Dynamical Analysis of
Socio-Economic Oscillations
Peter Turchin
University 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
Phase shifts between
oscillating variables
tell us whether their
interaction can
potentially drive the
observed cycles (here
illustrated with a
predator-prey system)
Turchin P. 2003.
Nature 424:257
England: population and "carrying capacity"
100
Population, mln (log-scale)
Population, mln
Est. carrying capacity
Wheat yield in bus/ac
10
1
1100
1200
1300
1400
1500
Year
1600
1700
1800
England: detrended population
100
Population, percent of K
80
60
40
20
0
1100
1200
1300
1400
1500
Year
1600
1700
1800
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
10
90
9
80
8
70
7
60
6
50
5
40
4
30
1800
1300
1400
1500
Year
1600
1700
Relative Population
Real wage
wage
population
England: 1250-1800
10
9
Real wages
8
7
6
5
4
40
50
60
Relative population
70
80
England: 1250 - 1800
Variables, log-scale, arbitrary const.
Inv. wage
Rel. Pop
1200
1300
1400
1500
Year
1600
1700
1800
England: 1350 - 1700
Populatiojn
Plague
Population (detrended)
100
60
50
40
10
30
1300
1400
1500
1600
1700
Plague incidence (log scale)
70
Plague incidence (log-scale)
100
10
30
40
50
60
Population (detrended)
70
80
Real wages and epidemics: conclusions
• Both variables fluctuate synchronously
with population
• Act as first-order factors
• Cannot drive the secular cycle
England: 1450 - 1800
Instability index
3
2
1
0
1450
1500
1550
1600
1650
Year
1700
1750
1800
0.3
Population (detrended)
0.2
0.1
0.0
-0.1
-0.2
1.55
1.60
1.65
1.70
Instability (detrended)
1.75
1.80
Compound annual growth rate
0.8
0.6
0.4
0.2
0.0
-0.2
-0.1
0.0
0.1
0.2
Instability Index (log-transformed)
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
1.8
Population
Internal War
1.6
0.5
1.4
1.2
-200
0.0
-100
0
100
Years
200
300
log Internal War
log Population
1.0
log Warfare
China (200 BCE - 300 CE): population and instability
log Population
Table 1. Comparing out-of-sample predictive
abilities of the inertial and interactive models
(from Turchin-Korotayev Supplement)
Source of
data
Dependent
variable
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
45
45
CBR
CDR
CBR smoothed
CDR smoothed
40
40
35
30
30
25
25
20
20
CBR
CDR
35
1550
1600
1650
1700
1750
1800
1850
30
1540
CDR smoothed
28
26
24
1870
22
20
28
30
32
34
CBR smoothed
36
38
England: 1540-1800
38
Crude birth rate at t+30
36
34
32
30
28
45
50
55
60
65
Relative population density at t
70
30
Crude death rate at t+70
29
28
27
26
25
24
45
50
55
60
65
Relative population density at t
70
Lag = 50 y
36
36
34
34
CBR
CBR
No lag
32
32
30
30
28
28
5
6
7
8
Real Wage at t
9
5
6
7
8
Real Wage at t-50
9
England: 1200-1800
1.8
0.3
log Population, Inverse Wage
1.7
0.2
0.1
1.6
0.0
1.5
-0.1
1.4
-0.2
1.3
1.2
1200
-0.3
-0.4
1300
1400
1500
Time
1600
1700
1800
log Homicide Rate
pop
inv. wage
crime
45
0.6
Neonaticide
Instability
40
0.4
35
30
0.2
25
0.0
20
-0.2
Neonaticide rate
35
Instability index
Neonaticide indictment rate
40
30
25
20
15
15
-0.4
10
10
1600
1650
1700
Year
1750
-0.6
1800
5
-0.6
-0.4
-0.2
0.0
Instability
0.2
0.4
0.6
England: population and climate
90
0.1
Pop (detr)
Climate
80
70
60
-0.1
50
-0.2
40
30
-0.3
1200
1300
1400
1500
Time
1600
1700
1800
Climate
Population (% of K)
0.0
Regression Analysis: r2 versus BD, logW, y, t, N, WAGE
r2 = population rate of change, tau = 20 y
BD = dummy variable for the Black Death
logW = instability, log-transformed
y = year (monotonic temporal trend)
t = temperature
N = population pressure (in relation to K)
WAGE = real wage
Predictor of r2
Constant
BD
logW
y
t
N
WAGE
Coef
-0.15804
-0.11646
-0.030560
0.00008885
-0.17738
-0.0005824
0.003293
SE Coef
0.03053
0.01118
0.003385
0.00002282
0.05676
0.0002043
0.001572
T
-5.18
-10.42
-9.03
3.89
-3.13
-2.85
2.10
R-Sq = 90.0% R-Sq(adj) = 89.0% R-Sq(pred) = 85.55%
P
0.000
0.000
0.000
0.000
0.003
0.006
0.041
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