40th IAEE International Conference
Interval Tests for Structural Breaks in the Dependence:
Empirical Evidence of Oil and Gold Markets
BingYue Liu
Department of Statistics and Finance
University of Science and Technology of China
June 18-21, Singapore
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Interval Tests for Structural Breaks
Introduction
 Crude Oil
•
•
•
Important commodity
Large trading volume
Low price
 Gold
•
•
Important precious metal
Hedge heaven
 Co-movement
•
•
Positive
Dynamic
 Guidance
•
•
Investment portfolio
Risk management
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Interval Tests for Structural Breaks
Literature
 Baffes (2007), Hammoudeh and Yuan (2008), Soytas et al.
(2009), Sari et al. (2010), Narayan et al. (2010), Zhang and
Wei (2010), Reboredo (2013)
•


•


Method
Static relationship
Linear relationship
Conclusion
Short-term or long-term relationship
Causal relationship
 But
•
•
Market volatility
Market mechanism change under extreme market shock
Structural (Large) change in dependence
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Interval Tests for Structural Breaks
Contribution
 Apply TVCopula to analyze dependence of oil & gold
•
•
•
Nonlinear
Dynamic
Small change in dependence
 Change point test to analyze dependence of oil & gold
•
•


Large change in dependence
But
Extreme shocks last not long
Hardly capture the dependent features in extreme risk period
 Propose change interval test
•
•
Capture the dependent features in an extreme period
Test the effect of extreme shocks on the co-movement across markets
• Empirical evidence of crude oil and gold markets
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Interval Tests for Structural Breaks
Dependent Relationship
Comparision between Change Point and Change Interval
0
t1 t2
t3
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t4 t5
T
5
Interval Tests for Structural Breaks
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Interval Tests for Structural Breaks
T
T
ˆi  arg max  log fi ,t  xi ,t | F t 1 ;i  , i  1, 2 , and then ˆ  arg max  log ct  uˆ1,t , uˆ2,t | F t 1 ;  
i

t 1
ˆIFM  ˆ1 , ˆ 2 , ˆ 

t 1

T ˆIFM   0  N  0, G 1  0  
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Interval Tests for Structural Breaks
Methodology
 Change point test (Dias and Embrechts, 2009)
• Random vector U = (U1, U2), joint distribution C  u; θ t 
• Test HT: H0: θ1  θ2   θT vs. HA: θ1   θk  θk+1 
 Likelihood ratio statistic
 k  
sup
 θT
 c  u ; θ  c  u ; θ
sup  c  u ; θ 
 θ,θ 1 t  k
t
k  t T
θ 1 t T
t
t
 Note ZT  1max
 2log   k   , then referring to Csörgő and Horváth (1997)
 k T
Pr  ZT
12
x p  exp   x 2 2  
1  h 1  l   p  log 1  h 1  l   4  O  1   , as x  
 x  p 2
  log
 4 
2    p 2 
hl
x2
hl
x2
 x 
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Interval Tests for Structural Breaks
Methodology
 Change interval test
•
Define parameter space

Q   θ1 , , θT  : θ1 
Q011   θ1 ,
, θT  : θ1  θ2 
 θk 2 , θk 2 1 
, θT  : θ1 
 θk1 , θk1 1 
 θT ,

, θT  : θ1 
 θk 2 , θk 2 1 
 θT .
Q013   θ1 ,

 θT ,
 θT  ,

Q012   θ1 ,
•
 θk1 , θk1 1 


Construct hypothesis test
HT11: H0: θ1 , , θT  Q011
vs.
HA:  θ1 , , θT  Q Q011 ,
HT12: H0: θ1 , , θT  Q012
vs.
HA:  θ1 , , θT  Q Q012 ,
HT13: H0: θ1 , , θT  Q013
vs.
HA:  θ1 , , θT  Q Q013 .
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Null hypothesis
 HT11: No change point
 HT12: No change point or
only one change point in k1
 HT13: No change point or
only one change point in k2
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Interval Tests for Structural Breaks
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Interval Tests for Structural Breaks
Empirical Analysis
Estimates for marginal distribution models
Model: rt    at , at   tt ,  t 2        I t 1   at 12     t 12 ,  t
Oil






1.07e-04
0.008* a
0.012**
0.961***
0.050***
8.465***
Model: rt    at , at   tt ,  t 2      at 12     t 12 ,  t
Gold
t  0,1
t  0,1





0.049**
0.013***
0.038***
0.954***
5.247***
Note: a *, **, *** denote the significant levels of 10%, 5% and 1%, respectively.
 Oil & Gold: volatility clustering
 Oil: volatility asymmetry
 Oil & Gold: innovation symmetric fat-tailed distribution
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Interval Tests for Structural Breaks
Empirical Analysis
Change point test over the whole sample period
n
T
zT 1 2
P  ZT 1 2  zT 1 2 
H0 (0.95)
Time of Change
1/4/2006–4/29/2016
1
2546
5.542
4.40e-06
R. a
10/9/2013
1/4/2006–10/9/2013
1
1913
1.851
1
--
--
10/10/2013–4/29/2016
1
633
2.765
0.165
--
--
1/4/2006–4/29/2016
2
2546
5.720
1.16e-05
R.
10/9/2013
1/4/2006–10/9/2013
2
1913
2.758
0.608
--
--
10/10/2013–4/29/2016
2
633
2.739
0.516
--
--
1/4/2006–4/29/2016
1
2546
4.749
2.22e-04
R.
11/7/2013
1/4/2006–11/7/2013
1
1934
2.347
0.484
--
--
11/8/2013–4/29/2016
1
612
2.183
0.525
--
--
Period
Normal
t
Clayton
Note: a R. denotes the null hypothesis is rejected at the 5% level.
Change point test can only check out one change point over the whole sample period.
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Interval Tests for Structural Breaks
Empirical Analysis
Change interval test for financial crisis over the whole sample period
Initial Period: 7/1/2008–6/30/2009
Copula
n
Period of Change
Change Interval Test Statistics
k1
k2
HT11
HT12
HT13
Normal
1
9/10/2008
4/23/2009
13.677** a
6.828**
13.140**
t
2
9/9/2008
4/23/2009
14.440**
6.046*
13.012**
Clayton
1
9/9/2008
6/11/2009
9.570**
6.872**
9.567**
All Copula
--
9/9/2008
4/23/2009
--
--
--
Note: a *, ** denote the significant levels of 5% and 1%, respectively.
Change interval period: 9/9/2008-4/23/2009
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Interval Tests for Structural Breaks
Empirical Analysis
Change point test for pre-crisis period
Period
n
T
zT 1 2
P  ZT 1 2  zT 1 2 
H0 (0.95)
Time of Change
Pre-crisis Period: 1/4/2006–9/9/2008
Normal
1/4/2006–9/9/2008
1
666
2.637
0.223
--
--
1/4/2006–9/9/2008
2
666
2.602
0.660
--
--
1
666
2.275
0.457
--
--
t
Clayton
1/4/2006–9/9/2008
Note: a R. denotes the null hypothesis is rejected at the 5% level.
No change point in the pre-crisis period.
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Change point test for post-crisis period
Period
n
T
zT 1 2
P  ZT 1 2  zT 1 2 
H0 (0.95)
Time of Change
Post-crisis Period: 4/24/2009–4/29/2016
Normal
4/24/2009–4/29/2016
1
1728
7.030
4.56e-10
R. a
4/27/2010
4/24/2009–4/27/2010
1
248
2.371
0.311
--
--
4/28/2010–4/29/2016
1
1480
4.007
0.004
R.
10/9/2013
4/28/2010–10/9/2013
1
847
3.434
0.028
R.
9/19/2011
10/10/2013–4/29/2016
1
633
2.765
0.165
--
--
4/28/2010–9/19/2011
1
343
2.542
0.238
--
--
9/20/2011–10/9/2013
1
504
2.344
0.382
--
--
4/24/2009–4/29/2016
2
1728
6.739
2.67e-08
R.
4/27/2010
4/24/2009–4/27/2010
2
248
2.328
0.793
--
--
4/28/2010–4/29/2016
2
1480
4.488
0.003
R.
10/9/2013
4/28/2010–10/9/2013
2
847
3.455
0.103
--
--
10/10/2013–4/29/2016
2
633
2.739
0.516
--
--
4/24/2009–4/29/2016
1
1728
5.707
1.68e-06
R.
5/5/2010
4/24/2009–5/5/2010
1
253
1.671
0.854
--
--
5/6/2010–4/29/2016
1
1475
3.869
0.007
R.
11/7/2013
5/6/2010–11/7/2013
1
863
2.205
0.539
--
--
11/8/2013–4/29/2016
1
612
2.183
0.525
--
--
t
Clayton
Note: a R. denotes the null hypothesis is rejected at the 5% level.
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Interval Tests for Structural Breaks
Test for the structural change between pre-crisis and post-crisis
Sample Period
Financial Crisis Period
1/4/2006–4/27/2010
9/10/2008–4/23/2009
Change Interval Test
n
Hypothesis Test T2
HT11
HT12
HT13
HT2
P-value
Normal
1
45.595**
45.303**
12.828**
23.510**
1.24e-06
t
2
43.889**
39.115**
13.012**
22.984**
1.02e-05
Clayton
1
36.274**
35.918**
12.119**
17.333**
3.14e-05
Note: a ** denotes the significant levels of 1%.
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Interval Tests for Structural Breaks
Empirical Analysis
Dynamic Changes in Dependence between Oil and Gold
0.7
time-varying
constant
0.6
Dependent Relationship
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
06
07
08
09
10
11
12
13
14
15
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• Change interval
9/10/2008-4/23/2009
• Change point
 4/27/2010
 10/9/2013
• Various period
 Market boom
 Financial crisis
 Rapid recovery
 Market reform
16
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Interval Tests for Structural Breaks
Conclusion
 Theory study
•
Propose a new change interval test
 Empirical study
•
•
Commodity properties are larger in economic boom period, and financial properties are
larger in extreme risk period
Exogenous economic shocks and the changes of internal mechanism may lead to structural
changes
 Other Application
•
•
Risk management
Financial contagion
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Thank you for your attention.
Questions and comments are welcome.
BingYue Liu
Department of Statistics and Finance
University of Science and Technology of China
Email: [email protected]
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