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 lt  :

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 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:

## Weighted Least Squares – Weights are Period Length

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:

### 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.

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:

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.