We begin by plotting a scattergram of the birth rate and GDP per capita data for 1990 to 2015 and for all 4 categories of country (100 observations total). The scatter suggests a functional form which has the following properties:
1) convex
2) Having a natural limit at birth rates of around 45, (beyond which human populations are generally unable to reproduce faster)
3) Tapering to some limit of birth rates of around 7.5 as income goes to infinity, reflecting perhaps the innate human desire to procreate, independent of income levels
We test the following functional form:
b = (45*min(Y)^a)/y^a
by taking logs:
ln(b) = ln(45*min(Y)) – a*ln(Y)
and estimating using OLS. Here are the regression results:
R Square 0.961571033
Coefficient of Intercept 6.25544297
t-stat of Intercept 96.5661
Cofficient of X Variable -0.36486599
t-stat of X Variable -49.5193
The intercept is closely approximated by the constant ln(45*min(Y)), the t-scores are high and the R Square shows a very good fit.
Here’s the fitted values against the actual values:
As with birth rates, we use data for 4 categories of countries from 1990 to 2015 (100 observations total). We have two explanatory variables, AGE and Y, where AGE is defined as the percentage of the population aged over 65 and Y is per capita GDP.
After eyeballing the scattergrams, we test the following functional form:
d = (minY^a)/Y^a * (1/AGE^g)
Where minY is the constant equal to the smallest value of Y in the series.
Logarithmic transformation gives:
ln(d) = ln(minY^a) – a*ln(Y) – g*ln(AGE)
which we test on the data using OLS. Here are the results:
Adjusted R square: 75.191
Intercept coefficient: 7.37384
t-Stat: 20.4011
Y coefficient: -1.01444
t-Stat: -13.1059
AGE coefficient: 2.0097
t-Stat: 11.5208
The estimated intercept is a good, but not perfect, approximation of ln(minY^a)
Here are the fitted against actual values of the scattergram for death rate against per capita GDP:
While the results are not as good as with the birth rates calculations, it is nevertheless a good enough fit and the explanatory variables have a strong enough confidence factor to be usable in our estimations.
We begin by examining the scatter of data for 100 observations of per capita GDP and per capita emissions for 4 categories of countries, over 25 years (1990 – 2015).
The scatter suggests a cubic functional form, so we test:
GHG = a + b*Y + c*Y^2 + d*Y^3
where GHG are per capita emissions of GHG, and Y is per capita GDP.
The results from OLS regression are:
Adjusted R square: 0.980438073
coefficient a: 1090
t-stat a: 3.06
coefficient b: 0.709310153
t-Stat b: 8.241453
coefficient c: -0.0000047025
t-Stat c: -1.01233
coefficient d: -0.000000000105314
t-Stat d: -1.47005
While the t-scores on the squared and cubed terms are low, the number of observations are also limited.
Here is the plot of the fitted against actual values:
