My seventh letter to you

By Graham

Dear Daniel,

It has been a while since I’ve written to you, but that is not to say I have stopped thinking about you. Oh, not at all. On the contrary, you are in my thoughts every day.

I hope now that the evenings are getting brighter you will have a chance to play outside a little bit more. The parks are certainly bustling these day. When I collect your little sister from creche I often stop at the park and she watches the big boys and girls playing on their scooters, bikes and roller skates. She’s eager to get going too, but at 22 months still too small for that kind of thing.

It’s Lent now, which is a good period of the year to reflect on things and take stock of the good. It’s sort of a spiritual springtime! I am using this time to try and get back in shape – lots of running in the morning and playing tennis.

How is your tennis going? I hope you are improving. One day, perhaps, we can go out to the courts together and play a few games. My backhand is still kind of weak but I have quite a strong forehand and a decent enough serve.

Anyway, enough for now. I recorded a little video message for you before Christmas. Hopefully you have had a chance to watch it.

As always, you have my love.

Category: Dear Daniel

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:

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.

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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:

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:

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