Case Study – Technology and Population Growth
Theory: Population is limited by technology change. One hypothesis is that technology
change increses with population, since there are more minds available to generate new
technology. Thus, large population should be associated with more technology change,
and thus more population growth. The Malthusian model believes population increases
after major technological advances.
Data: Population (in Millions) and subsequent annual growth rate for 37 time periods
from –1M BC through 1980.
Year
-1000000
-300000
-25000
-10000
-5000
-4000
-3000
-2000
-1000
-500
-200
1
200
400
600
800
1000
1100
1200
1300
1400
1500
1600
1650
1700
1750
1800
1850
1875
1900
1920
1930
1940
1950
1960
1970
1980
1990
Population (Mill)
.125
1.000
3.340
4.000
5.000
7.000
14.000
27.000
50.000
100.000
150.000
170.000
190.000
190.000
200.000
220.000
265.000
320.000
360.000
360.000
350.000
425.000
545.000
545.000
610.000
720.000
900.000
1200.000
1325.000
1625.000
1813.000
1987.000
2213.000
2516.000
3019.000
3693.000
4450.000
5333.000
Annual growth rate (prop)
.00000297
.00000439
.00003100
.00004500
.00033600
.00069300
.00065700
.00061600
.00138600
.00135200
.00062300
.00055900
.00000000
.00025600
.00047700
.00093100
.00188600
.00117800
.00000000
-.00028170
.00194200
.00248700
.00000000
.00225300
.00331600
.00446300
.00575400
.00396400
.00816400
.00830600
.00916400
.01077200
.01283200
.01822600
.02015100
.01864600
.01810100
.
Period Length
700000
250000
15000
5000
1000
1000
1000
1000
500
300
200
200
200
200
200
200
100
100
100
100
100
100
50
50
50
50
50
25
25
20
10
10
10
10
10
10
10
.
Explanation of annual growth rate (stated as proportion in previous table):
Going from 1980 to 1990:
P1980 (1  gr1980 ) pl1980  4450(1  .01810100)10  4450(1.1964888)  5324.4  P1990
In general, to get the growth rate between periods t and t+1, of length lt :
P 
grt   t 1 
 Pt 
1 / lt
1
Plot of growth rate versus population with ordinary least squares fit (population
converted to billions, growth rate converted to percent):
Linear Regression
grow thp = -0.00 + 0.52 * popb
R-Square = 0.92

2.00

growthp




1.00





0.00


 
  

 

 
 
 
0.00

1.00
2.00
3.00
popb
4.00
5.00
OLS Regression Analysis and Durbin-Watson Statistic:
Model Summaryb
Model
1
R
.960a
R Square
.922
Adjusted
R Square
.919
Std. Error of
the Estimate
.17287
Durbin-W
ats on
1.097
a. Predictors: (Constant), POPB
b. Dependent Variable: GROWTHP
Coeffi cientsa
Model
1
(Const ant)
POPB
Unstandardized
Coeffic ients
B
St d. Error
-.002
.036
.524
.026
St andardiz ed
Coeffic ients
Beta
.960
t
-.064
20.290
Sig.
.950
.000
a. Dependent Variable: GROW THP
Note that the critical values for the Durbin-Watson Statistic with 1 predictor and 37
observations are dL = 1.419 and dU = 1.530. Since 1.097 lies below both values, we reject
the null hypothesis of no autocorrelation among the residuals.
Plot of Residuals versus Time Order:
.6
.4
.2
-.0
-.2
-.4
-.6
1
3
5
7
9
Case Number
11 13 15 17 19 21 23 25 27 29 31 33 35 37
Weighted Least Squares – Weights are Period Length
Expecting less uncertainty in average growth rates over longer periods.
WLS Estimation:
^ W
^ W
n
Choose  0 and  1 that minimize Q   wi (Yi  (  0  1 X i )) 2
W
i 1
Create new variables: Yi*  wi Yi and X i*  wi X i where in this problem, the
weighting variable is period length. Then fit OLS on transformed Y and X.
WLS Regression Analysis:
Model Summaryb,c
Model
1
R
.918a
R Square
.844
Adjusted
R Square
.839
Std. Error of
the Estimate
1.14562
Durbin-W
ats on
.845
a. Predictors: (Constant), POPB
b. Dependent Variable: GROWTHP
c. Weighted Leas t Squares Regres sion - Weighted by PRDLNGTH
Coeffi cientsa,b
Model
1
(Const ant)
POPB
Unstandardized
Coeffic ients
B
St d. Error
.000
.001
.504
.037
St andardiz ed
Coeffic ients
Beta
.918
t
.336
13.741
a. Dependent Variable: GROW THP
b. W eighted Leas t Squares Regres sion - W eighted by PRDLNGTH
Sig.
.739
.000
Testing for Heteroskedasticity Among the Residuals
We might suspect that the variance of the residuals is proportional to the reciprocal of the
period length, since we can loosely think of having averaged over the years of the period.
That is : V (Y ) 
2
n
Also, we might expect that measurement error is worse the further back in time we go (at
least proportionally). To test for heteroskedasticity, we first collect the squared residuals
from the OLS and WLS regressions fit previously, then regress them on the reciprocal of
period length and year.
1) OLS Squared Residuals
a) Regressed on Inverse Period Length and Year:
Model Summaryb
Model
1
R
.569a
R Square
.324
Adjusted
R Square
.284
Std. Error of
the Estimate
.05616
Durbin-W
ats on
1.739
a. Predictors: (Constant), YEAR, INVPL
b. Dependent Variable: RES_OLS2
Coeffi cientsa
Model
1
Unstandardized
Coeffic ients
B
St d. Error
(Const ant) 1.698E-05
.012
INVPL
1.015
.255
YEAR
-4. 91E-11
.000
St andardiz ed
Coeffic ients
Beta
t
Sig.
.999
.000
.999
.001
3.977
-.001
.569
.000
a. Dependent Variable: RES_OLS2
b) Regressed on Inverse Period Length
Model Summaryb
Model
1
R
.569a
Adjusted
R Square
.304
R Square
.324
Std. Error of
the Estimate
.05535
Durbin-W
ats on
1.739
a. Predictors: (Constant), INVPL
b. Dependent Variable: RES_OLS2
Coeffi cientsa
Model
1
Unstandardized
Coeffic ients
B
St d. Error
(Const ant) 1.977E-05
.011
INVPL
1.015
.248
a. Dependent Variable: RES_OLS2
St andardiz ed
Coeffic ients
Beta
.569
t
.002
4.094
Sig.
.999
.000
2) WLS Squared Residuals
a) Regressed on Inverse Period Length and Year:
Model Summaryb
Model
1
R
.575a
Adjusted
R Square
.292
R Square
.331
Std. Error of
the Estimate
.05800
Durbin-W
ats on
1.701
a. Predictors: (Constant), YEAR, INVPL
b. Dependent Variable: RES_WLS2
Coeffi cientsa
Model
1
(Const ant)
INVPL
YEAR
Unstandardized
Coeffic ients
B
St d. Error
-.001
.012
1.066
.264
-9. 98E-10
.000
St andardiz ed
Coeffic ients
Beta
t
-.060
4.046
-.017
.576
-.002
Sig.
.952
.000
.986
a. Dependent Variable: RES_W LS2
b) Regressed on Inverse Period Length
Model Summaryb
Model
1
R
.575a
R Square
.331
Adjusted
R Square
.312
Std. Error of
the Estimate
.05716
Durbin-W
ats on
1.701
a. Predictors: (Constant), INVPL
b. Dependent Variable: RES_WLS2
Coeffi cientsa
Model
1
(Const ant)
INVPL
Unstandardized
Coeffic ients
B
St d. Error
-.001
.012
1.066
.256
St andardiz ed
Coeffic ients
Beta
.575
t
-.059
4.161
Sig.
.953
.000
a. Dependent Variable: RES_W LS2
Thus, we can see that for both OLS and WLS models, the squared residuals are
proportional to inverse period length.
Source: M. Kremer (1993). “Population Growth and Technological Change: One Million
B.C. to 1990”, Quarterly Journal of Economics, Vol. 108, #3,. pp681-716.