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 |
Population
(Mill) |
Annual growth
rate (prop) |
Period Length |
-1000000 |
.125 |
.00000297 |
700000 |
-300000 |
1.000 |
.00000439 |
250000 |
-25000 |
3.340 |
.00003100 |
15000 |
-10000 |
4.000 |
.00004500 |
5000 |
-5000 |
5.000 |
.00033600 |
1000 |
-4000 |
7.000 |
.00069300 |
1000 |
-3000 |
14.000 |
.00065700 |
1000 |
-2000 |
27.000 |
.00061600 |
1000 |
-1000 |
50.000 |
.00138600 |
500 |
-500 |
100.000 |
.00135200 |
300 |
-200 |
150.000 |
.00062300 |
200 |
1 |
170.000 |
.00055900 |
200 |
200 |
190.000 |
.00000000 |
200 |
400 |
190.000 |
.00025600 |
200 |
600 |
200.000 |
.00047700 |
200 |
800 |
220.000 |
.00093100 |
200 |
1000 |
265.000 |
.00188600 |
100 |
1100 |
320.000 |
.00117800 |
100 |
1200 |
360.000 |
.00000000 |
100 |
1300 |
360.000 |
-.00028170 |
100 |
1400 |
350.000 |
.00194200 |
100 |
1500 |
425.000 |
.00248700 |
100 |
1600 |
545.000 |
.00000000 |
50 |
1650 |
545.000 |
.00225300 |
50 |
1700 |
610.000 |
.00331600 |
50 |
1750 |
720.000 |
.00446300 |
50 |
1800 |
900.000 |
.00575400 |
50 |
1850 |
1200.000 |
.00396400 |
25 |
1875 |
1325.000 |
.00816400 |
25 |
1900 |
1625.000 |
.00830600 |
20 |
1920 |
1813.000 |
.00916400 |
10 |
1930 |
1987.000 |
.01077200 |
10 |
1940 |
2213.000 |
.01283200 |
10 |
1950 |
2516.000 |
.01822600 |
10 |
1960 |
3019.000 |
.02015100 |
10 |
1970 |
3693.000 |
.01864600 |
10 |
1980 |
4450.000 |
.01810100 |
10 |
1990 |
5333.000 |
. |
. |
Explanation of annual growth
rate (stated as proportion in previous table):
Going
from 1980 to 1990:
_{}
In
general, to get the growth rate between periods t and t+1, of length l_{t} :
_{}
Plot of growth rate versus population with ordinary least squares fit (population converted to billions, growth rate converted to percent):
OLS Regression Analysis and
Durbin-Watson Statistic:
Note
that the critical values for the Durbin-Watson Statistic with 1 predictor and
37 observations are d_{L} = 1.419 and d_{U} = 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:
Expecting
less uncertainty in average growth rates over longer periods.
WLS Estimation:
_{}
Create
new variables: _{} where in this problem,
the weighting variable is period length. Then fit OLS on transformed Y and X.
WLS Regression Analysis:
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.
_{}
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.
a) Regressed on Inverse Period Length and Year:
b) Regressed on Inverse Period Length
2) WLS Squared Residuals
a) Regressed on Inverse Period Length and Year:
b) Regressed on Inverse Period Length
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.