-
Bond UniversityResearch Repository
Does systematic sampling preserve granger causality with an
application to high frequencyfinancial data?
Rajaguru, Gulasekaran; Abeysinghe, Tilak; O'Neill, Michael
Published: 01/01/2017
Document Version:Other version
Licence:CC BY-NC-ND
Link to publication in Bond University research repository.
Recommended citation(APA):Rajaguru, G., Abeysinghe, T., &
O'Neill, M. (2017). Does systematic sampling preserve granger
causality with anapplication to high frequency financial data?.
Australian Conference of Economists: Economics for Better
Lives,Sydney, Australia.
http://ace2017.org.au/wp-content/uploads/2017/07/Gulasekaran-Rajaguru-Full-Paper.pdf
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Download date: 19 Jun 2021
https://research.bond.edu.au/en/publications/16f99087-c3c8-4057-b62d-156a935280b0http://ace2017.org.au/wp-content/uploads/2017/07/Gulasekaran-Rajaguru-Full-Paper.pdf
-
2017 Australian Conference of Economists, Sydney, 19-21 July
2017
Does Systematic Sampling Preserve Granger Causality with an
Application
to High Frequency Financial Data?
Tilak Abeysinghe, Michael O'Neill andGulasekaran Rajaguru
-
Motivation
There was a lot of high powered analysis of this topic, but I
came away from a reading of it with the feeling that it was one of
the most unfortunate turnings for econometricsin the last two
decades, and it has probably generated more nonsense results than
anything else during that time.
-Pagan (1989)
-
Motivation Financial Development (FD) and Economic
Growth (EG) Controversial causal results
FD EG: McKinnon (1973), King and Levine (1993a,b), Neusser and
Kugler (1998), and Levine et al. (2000), Christopoulos and Tsionas
(2004), Dimitris and Efthymios (2004), Shan (2006), and Zhang, Wang
and Wang (2012)
EG FD: Gurley and Shaw (1967), Goldsmith (1969), Jung (1986) and
Liang and Teng (2006)
Bidirectional: Shan, Morris, and Sun (2001) and Demetriades and
Hussein (1996), Calderón and Liu (2003), Hassan, Fung (2009),
Sanchez and Yu – (2011)and Kar et al. (2011)
-
Literature Review
Univariate ARIMA models: Wei(1990)
Dynamic relationships: Temporal Aggregation weakens the
distributed lag relationships, Telser(1967), Zellner and
Montmarquette(1971), Sims(1971), Wei and Tiao (1975), Tiao and
Wei(1976,1978), Wei and Metha (1980)
Forecasting: Lutkepohl (1987), Marcellino (1999)
Misspecification involved in cross- country regression: Ericsson
et al,(2000)
Causality: Wei (1982), Mamingi (1996), Marcellino (1999),
Rajaguru (2004), Rajaguru and Abeysinghe ( 2008), Abeysinghe and
Rajaguru (2012) and Ashley and Tsang(2014)
Temporal aggregation turns one-way causality into feedback
system
Systematic sampling preserves the causal directions
Temporal Aggregation / Systematic Sampling
Unit Roots: Pierce and Snell(1995) Co-integration:
Phillips(1991)
Literature ReviewTemporal Aggregation /
Systematic Sampling
Forecasting:
Lutkepohl (1987),
Marcellino (1999)
Causality:
Wei (1982),
Mamingi (1996),
Marcellino (1999),
Rajaguru (2004),
Rajaguru and Abeysinghe ( 2008),
Abeysinghe and Rajaguru (2012)
and Ashley and Tsang(2014)
Misspecification involved in cross- country regression: Ericsson
et al,(2000)
Dynamic relationships:
Temporal Aggregation weakens the distributed lag
relationships,
Telser(1967), Zellner and Montmarquette(1971), Sims(1971), Wei
and Tiao (1975), Tiao and Wei(1976,1978), Wei and Metha (1980)
Univariate ARIMA models: Wei(1990)
Unit Roots:
Pierce and Snell(1995)
Co-integration:
Phillips(1991)
Systematic sampling preserves the causal directions
Temporal aggregation turns one-way causality into feedback
system
-
Systematic Sampling• Let ),...,,( 21 ntttt zzzz = , t=1,2,…,T be
an
equally spaced n-variate basic disaggregated
series.
• Systematic sampling : ττ mzZ = (τ =1,2,…,N
and T=mN) - sampling from zt at every mth
interval (m is a positive integer).
·
Let , t=1,2,…,T be an equally spaced n-variate basic
disaggregated series.
·
Systematic sampling : (=1,2,…,N and T=mN) - sampling from zt at
every mth interval (m is a positive integer).
)
,...,
,
(
2
1
nt
t
t
t
z
z
z
z
=
t
t
m
z
Z
=
t
-
Systematic Sampling• Let wt = ),...,,( 21 nttt www , jt
djt zLw j)1( −= , be a weakly
stationary process.
• where γwii(k) is the autocovariance of the i-th component, wit
at
lag k.
• γwij(k) is the cross covariance between i-th and j-th
components
• γwii(0) is the variance of the i-th series
·
Let wt =, , be a weakly stationary process.
· where wii(k) is the autocovariance of the i-th component, wit
at lag k.
· wij(k) is the cross covariance between i-th and j-th
components
· wii(0) is the variance of the i-th series
)
,...,
,
(
2
1
nt
t
t
w
w
w
jt
d
jt
z
L
w
j
)
1
(
-
=
-
Relationship between disaggregated and Systematic
Sampled Series
ττττ jmdm
mjdm
jd wLLzLZL jjj ).....1( )1()'1( W 1j
−+++=−=−= . • The dj-th difference of the systematically
sampled
series (j-th component) is simply the weighted sum of the dj-th
difference of the basic series.
.
· The dj-th difference of the systematically sampled series
(j-th component) is simply the weighted sum of the dj-th difference
of the basic series.
t
t
t
t
jm
d
m
m
j
d
m
j
d
w
L
L
z
L
Z
L
j
j
j
)
.....
1
(
)
1
(
)
'
1
(
W
1
j
-
+
+
+
=
-
=
-
=
-
Relationship between cross-covariances of disaggregated and
Systematic Sampled SeriesProposition 1
The cross covariance between i-th and j-th components of the
systematically sampled series Wiτ and Wjτ-k can be expressed in
terms of cross
covariances of the i-th and j-th components of the basic
disaggregated series itw
and jtw , that is,
))1(()......1( ),( )( 12 −+++++== +−− mdmkLLLWWCovk jwij
ddmkji
Wij
ji γγ ττ (1)
))1(()......1( ),( )( 12 −+++++== +−− mdmkLLLWWCovk iwji
ddmkij
Wji
ji γγ ττ (2)
Proposition 1
The cross covariance between i-th and j-th components of the
systematically sampled series Wi and Wj-k can be expressed in terms
of cross covariances of the i-th and j-th components of the basic
disaggregated series and , that is,
(1)
(2)
))
1
(
(
)
......
1
(
)
,
(
)
(
1
2
-
+
+
+
+
+
=
=
+
-
-
m
d
mk
L
L
L
W
W
Cov
k
j
w
ij
d
d
m
k
j
i
W
ij
j
i
g
g
t
t
))
1
(
(
)
......
1
(
)
,
(
)
(
1
2
-
+
+
+
+
+
=
=
+
-
-
m
d
mk
L
L
L
W
W
Cov
k
i
w
ji
d
d
m
k
i
j
W
ji
j
i
g
g
t
t
it
w
jt
w
-
Systematic Sampling and Granger Causality
consider the following bivariate VAR(1) system with )(~ 11 dIz
t
and )(~ 22 dIz t such that itdit zLw i)1( −= for i=1,2:
+
=
−
−
t
t
t
t
t
t
ee
ww
ww
2
1
12
11
2221
1211
2
1
ϕϕϕϕ
,
22
21
2
1
00
,00
~σ
σN
ee
t
t ,
• 012 ≠ϕ implying Granger causality from 2w to 1w
• 021 ≠ϕ implying Granger causality from 1w to 2w
consider the following bivariate VAR(1) system with and such
that for i=1,2:
, ,
·
implying Granger causality from to
·
implying Granger causality from to
it
d
it
z
L
w
i
)
1
(
-
=
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
=
÷
÷
ø
ö
ç
ç
è
æ
-
-
t
t
t
t
t
t
e
e
w
w
w
w
2
1
1
2
1
1
22
21
12
11
2
1
j
j
j
j
÷
÷
ø
ö
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
2
2
2
1
2
1
0
0
,
0
0
~
s
s
N
e
e
t
t
0
12
¹
j
2
w
1
w
0
21
¹
j
2
w
)
(
~
1
1
d
I
z
t
)
(
~
2
2
d
I
z
t
-
Systematic Sampling and Granger Causality
consider the following bivariate VAR(1) system based on
systematically sampled series:
+
=
−
−
τ
τ
τ
τ
τ
τ
ϕϕϕϕ
2
1
12
11*22
*21
*12
*11
2
1
EE
WW
WW
,
( ) ( )* *11 22 12 12 12 11 11 12
11 122 2
11 22 12 11 22 12
(1) (0) (1) (0) (1) (0) (1) (0)ˆ ˆlim , lim(0) (0) (0) (0) (0)
(0)
W W W W W W W W
W W W W W Wp pγ γ γ γ γ γ γ γφ φ
γ γ γ γ γ γ
− −= =
− − ,
( )212221112222221*
21)0()0()0(
)0()1()0()1(ˆlimWWW
WWWW
pγγγ
γγγγϕ−
−= , ( )2122211
12211122*22
)0()0()0(
)0()1()0()1(ˆlimWWW
WWWW
pγγγ
γγγγϕ
−
−=
consider the following bivariate VAR(1) system based on
systematically sampled series:
,
,
,
(
)
2
12
22
11
12
22
22
21
*
21
)
0
(
)
0
(
)
0
(
)
0
(
)
1
(
)
0
(
)
1
(
ˆ
lim
W
W
W
W
W
W
W
p
g
g
g
g
g
g
g
j
-
-
=
(
)
2
12
22
11
12
21
11
22
*
22
)
0
(
)
0
(
)
0
(
)
0
(
)
1
(
)
0
(
)
1
(
ˆ
lim
W
W
W
W
W
W
W
p
g
g
g
g
g
g
g
j
-
-
=
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
=
÷
÷
ø
ö
ç
ç
è
æ
-
-
t
t
t
t
t
t
j
j
j
j
2
1
1
2
1
1
*
22
*
21
*
12
*
11
2
1
E
E
W
W
W
W
(
)
(
)
**
1122121212111112
1112
22
112212112212
(1)(0)(1)(0)(1)(0)(1)(0)
ˆˆ
lim,lim
(0)(0)(0)(0)(0)(0)
WWWWWWWW
WWWWWW
pp
gggggggg
ff
gggggg
--
==
--
-
Systematic Sampling and Granger Causality
Parameters of the Systematically Sampled Series
Cross covariance of the systematically sample series
Cross covariance of the basic series
Parameters of the basic series
Parameters of the Systematically Sampled Series
Cross covariance of the systematically sample series
Cross covariance of the basic series
Parameters of the basic series
-
Case 1: No Granger causality between the variables in the
disaggregated form
+
=
−
−
t
t
t
t
t
t
ee
ww
ww
2
1
12
11
2221
1211
2
1
ϕϕϕϕ
Here 02112 == ϕϕ and with 012 =σ Proposition 2
If there does not exist Granger causality between the basic
series
then the Granger causality between the systematically
sampled
series is also absent.
Here and with
Proposition 2
If there does not exist Granger causality between the basic
series then the Granger causality between the systematically
sampled series is also absent.
0
12
=
s
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
=
÷
÷
ø
ö
ç
ç
è
æ
-
-
t
t
t
t
t
t
e
e
w
w
w
w
2
1
1
2
1
1
22
21
12
11
2
1
j
j
j
j
0
21
12
=
=
j
j
-
Case 2: Causality between the disaggregated series is
one-sided
+
=
−
−
t
t
t
t
t
t
ee
ww
ww
2
1
12
11
2221
1211
2
1
ϕϕϕϕ
Here 12 210 and 0φ φ= ≠ and with 012 =σ Theorem 1
Systematic sampling induces spurious bi-directional Granger
causality among
the variables if the uni-directional causality runs from a
non-stationary series to
either a stationary or a non-stationary series.
Equivalently, systematic sampling induces spurious
bi-directional Granger
causality among the variables if 01 >d .
Here and with
Theorem 1
Systematic sampling induces spurious bi-directional Granger
causality among the variables if the uni-directional causality runs
from a non-stationary series to either a stationary or a
non-stationary series.
Equivalently, systematic sampling induces spurious
bi-directional Granger causality among the variables if .
0
12
=
s
0
1
>
d
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
=
÷
÷
ø
ö
ç
ç
è
æ
-
-
t
t
t
t
t
t
e
e
w
w
w
w
2
1
1
2
1
1
22
21
12
11
2
1
j
j
j
j
1221
0 and 0
ff
=¹
-
Case 3: Causality between the disaggregated series is
bi-directional
+
=
−
−
t
t
t
t
t
t
ee
ww
ww
2
1
12
11
2221
1211
2
1
ϕϕϕϕ
Here 12 210 and 0φ φ≠ ≠ and with 012 =σ • Bi-directional causal
system becomes uni-directional at the lower level of
aggregation (systematic sampling) and subsequently becomes
no-causality
among the variables of interest
• All causal informations concentrate on contemporaneous
relationships at
the higher level of aggregation
Here and with
· Bi-directional causal system becomes uni-directional at the
lower level of aggregation (systematic sampling) and subsequently
becomes no-causality among the variables of interest
· All causal informations concentrate on contemporaneous
relationships at the higher level of aggregation
0
12
=
s
÷
÷
ø
ö
ç
ç
è
æ
+
÷
÷
ø
ö
ç
ç
è
æ
÷
÷
ø
ö
ç
ç
è
æ
=
÷
÷
ø
ö
ç
ç
è
æ
-
-
t
t
t
t
t
t
e
e
w
w
w
w
2
1
1
2
1
1
22
21
12
11
2
1
j
j
j
j
1221
0 and 0
ff
¹¹
-
Case 3: Monte Carlo Simulation*12ˆlimϕp from a feedback system
when 02211 == ϕϕ , m=3 and 021 == dd
-1-0.5
00.5
1
-1-0.5
0
0.51
-20
-10
0
10
20
12ϕ 21ϕ
)ˆ( *12ϕt
-1-0.5
00.5
1
-1-0.5
0
0.51
-1.5
-1
-0.5
0
0.5
1
1.5
12ϕ
*12ˆlimϕp
21ϕ
m=3 m=3
from a feedback system when , m=3 and
0
2
1
=
=
d
d
*
12
ˆ
lim
j
p
0
22
11
=
=
j
j
21
j
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-20
-10
0
10
20
)
ˆ
(
*
12
j
t
12
j
12
j
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1.5
-1
-0.5
0
0.5
1
1.5
*
12
ˆ
lim
j
p
21
j
-
Case 3: Monte Carlo Simulation*
12ˆ( )t φ from a feedback system when 02211 == ϕϕ , m=12 and 60
and 021 == dd
-1-0.5
00.5
1
-1-0.5
0
0.51
-5
0
5
12ϕ 21ϕ
)ˆ( *12ϕt
-1-0.5
00.5
1
-1-0.5
0
0.51
-4
-2
0
2
4
12ϕ 21ϕ
)ˆ( *12ϕt
m=12 m=60
from a feedback system when , m=12 and 60 and
0
2
1
=
=
d
d
*
12
ˆ
()
t
f
0
22
11
=
=
j
j
12
j
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-5
0
5
)
ˆ
(
*
12
j
t
21
j
12
j
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-4
-2
0
2
4
)
ˆ
(
*
12
j
t
21
j
-
Case 3: Monte Carlo Simulationˆ( )t c from a feedback system
when 02211 == ϕϕ , m=12 and 60 and 1 2 0d d= =
m=12 m=60
-1-0.5
00.5
1
-1-0.5
0
0.51
-30
-20
-10
0
10
20
30
12ϕ 21ϕ
)ˆ(ct
-1-0.5
00.5
1
-1-0.5
0
0.51
-15
-10
-5
0
5
10
15
21ϕ 12ϕ
)ˆ(ct
Contemporaneous Regression
1 2t tw cw v= +
from a feedback system when , m=12 and 60 and
12
0
dd
==
ˆ
()
tc
0
22
11
=
=
j
j
12
j
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-30
-20
-10
0
10
20
30
)
ˆ
(
c
t
21
j
21
j
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-15
-10
-5
0
5
10
15
)
ˆ
(
c
t
12
j
Contemporaneous Regression
12
tt
wcwv
=+
-
Applications
Bi-directional Granger causality has been highlighted in studies
using high frequency data (Frijns et al, 2015; Bollen, O’Neill and
Whaley, 2016).
Data sourced from Jan 2010 to Dec 2014 from Thompson Reuters
SIRCA portal and Bloomberg:-US equities: S&P 500 index (SPX).
-Equity index futures: E-Mini futures index (SC1/ES1). -“Investor
fear gauge”: CBOE Volatility Index (VIX).-Futures on VIX: S&P
VIX futures short term index
(SPVXSTR/VST)
-
Application 1: SPX vs VIX I(0)/I(0)
1-minute 5-minutes 10-minutes
Both SPX VIX None Both SPX VIX None Both SPX VIX None
15 S
ec Both 63 38 20 3 42 33 16 33 26 12 4 82
SPX 81 590 14 96 40 388 8 345 22 107 0 652 VIX 11 12 59 15 5 5
34 53 0 1 9 87 None 7 19 21 200 2 11 10 224 0 3 4 240
Note: Rejection frequencies of Granger non-causality at the 5%
level of significance
+
=
−
−
t
t
t
t
t
t
ee
VIXSPX
VIXSPX
2
1
1
1
2221
1211
ϕϕϕϕ
1-minute
5-minutes
10-minutes
Both
SPX
VIX
None
Both
SPX
VIX
None
Both
SPX
VIX
None
15 Sec
Both
63
38
20
3
42
33
16
33
26
12
4
82
SPX
81
590
14
96
40
388
8
345
22
107
0
652
VIX
11
12
59
15
5
5
34
53
0
1
9
87
None
7
19
21
200
2
11
10
224
0
3
4
240
Note: Rejection frequencies of Granger non-causality at the 5%
level of significance
-
Application 2:VIX vs SPVXSTR I(0)/I(1)
1-minute 5-minutes 10-minutes
Both VIX VST None Both VIX VST None Both VIX VST None
15 S
ec Both 566 118 322 23 32 176 335 486 14 226 143 646
VIX 2 57 1 2 0 49 0 13 0 35 0 27 VST 9 21 7 2 4 23 4 8 1 17 2 19
None 1 3 2 16 0 2 0 20 0 1 0 21
Note: Rejection frequencies of Granger non-causality at the 5%
level of significance
+
∆
=
∆ −
−
t
t
t
t
t
t
ee
VSTVIX
VSTVIX
2
1
1
1
2221
1211
ϕϕϕϕ
1-minute
5-minutes
10-minutes
Both
VIX
VST
None
Both
VIX
VST
None
Both
VIX
VST
None
15 Sec
Both
566
118
322
23
32
176
335
486
14
226
143
646
VIX
2
57
1
2
0
49
0
13
0
35
0
27
VST
9
21
7
2
4
23
4
8
1
17
2
19
None
1
3
2
16
0
2
0
20
0
1
0
21
Note: Rejection frequencies of Granger non-causality at the 5%
level of significance
-
Application 3: ES1 vs SPVXSTR I(1)/I(1)
5-minutes 10-minutes
Both ES1 VST NONE Both ES1 VST NONE
1-m
inut
e Both 219 383 237 147 63 176 198 549 ES1 8 54 6 26 5 21 7 61
VST 7 6 19 14 1 2 11 32 NONE 1 0 5 20 0 0 1 25
Note: Rejection frequencies of Granger non-causality at the 5%
level of significance
+
∆∆
=
∆∆
−
−
t
t
t
t
t
t
ee
VSTES
VSTES
2
1
1
1
2221
1211 11ϕϕϕϕ
5-minutes
10-minutes
Both
ES1
VST
NONE
Both
ES1
VST
NONE
1-minute
Both
219
383
237
147
63
176
198
549
ES1
8
54
6
26
5
21
7
61
VST
7
6
19
14
1
2
11
32
NONE
1
0
5
20
0
0
1
25
Note: Rejection frequencies of Granger non-causality at the 5%
level of significance
-
Conclusion If there does not exist Granger causality between
the
basic series then the Granger causality between the
systematically sampled series is also absent.
Systematic sampling induces spurious bi-directional Granger
causality among the variables if the uni-directional causality runs
from a non-stationary series to either a stationary or a
non-stationary series.
As m increases VAR(1) becomes VAR(0). All causal inferences
concentrate on contemporaneous
relationship among the variables due to systematic sampling of
integrated process. However, interestingly, the spurious
contemporaneous relationships do not disappear even if the sampling
interval is larger.
-
Thank you!
��2017 Australian Conference of Economists, Sydney, 19-21 July
2017��Does Systematic Sampling Preserve Granger Causality with an
Application to High Frequency Financial
Data?MotivationMotivationSlide Number 4Systematic
SamplingSystematic SamplingRelationship between disaggregated and
Systematic Sampled SeriesRelationship between cross-covariances of
disaggregated and Systematic Sampled SeriesSystematic Sampling and
Granger CausalitySystematic Sampling and Granger
CausalitySystematic Sampling and Granger CausalityCase 1: No
Granger causality between the variables in the disaggregated
form�Case 2: Causality between the disaggregated series is
one-sidedCase 3: Causality between the disaggregated series is
bi-directionalCase 3: Monte Carlo Simulation�Case 3: Monte Carlo
Simulation�Case 3: Monte Carlo Simulation�ApplicationsApplication
1: SPX vs VIX I(0)/I(0)Application 2:VIX vs SPVXSTR I(0)/I(1)
Application 3: ES1 vs SPVXSTR I(1)/I(1) ConclusionSlide Number
23
