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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 (quater) 150 200 US 0.04 0.02 0.00 0.02 0 50 100 (quater) 150 200 Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University 0 50 100 (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 (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 as a imaginary part. Consequently a phase time series is obtained. = + = ∞ 1 = = � −∞ − (PV: the Cauchy principal value) Example: = = = = + = + = 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 and the intercept � are estimated by fitting the time-series of the phase using = + � , where i indicates the country. The estimated angular frequencies 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 40 20 0 0 50 100 150 200 (quater) <ω>=0.359rad/Q → T=2π/<ω>=52.4months 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 9 Partial Phase Locking The condition of the partial phase locking ≪ is satisfied. 1.0 1 ∑ − − =1 1 = � − =1 1⁄2 0.8 0.6 = 2 0.4 0.2 0.0 0 50 100 150 200 (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 =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 0.05 0.00 0 50 100 150 200 0 50 100 150 200 0.5 0.0 0.5 (quater) 2009 cos 1 = � cos 0.10 1997 =1 0.15 1985 =1 1 1 = � = � cos 0.20 1973 0.25 1961 0.30 cos 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 δ = cos − cos 100 150 200 50 100 150 200 1.5 1.0 0.5 δ Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University 0.0 0.5 1.0 0 (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 =1 1985 0.5 1961 cos 1 = � cos 0.5 cos Economic shocks fall into the general classification of common shocks cos and individual shocks δ . ©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 sin Δ Labor Money flow ′ sin Δ ′ Domestic sales revenue Export to other countries If the power is balanced, the oscillator rotates with a constant speed ̇ . Graduate School of Advanced Integrated Studies in Human Survivability, Kyoto University “Power” balance equation (change in “kinetic energy”=summed “power”) 1 2 I ̇ 2 2 = − − ̇ + � sin the Kuramoto oscillator (̈ ≪ ̇ ) ̇ = − + � sin ⇒̇ = + ∑ sin ©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 ,+1 = , + + � sin R2 R2 Aus:1 0.9968 0.4128 0.9995 UK:4 0.9969 0.4275 0.9993 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 24 62 46 12 25 26 US 61 65 51 Ita 63 36 31 53 35 Fra The edge between and is shown, if the confidence interval of 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 = + � sin 10 κ = 0.002 10 − 10 = [0.35,0.45] 0 = [− , ] 2 2 0 10 30 0 200 400 600 800 1000 − 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 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