dynamical analysis of socio-economic oscillations peter turchin university of connecticut to be...
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Dynamical Analysis ofSocio-Economic Oscillations
Peter TurchinUniversity of Connecticut
To be presented at the Santa Fe Workshop
April-May 2004
Analytical approaches
• Graphical analysis: time and phase plots
• Fitting models: ΔYt = f(Xt) + et
– Xt : predictor variable(s)
– ΔYt = Yt+τ – Yt : rate of change (response)
– τ : time lag– more on this in the Turchin-Korotayev
supplement, also see Historical Dynamics
Turchin P. 2003. Nature 424:257
Phase shifts betweenoscillating variables tell us whether their interaction canpotentially drive the observed cycles (hereillustrated with a predator-prey system)
England: population and "carrying capacity"
Year
1100 1200 1300 1400 1500 1600 1700 1800
Pop
ulat
ion,
mln
(lo
g-sc
ale)
1
10
100
Population, mlnEst. carrying capacityWheat yield in bus/ac
England: detrended population
Year
1100 1200 1300 1400 1500 1600 1700 1800
Pop
ula
tion,
per
cent
of
K
0
20
40
60
80
100
Time-series analysis results
• Periodicity is statistically significant– average period of 3.2 centuries– “secular cycle”
• Second-order system– with a strong endogenous (deterministic)
component
• Q: what is the identity of the second-order factor(s) that drive the cycle?
England: 1250-1800
Year
1300 1400 1500 1600 1700 1800
Re
al w
age
4
5
6
7
8
9
10
Re
lativ
e P
op
ula
tion
30
40
50
60
70
80
90
wagepopulation
England: 1250-1800
Relative population
40 50 60 70 80
Re
al w
age
s
4
5
6
7
8
9
10
England: 1250 - 1800
Year
1200 1300 1400 1500 1600 1700 1800
Var
iabl
es, l
og-s
cale
, arb
itra
ry c
ons
t.Inv. wageRel. Pop
England: 1350 - 1700
1300 1400 1500 1600 1700
Po
pu
latio
n (
de
tre
nd
ed
)
30
40
50
60
70
Pla
gue
inci
de
nce
(lo
g sc
ale
)
10
100
PopulatiojnPlague
Population (detrended)
30 40 50 60 70 80
Pla
gue
inci
denc
e (lo
g-sc
ale)
10
100
Real wages and epidemics: conclusions
• Both variables fluctuate synchronously with population
• Act as first-order factors
• Cannot drive the secular cycle
England: 1450 - 1800
Year
1450 1500 1550 1600 1650 1700 1750 1800
Inst
abi
lity
ind
ex
0
1
2
3
Instability (detrended)
1.55 1.60 1.65 1.70 1.75 1.80
Pop
ula
tion
(de
tren
ded
)
-0.2
-0.1
0.0
0.1
0.2
0.3
Instability Index (log-transformed)
-0.2 -0.1 0.0 0.1 0.2
Co
mpo
un
d a
nnu
al g
row
th r
ate
0.0
0.2
0.4
0.6
0.8
Population and sociopolitical instability
• Instability as a second-order factor– correct phase shift
• Effect very strong– explains 80% of variance in compound annual
growth rate (Schofield et al data)
• Analysis results are consistent with the hypothesis that interaction between population and instability drives the secular cycle
China (200 BCE - 300 CE): population and instability
Years
-200 -100 0 100 200 300
log
Pop
ula
tion
1.2
1.4
1.6
1.8
log
Inte
rna
l Wa
r
0.0
0.5
1.0
PopulationInternal War
China (200 BCE - 300 CE): population and instability
log Population
log
War
fare
Table 1. Comparing out-of-sample predictive abilities of the inertial and interactive models (from Turchin-Korotayev Supplement)
Source of data
Dependentvariable
Correlation between predicted and observed
1st half => 2nd half 2nd half => 1st half
inertial interactive inertial interactive
England population –0.57 0.94 –0.07 0.44
England instability –0.13 0.80 –0.53 0.89
Han China population 0.45 0.57 0.73 0.48
Han China instability 0.39 0.87 0.37 0.68
Tang China population 0.56 0.80 0.61 0.90
Tang China instability 0.57 0.78 0.66 0.92
Some other analyses
• Vital rates (fertility, mortality)
• Crime statistics
• Climate change
England
1550 1600 1650 1700 1750 1800 1850
CB
R
20
25
30
35
40
45
CD
R
20
25
30
35
40
45
CBRCDRCBR smoothedCDR smoothed
CBR smoothed
28 30 32 34 36 38
CD
R s
moo
the
d
20
22
24
26
28
30
1540
1870
England: 1540-1800
Relative population density at t
45 50 55 60 65 70
Cru
de b
irth
rate
at
t+30
28
30
32
34
36
38
Relative population density at t
45 50 55 60 65 70
Cru
de d
eat
h r
ate
at
t+70
24
25
26
27
28
29
30
No lag
Real Wage at t
5 6 7 8 9
CB
R
28
30
32
34
36
Lag = 50 y
Real Wage at t-50
5 6 7 8 9
CB
R
28
30
32
34
36
England: 1200-1800
Time
1200 1300 1400 1500 1600 1700 1800
log
Pop
ulat
ion,
Inv
erse
Wag
e
1.2
1.3
1.4
1.5
1.6
1.7
1.8
log
Hom
icid
e R
ate
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
popinv. wagecrime
Year
1600 1650 1700 1750 1800
Neo
natic
ide
indi
ctm
ent
ra
te
10
15
20
25
30
35
40
Inst
abili
ty in
dex
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
NeonaticideInstability
Instability
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
Ne
onat
icid
e ra
te
5
10
15
20
25
30
35
40
45
England: population and climate
Time
1200 1300 1400 1500 1600 1700 1800
Po
pu
latio
n (
% o
f K
)
30
40
50
60
70
80
90
Clim
ate
-0.3
-0.2
-0.1
0.0
0.1
Pop (detr)Climate
Regression Analysis: r2 versus BD, logW, y, t, N, WAGEr2 = population rate of change, tau = 20 yBD = dummy variable for the Black DeathlogW = instability, log-transformedy = year (monotonic temporal trend)t = temperatureN = population pressure (in relation to K)WAGE = real wage
Predictor of r2 Coef SE Coef T PConstant -0.15804 0.03053 -5.18 0.000BD -0.11646 0.01118 -10.42 0.000logW -0.030560 0.003385 -9.03 0.000y 0.00008885 0.00002282 3.89 0.000t -0.17738 0.05676 -3.13 0.003N -0.0005824 0.0002043 -2.85 0.006WAGE 0.003293 0.001572 2.10 0.041
R-Sq = 90.0% R-Sq(adj) = 89.0% R-Sq(pred) = 85.55%
General conclusions: regression analysis of population rate of change
• Strong effect of the Black Death– not surprising!
• Strong effect of instability• Moderate temporal trend • Moderate effect of temperature
– but the sign is negative! (expect positive)
• Moderate effect of population pressure• Weak effect of wage
– but without pop. pressure in the model, effect of wage strengthens, t = 2.5, P < 0.016