My Christmas letter to you, 2024

Dear Daniel,

Merry Christmas! I hope that through the gloom of the dark winter the joy of Christmas is shining brightly upon you.

I have always loved Christmas, for many reasons. First, because it is the birth of Jesus, a baby who comes into this world to give us new hope and to save us from our sins. I believe that even those who are most sinful can find a path to justice, and Christmas is the start of that path.

But I also love Christmas as a pagan holiday. It is perfectly timed, just after the Winter Solstice (December 21st), to symbolise the time in the year when the days are getting brighter and spring is coming (the ‘Yuletide’). You can’t feel it, but it is coming, and the world will be renewed again!

Finally, I love Christmas because it is a time for family to come together. Of course, in your case, it means I miss you. But it has been nice spending time with your sister Daphne. And I hope you have been loved and have been able to love this Yuletide.

So Happy Christmas, my lovely son. I love you,

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: