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