Direct Evidence for Synchronization in
International Business Cycle
Y. Ikeda and H. Aoyama* (Kyoto University)
18072013
* Faculty Fellow, Research Institute of Economy, Trade & Industry, IAA
Outline
 Limit Cycle in Macro Economic System
 Evidence of the Synchronization:
Frequency Entrainment and Partial Phase Locking
 Common and Individual Shocks
 Coupled Limit Cycle Oscillator Model
Origin of Synchronization: International Trade
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
©2013 Y. Ikeda
2
Background
 The synchronization in international business cycle attracts
economists and physicist as an example of the selforganization in the time domain*.
* P. R. Krugman, “The Self-Organizing Economy”, Cambridge,
Mass., and Oxford: Blackwell Publishers (1996).
 Synchronization of the business cycle has been discussed
using correlation coefficients between GDP time series**.
However more definitive discussion using a suitable quantity
describing the business cycle is needed.
** J. H. Stock and M. W. Watson, “Understanding Changes in
International Business Cycle Dynamics”, Journal of the European
Economic Association, 3 (5) pp.968-1006 (2005).
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
©2013 Y. Ikeda
3
Purpose
The quarterly GDP time series is available during the last
50 years for Australia, Canada, France, UK, Italy, and US.
For these 6 countries we study international business
cycle in order to answer the following questions:
 What is the direct evidence for the synchronization?
 What is the origin of the synchronization?
 What is an appropriate model of the business cycle?
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
©2013 Y. Ikeda
4
Data
10000
8000
6000
4000
2000
0
0
Growth rate of GDP
Changes In
Inventories
US
12000
50
0.06
US
100
14000
GDP
 We analyze the quarterly GDP
time series (OECD Quarterly
National Accounts, QNA) for
Australia, Canada, France,
Italy, UK, and US from
1960/2Q to 2010/1Q to study
the synchronization in
international Business Cycle.
50
0
50
100
𝑡𝑡 (quater)
150
200
US
0.04
0.02
0.00
0.02
0
50
100
𝑡𝑡 (quater)
150
200
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
0
50
100
𝑡𝑡 (quater)
150
200
©2013 Y. Ikeda
5
GDP and Inventory
0.004
0.002
20
0.000
0.002
0.004
US
(T=2-10years)
0.006
0.008
change in inventory
0
50
100
150
200
20
10
change in inventory
growth rate of GDP
 The business cycle is due to the inventory adjustment.
The existence of a limit cycle is suggested.
10
0
10
20
0
30
10
20
0.008
US
(T=2-10years)
30
0
50
100
𝑡𝑡 (quater)
0.006
0.004
0.002
0.000
0.002
0.004
growth rate of GDP
150
200
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
©2013 Y. Ikeda
6
Stock=
fixed asset,
or inventory
expenditure
production
(a) Linear expenditure function
∆ stock<0
4
∆ invst>0 3
1
2
∆ stock>0
∆ invst<0
production
(c) expenditure as a function of stock
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
expenditure
Equilibrium
(stable)
Equilibrium
(unstable)
production
(b) S-shape expenditure function
production
Production=
investment+
consumption
expenditure
Limit Cycle (Kaldor 1940)
1
2
4
3
For the
international
case, limit cycle
oscillators will be
coupled. Let us
analyze data.
stock
(d) Limit cycle
©2013 Y. Ikeda
7
Hilbert Transformation
 A complex time series is obtained by adopting the time series
𝑦𝑦 𝑡𝑡 as a imaginary part. Consequently a phase time series
𝜃𝜃 𝑡𝑡 is obtained.
𝑧𝑧 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 + 𝑖𝑖𝑦𝑦 𝑡𝑡 = 𝐴𝐴 𝑡𝑡 𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝜃𝜃 𝑡𝑡
∞
1
𝑥𝑥 𝑠𝑠
𝑦𝑦 𝑡𝑡 = 𝐻𝐻 𝑥𝑥 𝑡𝑡 = 𝑃𝑃𝑃𝑃 �
𝑑𝑑𝑑𝑑
𝜋𝜋
−∞ 𝑡𝑡 − 𝑠𝑠
(PV: the Cauchy
principal value)
 Example:
𝑥𝑥 𝑡𝑡 = 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔𝑡𝑡
𝑦𝑦 𝑡𝑡 = 𝐻𝐻 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔𝑡𝑡
= 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔𝑡𝑡
𝑧𝑧 𝑡𝑡 = 𝑥𝑥 𝑡𝑡 + 𝑖𝑖𝑦𝑦 𝑡𝑡 = 𝑐𝑐𝑐𝑐𝑐𝑐 𝜔𝜔𝑡𝑡 + 𝑖𝑖 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔𝑡𝑡 = 𝐴𝐴 𝑡𝑡 𝑒𝑒𝑒𝑒𝑒𝑒 𝑖𝑖𝜃𝜃 𝑡𝑡
D. Gabor (1945) “Theory of Communication, Part 1”
C.W.J. Granger and M. Hatanaka (1964) “Spectral Analysis of
Economic Time Series”
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
©2013 Y. Ikeda
8
Frequency Entrainment
 The angular frequency 𝜔𝜔𝑖𝑖 and the
intercept 𝜃𝜃�𝑖𝑖 are estimated by fitting
the time-series of the phase
𝜃𝜃𝑖𝑖 𝑡𝑡 using 𝜃𝜃𝑖𝑖 𝑡𝑡 = 𝜔𝜔𝑖𝑖 𝑡𝑡 + 𝜃𝜃�𝑖𝑖 , where i
indicates the country. The estimated
angular frequencies 𝜔𝜔𝑖𝑖 for all the 6
countries are plotted.
 The estimated angular frequencies
using the Hilbert transformation are
almost identical for these 6
countries.
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
80
US
(T=2-10years)
60
𝜃𝜃𝑖𝑖 𝑡𝑡
40
20
0
0
50
100
150
200
𝑡𝑡 (quater)
<ω>=0.359rad/Q
→ T=2π/<ω>=52.4months
𝜔𝜔𝑖𝑖
 The angular frequencies are
estimated using the GDP growth
rate and the Hilbert transformation
of the GDP growth rate.
Aus
Can
Fra
UK
Ita
US
©2013 Y. Ikeda
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Partial Phase Locking
 The condition of the partial phase locking 𝜎𝜎 𝑡𝑡 ≪ 𝜔𝜔𝑖𝑖 is
satisfied.
1.0
1 𝑁𝑁 𝑑𝑑
∑
𝜃𝜃 𝑡𝑡 −𝜔𝜔𝑖𝑖 𝑡𝑡 −𝜇𝜇 𝑡𝑡
𝑁𝑁 𝑖𝑖=1 𝑑𝑑𝑑𝑑 𝑖𝑖
𝑁𝑁
1
𝑑𝑑
𝜇𝜇 𝑡𝑡 = �
𝜃𝜃𝑖𝑖 𝑡𝑡 − 𝜔𝜔𝑖𝑖 𝑡𝑡
𝑁𝑁
𝑖𝑖=1 𝑑𝑑𝑑𝑑
1⁄2
0.8
0.6
𝜎𝜎 𝑡𝑡
𝜎𝜎 𝑡𝑡 =
2
0.4
0.2
0.0
0
50
100
150
200
𝑡𝑡 (quater)
 The frequency entrainment and the phase locking are
the direct evidence of the synchronization in the
international business cycle.
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
©2013 Y. Ikeda
10
Amplitude, Phase, and Recessions
𝑖𝑖=1
 NBER Business Cycles (Trough)
(1)
(2)
(3)
(4)
1961/1Q
1970/4Q
1975/1Q
1980/3Q
(5)
(6)
(7)
(8)
1982/4Q
1991/1Q
2001/4Q
2009/2Q
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
𝐴𝐴 𝑡𝑡
0.05
0.00
0
50
100
150
200
0
50
100
150
200
0.5
0.0
0.5
𝑡𝑡 (quater)
2009
cos 𝜃𝜃 𝑡𝑡
𝑁𝑁
1
= � cos 𝜃𝜃𝑖𝑖 𝑡𝑡
𝑁𝑁
0.10
1997
𝑖𝑖=1
0.15
1985
𝑖𝑖=1
1
1
𝑥𝑥𝑖𝑖 𝑡𝑡
= � 𝐴𝐴𝑖𝑖 𝑡𝑡 = �
𝑁𝑁
𝑁𝑁
cos 𝜃𝜃𝑖𝑖 𝑡𝑡
0.20
1973
𝑁𝑁
0.25
1961
𝐴𝐴 𝑡𝑡
𝑁𝑁
0.30
cos 𝜃𝜃 𝑡𝑡
 Phase time series identifies
the 8 recessions after 1960.
phase time series is more
sensitive to the recessions
compared with the amplitude.
©2013 Y. Ikeda
11
Common and Individual Shocks
0.0
0
δ𝑖𝑖 𝑡𝑡 = cos 𝜃𝜃𝑖𝑖 𝑡𝑡 − cos 𝜃𝜃 𝑡𝑡
100
150
200
50
100
150
200
1.5
1.0
0.5
δ𝑖𝑖 𝑡𝑡
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
0.0
0.5
1.0
0
𝑡𝑡 (quater)
2009
1.5
1973
 We have many individual shocks
all the time, and many of them
seem to occur randomly. More
noteworthy are contraction of
the individual shocks at the
recessions. All countries were
exposed to the common shocks.
50
1997
𝑖𝑖=1
1985
0.5
1961
cos 𝜃𝜃 𝑡𝑡
𝑁𝑁
1
= � cos 𝜃𝜃𝑖𝑖 𝑡𝑡
𝑁𝑁
0.5
cos 𝜃𝜃 𝑡𝑡
 Economic shocks fall into the
general classification of common
shocks cos 𝜃𝜃 𝑡𝑡 and individual
shocks δ𝑖𝑖 𝑡𝑡 .
©2013 Y. Ikeda
12
GDP and International Trade
 Cycle of the inventory adjustment may differ from one country to
another. Why dose the business cycle synchronize?
 Trade data shows that the import (export) to GDP ratio is high except
for US. Trade may affects the cycle and phase of the business cycle.
0.35
0.35
Australia
0.30
France
0.30
0.25
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
0.00
0.00
0
0.35
50
100
150
0.35
UK
0.30
0
200
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05
0.00
0.00
0
50
100
150
200
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
100
US
0.30
0.25
50
0
150
200
Import/GDP
Export/GDP
50
100
150
200
©2013 Y. Ikeda
13
Coupled Limit-cycle Oscillator Model
Import
from
other
country
𝑘𝑘 sin Δ𝜃𝜃
𝜃𝜃𝑖𝑖
𝐿𝐿 Labor
Money
flow
𝑘𝑘 ′ sin Δ𝜃𝜃 ′
𝑅𝑅
Domestic sales
revenue
Export to
other
countries
If the power is balanced, the oscillator
rotates with a constant speed 𝜃𝜃̇𝑖𝑖 .
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
 “Power” balance equation
(change in “kinetic
energy”=summed “power”)
𝑑𝑑 1
2
I𝑖𝑖 𝜃𝜃̇𝑖𝑖
𝑑𝑑𝑑𝑑 2
2
= 𝑅𝑅𝑖𝑖 − 𝐿𝐿𝑖𝑖 − 𝐾𝐾𝑑𝑑 𝜃𝜃̇𝑖𝑖
+ � 𝑘𝑘𝑗𝑗𝑗𝑗 sin 𝛥𝛥𝜃𝜃𝑗𝑗𝑗𝑗
𝑗𝑗
 the Kuramoto oscillator
(𝜃𝜃̈𝑖𝑖 ≪ 𝛼𝛼𝑖𝑖 𝜃𝜃̇𝑖𝑖 )
𝐾𝐾𝑑𝑑 𝜃𝜃̇𝑖𝑖 = 𝑅𝑅𝑖𝑖 − 𝐿𝐿𝑖𝑖 + � 𝑘𝑘𝑗𝑗𝑗𝑗 sin 𝛥𝛥𝜃𝜃𝑗𝑗𝑗𝑗
𝑗𝑗
⇒𝜃𝜃̇𝑖𝑖 = 𝑄𝑄𝑖𝑖 + ∑𝑗𝑗 𝜅𝜅𝑗𝑗𝑗𝑗 sin 𝛥𝛥𝜃𝜃𝑗𝑗𝑗𝑗
©2013 Y. Ikeda
14
Parameter Estimation
 The coupled limit-cycle oscillator fits the phase time series of
the GDP growth rate very well. This means that the origin of
the synchronization is interaction due to the international trade.
Aus
𝜃𝜃𝑖𝑖,𝑡𝑡+1 = 𝜆𝜆𝑖𝑖 𝜃𝜃𝑖𝑖,𝑡𝑡 + 𝑄𝑄𝑖𝑖 + � 𝜅𝜅𝑗𝑗𝑗𝑗 sin 𝛥𝛥𝜃𝜃𝑗𝑗𝑗𝑗
𝜆𝜆
𝑄𝑄
R2
𝜆𝜆
𝑄𝑄
R2
Aus:1
0.9968
0.4128
0.9995
UK:4
0.9969
0.4275
0.9993
𝑗𝑗
Can:2
0.9969
0.4042
0.9997
Fra:3
0.9954
0.3973
0.9993 UK
Ita:5 US:6
0.9996 0.9982
0.3309 0.4506
0.9999 0.9999
Can
𝜅𝜅24
𝜅𝜅62
𝜅𝜅46
𝜅𝜅12
𝜅𝜅25
𝜅𝜅26
US
𝜅𝜅61
𝜅𝜅65
𝜅𝜅51
Ita
𝜅𝜅63
𝜅𝜅36
𝜅𝜅31
𝜅𝜅53
𝜅𝜅35
Fra
The edge between 𝑖𝑖 and 𝑗𝑗 is shown, if the
confidence interval of 𝜅𝜅𝑖𝑖𝑖𝑖 dose not cross zero.
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
©2013 Y. Ikeda
15
Mechanism of Synchronization
 The synchronization is emerged as a consequence of the interactions.
κ = 0.
20
20
0
0
10
10
20
20
30
30
0
200
400
600
800
1000
5
𝑑𝑑
𝜃𝜃 = 𝑄𝑄𝑖𝑖 + � 𝜅𝜅 sin 𝛥𝛥𝜃𝜃𝑗𝑗𝑗𝑗
𝑑𝑑𝑑𝑑 𝑖𝑖
10
κ = 0.002
10
𝜃𝜃𝑖𝑖 − 𝜃𝜃𝑖𝑖
10
𝑄𝑄𝑖𝑖 = 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈[0.35,0.45]
𝜋𝜋 𝜋𝜋
𝜃𝜃𝑖𝑖 0 = 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈[− , ]
2 2
0
10
30
0
200
400
600
800
1000
𝜃𝜃𝑖𝑖 − 𝜃𝜃𝑖𝑖
200
400
600
800
1000
κ = 0.008
0.5
2
5
0.0
4
10
0.5
6
15
0
0
1.0
0
20
κ = 0.004
20
2
0
𝑗𝑗
κ = 0.006
200
400
8
600
800
1000
0
κ = 0.007
200
400
1.0
600
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
800
1000
0
200
400
600
800
©2013 Y. Ikeda
1000
16
Summary
We analyzed the quarterly GDP time series for Australia,
Canada, France, Italy, UK, and US from 1960/2Q to
2010/1Q to study the synchronization in international
Business Cycle. The followings are obtained:
 The frequency entrainment and the phase locking are
observed as the direct evidence of the synchronization
in the international business cycle.
 The business cycle due to stock adjustment is described
using a limit cycle. A coupled limit cycle oscillator model
explains the mechanism of synchronization.
 The origin of the synchronization is interaction due to
the international trade.
Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University
©2013 Y. Ikeda
17