Bonne rentrée – 17ieme lettre

Dear Daniel,

I am writing to you in English in the hope that you will have learned some English at school by now and can read this.

The summer is over! I hope you had a nice break and got plenty of sunshine (but not too much heat!) in Southern Europe. We avoided going south this year because the heat is just too much. Instead, I took your sister horseriding in Ireland and we visited England and also Normandy. We also went camping in the Ardennes (which is full of Dutch people).

Anyway, now it’s back to school. I hope you have a great start to the year – good teachers or at least tolerable teachers – and that you are enjoying seeing your friends again.

As always, I think of you a lot – especially when I am doing things like cutting wood or building the woodshed at home. It makes me a little sad that I can’t do these things with you, as a father really should.

Maybe one day.

Meanwhile, all the best from me.

Love,

Dad

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: