointegration Tests
Hypothesized
Trace
Max-Eigen
Critical Values (5%)
No. of CE(s)
Statistic
Statistic
Trace
Max-Eigen
r=0
r≤1
r≤2
r≤3
51.9884**
20.860
6.0078
0.9843
31.1284**
14.852
5.0235
0.9843
47.856
29.797
15.495
3.8415
27.584
21.132
14.265
3.8415
Note: ** denotes significant at 5% significance levels.
STEP 2.3: VECM Model
If the model contains cointegration relationship among the variables, then we can proceed
to VECM and the long run equation is:
LM3t-1 = - 5.0146 + 3.6031 LPt-1 + 0.2233 LRt-1+ 0.3357 LYt-1
s.e
(0.2872)
(0.0494)
(0.0946)
t-stat
[12.5459]
[4.5212]
[3.5477]
All variables are positively significant at 5% significance level.
17
You can write this
equation by referring to
page 15 (but the
coefficient signs are now
reversed, why?)
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
STEP 2.4: Granger causality test
15. After estimating the long-run VECM model, then we proceed to the short run Granger
causality test. Click “View” – “Lag Structure” – “Granger Causality/Block Exogeneity
Tests”.
EViews Output:
VEC Granger Causality/Block Exogeneity Wald Tests
Sample: 1980M01 2008M12
Included observations: 335
Dependent variable: D(LM3)
Excluded
Chi-sq
df
Prob.
D(LP)
D(LR)
D(LY)
30.22932
14.72288
21.50639
12
12
12
0.0026
0.2569
0.0434
All
67.21100
36
0.0012
18
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
Dependent variable: D(LP)
Excluded
Chi-sq
df
Prob.
D(LM3)
D(LR)
D(LY)
23.98443
23.39792
23.92561
12
12
12
0.0204
0.0245
0.0208
All
63.26058
36
0.0033
Dependent variable: D(LR)
Excluded
Chi-sq
df
Prob.
D(LM3)
D(LP)
D(LY)
15.82326
12.32216
13.98317
12
12
12
0.1995
0.4202
0.3018
All
36.82666
36
0.4305
Dependent variable: D(LY)
Excluded
Chi-sq
df
Prob.
D(LM3)
D(LP)
D(LR)
17.93616
16.41443
10.79759
12
12
12
0.1176
0.1730
0.5463
All
48.57377
36
0.0786
With Cointegration, the dynamic causal interactions among the variables should be
phrased in a vector error correction form. This allows us to assess both long-run and
short-run causality, respectively, on the  2 -test of the lagged first differenced terms for
each right-hand-side variable and the t-test of the error correction term. The results of the
test are presented in Table 3.
19
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
Table 3: Granger Causality Results based on VECM
Independent Variables
 2 -statistics of lagged 1st differenced term
Dependent
Variable
ΔLM3
ΔLM3
--
ΔLP
ΔLR
ΔLY
ECTt-1
coefficient
[p-value]
ΔLP
ΔLR
30.23***
14.72
[0.003]
[0.257]
ΔLY
21.51**
[0.043]
(t-ratio)
-0.028**
(-3.533)
23.98**
[0.020]
--
23.39**
[0.024]
23.93**
[0.021]
0.009**
(2.800)
15.82
[0.199]
12.32
[0.420]
--
13.98
[0.302]
0.124
(1.947)
17.93
[0.118]
16.41
[0.173]
10.79
[0.546]
--
0.093
(2.052)
Note: *** and ** denotes significant at 1% and 5% significance level, respectively. The figure in the
parenthesis (…) denote as t-statistic and the figure in the squared brackets […] represent as p-value.
and the causal channels can be summarized as below:
LM3
LP
LY
LR
20
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
STEP 2.5: Variance decomposition (VDC)
16. The result of VECM indicates the exogeneity or endogeneity of a variable in the system
and the direction of Granger-causality within the sample period. However, it does not
provide us with the dynamic properties of the system. The analysis of the dynamic
interactions among the variables in the post-sample period is conducted through variance
decompositions (VDCs) and impulse response functions (IRFs).
Click “View” – “Variance
Decompositions” – “Table”
EViews Output:
Variance Decomposition of
LM3:
Period
S.E.
LM3
LP
LR
LY
1
2
3
4
5
6
7
8
9
10
0.009207
0.012581
0.015458
0.018050
0.020777
0.023737
0.026313
0.028367
0.030475
0.032785
100.0000
99.59571
99.50123
98.21271
94.57704
91.01041
88.11711
84.90036
81.38129
76.88578
0.000000
0.237534
0.242707
0.478537
1.376087
2.386103
2.976382
3.376980
2.986563
2.630474
0.000000
7.15E-06
0.022763
0.134646
1.117796
1.674290
2.326990
2.783734
3.496885
4.784743
0.000000
0.166753
0.233302
1.174103
2.929073
4.929200
6.579520
8.938924
12.13526
15.69901
Variance Decomposition of LP:
Period
S.E.
LM3
LP
LR
LY
21
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
1
2
3
4
5
6
7
8
9
10
0.003904
0.006102
0.007787
0.009032
0.010067
0.010816
0.011544
0.012314
0.012968
0.013545
0.057833
0.232518
0.493569
0.723986
1.013711
2.379293
4.710690
6.518148
7.566270
8.302143
99.94217
99.30812
99.04644
98.88313
98.49578
96.86372
94.19444
91.82371
90.49069
89.10389
0.000000
0.034317
0.076551
0.067843
0.206479
0.504991
0.867165
1.438480
1.743578
2.390148
0.000000
0.425045
0.383441
0.325045
0.284028
0.251992
0.227708
0.219661
0.199466
0.203815
Variance Decomposition of LR:
Period
S.E.
LM3
LP
LR
LY
1
2
3
4
5
6
7
8
9
10
0.072920
0.111546
0.139640
0.161746
0.176165
0.191569
0.208351
0.222819
0.236249
0.251486
2.995915
3.619142
4.349874
4.672351
5.039088
5.355593
4.742339
4.361189
3.960742
3.503003
0.019765
0.098381
0.076814
0.223168
0.224267
0.193439
0.306652
0.434910
0.732402
0.914070
96.98432
96.28206
95.55480
95.05622
94.69537
94.34811
94.75085
94.95968
94.78659
94.85731
0.000000
0.000418
0.018517
0.048258
0.041277
0.102858
0.200158
0.244217
0.520266
0.725617
Variance Decomposition of LY:
Period
S.E.
LM3
LP
LR
LY
1
2
3
4
5
6
7
8
9
10
0.051559
0.060572
0.072073
0.082283
0.089036
0.094439
0.098761
0.104042
0.108643
0.112538
0.724280
0.608491
0.473792
1.219285
1.136213
1.235009
1.619555
2.296307
2.798176
3.291952
0.570742
0.768220
1.096219
1.277598
1.872747
1.895860
2.445372
4.431683
4.585921
5.643364
0.577937
0.439926
0.640465
0.572537
0.548866
0.577269
0.530370
0.639905
1.005189
1.336476
98.12704
98.18336
97.78952
96.93058
96.44217
96.29186
95.40470
92.63211
91.61071
89.72821
Cholesky Ordering: LM3 LP LR
LY
22
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
STEP 2.6: Impulse response functions (IRFs)
17. Estimate the impulse response functions (IRFs), click “Estimate” and change the “Vector
Error Correction” to “Unrestricted VAR” and increase one more lag for the model from
lag 12 to lag 13.
Select “Impulse”
23
Unit Root, Cointegration, VECM, Variance Decomposition and Impulse Response Functions
Response to Cholesky One S.D. Innovations ± 2 S.E.
Response of LM3 to LM3
Response of LM3 to LP
Response of LM3 to LR
Response of LM3 to LY
.015
.015
.015
.015
.010
.010
.010
.010
.005
.005
.005
.005
.000
.000
.000
.000
-.005
-.005
-.005
-.005
-.010
-.010
1
2
3
4
5
6
7
8
9
-.010
1
10
2
Response of LP to LM3
3
4
5
6
7
8
9
-.010
1
10
2
Response of LP to LP
3
4
5
6
7
8
9
1
10
.006
.006
.006
.004
.004
.004
.004
.002
.002
.002
.002
.000
.000
.000
.000
-.002
1
2
3
4
5
6
7
8
9
-.002
1
10
2
Response of LR to LM3
3
4
5
6
7
8
9
2
Response of LR to LP
3
4
5
6
7
8
9
1
10
.12
.12
.08
.08
.08
.08
.04
.04
.04
.04
.00
.00
.00
.00
-.04
2
3
4
5
6
7
8
9
-.04
1
10
2
Response of LY to LM3
3
4
5
6
7
8
9
2
Response of LY to LP
3
4
5
6
7
8
9
1
10
.06
.06
.06
.04
.04
.04
.02
.02
.02
.02
.00
.00
.00
.00
-.02
-.02
-.02
-.02
-.04
2
3
4
5
6
7
8
9
10
-.04
1
2
3
4
5
6
7
8
9
10
24
7
8
9
10
3
4
5
6
7
8
9
10
3
4
5
6
7
8
9
10
9
10
Response of LY to LY
.04
1
2
Response of LY to LR
.06
-.04
6
-.04
1
10
5
Response of LR to LY
.12
1
2
Response of LR to LR
.12
-.04
4
-.002
1
10
3
Response of LP to LY
.006
-.002
2
Response of LP to LR
-.04
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8