1.

Stationary:
LNZSE
9.6

9.2

8.8

8.4

8.0

7.6

7.2
02 04 06 08 10 12 14 16 18

LNZ$
-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

-0.7

-0.8

-0.9

-1.0
02 04 06 08 10 12 14 16 18

2. Level of Integration:

a. (LNZSE):

Null Hypothesis: LNZSE has a unit root

Exogenous: Constant, Linear Trend
Lag Length: 1 (Automatic - based on SIC, maxlag=31)

t-Statistic Prob.*
Augmented Dickey-Fuller test statistic -1.050992 0.9353
Test critical values: 1% level -3.959915
5% level -3.410724
10% level -3.127150

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LNZSE)
Method: Least Squares
Date: 08/26/19 Time: 20:18
Sample (adjusted): 1/04/2001 7/31/2019
Included observations: 4887 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

LNZSE(-1) -0.000595 0.000566 -1.050992 0.2933

D(LNZSE(-1)) 0.048715 0.014297 3.407308 0.0007
C 0.004721 0.004259 1.108361 0.2678
@TREND("1/02/2001") 2.29E-07 1.87E-07 1.224269 0.2209

R-squared 0.002664 Mean dependent var 0.000379

Adjusted R-squared 0.002051 S.D. dependent var 0.006693
S.E. of regression 0.006686 Akaike info criterion -7.176855
Sum squared resid 0.218267 Schwarz criterion -7.171540
Log likelihood 17540.65 Hannan-Quinn criter. -7.174990
F-statistic 4.346977 Durbin-Watson stat 1.997455
Prob(F-statistic) 0.004583

Result: Data is not stationary or has Unit Root

Null Hypothesis: D(LNZSE) has a unit root

Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic - based on SIC, maxlag=31)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -66.58528 0.0000

Test critical values: 1% level -3.959915
5% level -3.410724
10% level -3.127150

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LNZSE,2)
Method: Least Squares
Date: 08/26/19 Time: 20:23
Sample (adjusted): 1/04/2001 7/31/2019
Included observations: 4887 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

D(LNZSE(-1)) -0.951675 0.014293 -66.58528 0.0000

 C 0.000249 0.000191 1.300260 0.1936
@TREND("1/02/2001") 4.58E-08 6.78E-08 0.675108 0.4996

R-squared 0.475831 Mean dependent var -3.99E-07

Adjusted R-squared 0.475616 S.D. dependent var 0.009233
S.E. of regression 0.006686 Akaike info criterion -7.177038
Sum squared resid 0.218317 Schwarz criterion -7.173052
Log likelihood 17540.09 Hannan-Quinn criter. -7.175639
F-statistic 2216.800 Durbin-Watson stat 1.997434
Prob(F-statistic) 0.000000

Result:
I. Data is stationary.
II. LNZSE I(1)

b. (LNZ$):

Null Hypothesis: LNZ$ has a unit root

Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic - based on SIC, maxlag=31)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -1.989889 0.6062

Test critical values: 1% level -3.959915
5% level -3.410724
10% level -3.127150

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LNZ$)
Method: Least Squares
Date: 08/26/19 Time: 20:26
Sample (adjusted): 1/03/2001 7/31/2019
Included observations: 4888 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

LNZ$(-1) -0.001583 0.000796 -1.989889 0.0467

C -0.000562 0.000515 -1.090705 0.2755
@TREND("1/02/2001") 8.23E-09 1.02E-07 0.080679 0.9357

R-squared 0.001193 Mean dependent var 7.95E-05

Adjusted R-squared 0.000784 S.D. dependent var 0.008097
S.E. of regression 0.008094 Akaike info criterion -6.794726
Sum squared resid 0.320046 Schwarz criterion -6.790740
Log likelihood 16609.31 Hannan-Quinn criter. -6.793327
F-statistic 2.916874 Durbin-Watson stat 2.043132
Prob(F-statistic) 0.054197

Result: Data is not stationary or has Unit Root

Null Hypothesis: D(LNZ$) has a unit root
Exogenous: Constant, Linear Trend
Lag Length: 0 (Automatic - based on SIC, maxlag=31)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -71.52011 0.0000

Test critical values: 1% level -3.959915
5% level -3.410724
10% level -3.127150

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(LNZ$,2)
Method: Least Squares
Date: 08/26/19 Time: 20:29
Sample (adjusted): 1/04/2001 7/31/2019
Included observations: 4887 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

D(LNZ$(-1)) -1.022841 0.014301 -71.52011 0.0000

C 0.000375 0.000232 1.617506 0.1058
@TREND("1/02/2001") -1.19E-07 8.21E-08 -1.446729 0.1480

R-squared 0.511558 Mean dependent var 1.40E-06

Adjusted R-squared 0.511358 S.D. dependent var 0.011578
S.E. of regression 0.008093 Akaike info criterion -6.794987
Sum squared resid 0.319896 Schwarz criterion -6.791000
Log likelihood 16606.55 Hannan-Quinn criter. -6.793588
F-statistic 2557.566 Durbin-Watson stat 1.997426
Prob(F-statistic) 0.000000

Result:
I. Data is stationary.
II. LNZ$ I(1)

Both series NZSE & NZ$ are I (1) so JJ Approach is selected.

JJ Approach:
Date: 08/26/19 Time: 20:31
Sample (adjusted): 1/04/2001 7/31/2019
Included observations: 4887 after adjustments
Trend assumption: Linear deterministic trend
Series: LNZSE LNZ$
Lags interval (in first differences): 1 to 1

Unrestricted Cointegration Rank Test (Trace)

Hypothesized Trace 0.05

No. of CE(s) Eigenvalue Statistic Critical Value Prob.**
 None 0.001261 6.485339 15.49471 0.6381
At most 1 6.56E-05 0.320516 3.841466 0.5713

Trace test indicates no cointegration 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 Max-Eigen 0.05

No. of CE(s) Eigenvalue Statistic Critical Value Prob.**

None 0.001261 6.164822 14.26460 0.5922

At most 1 6.56E-05 0.320516 3.841466 0.5713

Max-eigenvalue test indicates no cointegration at the 0.05 level

* denotes rejection of the hypothesis at the 0.05 level
**MacKinnon-Haug-Michelis (1999) p-values

Unrestricted Cointegrating Coefficients (normalized by b'*S11*b=I):

LNZSE LNZ$
0.318188 5.045668
2.529853 -4.208631

Unrestricted Adjustment Coefficients (alpha):

D(LNZSE) 2.86E-06 5.36E-05

D(LNZ$) -0.000287 2.69E-06

1 Co integrating Equation(s): Log likelihood 34198.12

Normalized co integrating coefficients (standard error in parentheses)

LNZSE LNZ$
1.000000 15.85752
(7.00219)

Adjustment coefficients (standard error in parentheses)

D(LNZSE) 9.11E-07
(3.0E-05)
D(LNZ$) -9.13E-05
(3.7E-05)

Result: No Co integration exits. So VAR is selected.

Vector Auto Regressive Model (VAR):

Vector Autoregression Estimates

Date: 08/26/19 Time: 20:41
Sample (adjusted): 1/04/2001 7/31/2019
Included observations: 4887 after adjustments
Standard errors in ( ) & t-statistics in [ ]

LNZSE LNZ$
 LNZSE(-1) 1.044816 0.011070
(0.01416) (0.01730)
[ 73.7771] [ 0.63981]

LNZSE(-2) -0.044679 -0.011154

(0.01417) (0.01731)
[-3.15420] [-0.64455]

LNZ$(-1) 0.114496 0.976093

(0.01171) (0.01431)
[ 9.77609] [ 68.2174]

LNZ$(-2) -0.114707 0.022447

(0.01171) (0.01431)
[-9.79473] [ 1.56889]

C -0.000859 0.000206
(0.00214) (0.00262)
[-0.40115] [ 0.07890]

R-squared 0.999798 0.997997

Adj. R-squared 0.999798 0.997995
Sum sq. resids 0.214126 0.319604
S.E. equation 0.006623 0.008091
F-statistic 6051010. 608081.5
Log likelihood 17587.45 16608.78
Akaike AIC -7.195601 -6.795082
Schwarz SC -7.188957 -6.788437
Mean dependent 8.270658 -0.392362
S.D. dependent 0.466173 0.180708

Determinant resid covariance (dof adj.) 2.87E-09

Determinant resid covariance 2.86E-09
Log likelihood 34198.28
Akaike information criterion -13.99152
Schwarz criterion -13.97823

VAR Lag Order Selection Criteria

Endogenous variables: LNZSE LNZ$
Exogenous variables: C
Date: 08/26/19 Time: 21:13
Sample: 1/02/2001 7/31/2019
Included observations: 4881

Lag LogL LR FPE AIC SC HQ

0 -928.0708 NA 0.005018 0.381098 0.383759 0.382032

1 34105.99 70025.06 2.93e-09 -13.97254 -13.96456 -13.96974
2 34160.91 109.7236 2.87e-09 -13.99341 -13.98010* -13.98874*
3 34165.01 8.191908 2.87e-09 -13.99345 -13.97482 -13.98691
4 34170.10 10.16106 2.87e-09* -13.99390* -13.96995 -13.98549
5 34171.19 2.164257 2.87e-09 -13.99270 -13.96343 -13.98243
6 34174.95 7.507078 2.87e-09 -13.99260 -13.95802 -13.98047
7 34176.87 3.836761 2.87e-09 -13.99175 -13.95184 -13.97775
8 34182.69 11.60166* 2.87e-09 -13.99250 -13.94727 -13.97663

* indicates lag order selected by the criterion

LR: sequential modified LR test statistic (each test at 5% level)
FPE: Final prediction error
AIC: Akaike information criterion
SC: Schwarz information criterion
HQ: Hannan-Quinn information criterion

Panchura Principle:

Date: 08/26/19 Time: 21:14

Sample: 1/02/2001 7/31/2019
Included observations: 4884
Series: LNZSE LNZ$
Lags interval: 1 to 4

Selected
(0.05 level*)
Number of
Cointegrating
Relations by
Model

Data Trend: None None Linear Linear Quadratic

Test Type No Intercept Intercept Intercept Intercept Intercept
No Trend No Trend No Trend Trend Trend
Trace 2 0 0 0 0
Max-Eig 2 0 0 0 0

*Critical values based on MacKinnon-Haug-Michelis (1999)

Information
Criteria by
Rank and
Model

Data Trend: None None Linear Linear Quadratic

Rank or No Intercept Intercept Intercept Intercept Intercept
No. of CEs No Trend No Trend No Trend Trend Trend

Log
Likelihood by
Rank (rows)
and Model
(columns)
0 34179.89 34179.89 34186.93 34186.93 34188.36
1 34187.05 34187.12 34189.72 34189.96 34190.84
2 34189.79 34189.88 34189.88 34191.03 34191.03

Akaike
Information
Criteria by
Rank (rows)
and Model
(columns)
0 -13.99013 -13.99013 -13.99219* -13.99219* -13.99196
1 -13.99142 -13.99104 -13.99169 -13.99138 -13.99134
2 -13.99090 -13.99012 -13.99012 -13.98977 -13.98977

Schwarz
Criteria by
Rank (rows)
and Model
(columns)
 0 -13.96885* -13.96885* -13.96826 -13.96826 -13.96537
1 -13.96483 -13.96312 -13.96244 -13.96080 -13.95943
2 -13.95899 -13.95555 -13.95555 -13.95255 -13.95255

Result: Model 1 is selected because relationship exists so VECM is selected.

VECM (Vector Error Correction model):

Vector Error Correction Estimates

Date: 08/26/19 Time: 21:35
Sample (adjusted): 1/09/2001 7/31/2019
Included observations: 4884 after adjustments
Standard errors in ( ) & t-statistics in [ ]

Cointegrating Eq: CointEq1

LNZSE(-1) 1.000000

LNZ$(-1) 11.69114
(5.62266)
[ 2.07929]

C -3.685652

Error Correction: D(LNZSE) D(LNZ$)

CointEq1 2.87E-06 -0.000114

(4.0E-05) (4.8E-05)
[ 0.07243] [-2.35409]

D(LNZSE(-1)) 0.046713 0.010612

(0.01433) (0.01749)
[ 3.26044] [ 0.60677]

D(LNZSE(-2)) -0.028104 0.039516

(0.01433) (0.01750)
[-1.96058] [ 2.25828]

D(LNZSE(-3)) 0.029723 0.005305

(0.01434) (0.01751)
[ 2.07246] [ 0.30303]

D(LNZSE(-4)) -0.011675 0.002298

(0.01419) (0.01732)
[-0.82275] [ 0.13266]

D(LNZ$(-1)) 0.114005 -0.022491

(0.01173) (0.01432)
[ 9.71757] [-1.57044]

D(LNZ$(-2)) 0.005583 -0.003479

(0.01184) (0.01445)
[ 0.47163] [-0.24078]

D(LNZ$(-3)) -0.005589 -0.035004

(0.01183) (0.01444)
[-0.47249] [-2.42414]
 D(LNZ$(-4)) -0.003309 0.017654
(0.01183) (0.01444)
[-0.27981] [ 1.22280]

C 0.000356 5.76E-05
(9.5E-05) (0.00012)
[ 3.73182] [ 0.49528]

R-squared 0.023288 0.004088

Adj. R-squared 0.021484 0.002249
Sum sq. resids 0.213626 0.318333
S.E. equation 0.006620 0.008082
F-statistic 12.91238 2.222959
Log likelihood 17580.86 16606.82
Akaike AIC -7.195276 -6.796405
Schwarz SC -7.181980 -6.783109
Mean dependent 0.000378 7.59E-05
S.D. dependent 0.006693 0.008091

Determinant resid covariance (dof adj.) 2.86E-09

Determinant resid covariance 2.85E-09
Log likelihood 34189.72
Akaike information criterion -13.99169
Schwarz criterion -13.96244

Variance Decomposition Analysis:

Vector Autoregression Estimates
Date: 08/26/19 Time: 21:54
Sample (adjusted): 1/09/2001 7/31/2019
Included observations: 4884 after adjustments
Standard errors in ( ) & t-statistics in [ ]

D(LNZSE) D(LNZ$)

D(LNZSE(-1)) 0.046718 0.010398

(0.01433) (0.01750)
[ 3.26119] [ 0.59427]

D(LNZSE(-2)) -0.028099 0.039313

(0.01433) (0.01751)
[-1.96045] [ 2.24563]

D(LNZSE(-3)) 0.029729 0.005065

(0.01434) (0.01752)
[ 2.07312] [ 0.28916]

D(LNZSE(-4)) -0.011668 0.002021

(0.01419) (0.01733)
[-0.82236] [ 0.11663]

D(LNZ$(-1)) 0.114010 -0.022688

(0.01173) (0.01433)
[ 9.71916] [-1.58354]

D(LNZ$(-2)) 0.005586 -0.003606

 (0.01184) (0.01446)
[ 0.47195] [-0.24945]

D(LNZ$(-3)) -0.005587 -0.035100

(0.01183) (0.01445)
[-0.47233] [-2.42974]

D(LNZ$(-4)) -0.003307 0.017567

(0.01183) (0.01444)
[-0.27965] [ 1.21619]

C 0.000356 5.80E-05
(9.5E-05) (0.00012)
[ 3.73210] [ 0.49844]

R-squared 0.023287 0.002956

Adj. R-squared 0.021684 0.001319
Sum sq. resids 0.213626 0.318695
S.E. equation 0.006620 0.008085
F-statistic 14.52873 1.806426
Log likelihood 17580.86 16604.05
Akaike AIC -7.195684 -6.795678
Schwarz SC -7.183718 -6.783712
Mean dependent 0.000378 7.59E-05
S.D. dependent 0.006693 0.008091

Determinant resid covariance (dof adj.) 2.86E-09

Determinant resid covariance 2.85E-09
Log likelihood 34186.93
Akaike information criterion -13.99219
Schwarz criterion -13.96826

Table:

Variance Decomposition of D(LNZSE):

Period S.E. D(LNZSE) D(LNZ$)

1 0.006620 100.0000 0.000000

2 0.006692 98.10409 1.895910
3 0.006694 98.09529 1.904714
4 0.006698 98.08621 1.913789
5 0.006698 98.08413 1.915870
6 0.006698 98.08395 1.916050
7 0.006698 98.08395 1.916048
8 0.006698 98.08395 1.916047
9 0.006698 98.08395 1.916049
10 0.006698 98.08395 1.916049

Variance Decomposition of D(LNZ$):

Period S.E. D(LNZSE) D(LNZ$)

1 0.008085 0.082914 99.91709

2 0.008088 0.089040 99.91096
3 0.008092 0.193340 99.80666
4 0.008096 0.194598 99.80540
5 0.008097 0.194710 99.80529
6 0.008097 0.194711 99.80529
7 0.008097 0.194717 99.80528
8 0.008097 0.194717 99.80528
9 0.008097 0.194717 99.80528
10 0.008097 0.194717 99.80528

Cholesky Ordering: D(LNZSE) D(LNZ$)