## My Second Letter to you

##### By Graham

Dear Daniel,

How are you?

I guess school is out now and you are off for the summer. From what I gather from the legal stuff, you and your mom are off for the summer. I hope you get out into the parks, go swimming and have a good time. Maybe you will even get to go to the beach? Of course, I wish I could take you there. Maybe one day I will, who knows?

It has been an eventful summer so far, with the birth of your new little sister, Daphne, during an awful heat wave. Naturally, it has not been easy for her to adjust to that kind of weather, and she’s quite the demanding little miss, I have to say.

Of course, only naturally it takes me back to when you were a baby. Almost automatically, I fall into the same patterns of speech, of carrying her around. I have different nicknames for her than for you, of course. (You were always known as ‘my main man’, or ‘Danko Panko’, or other such names…). I remember giving you your ‘biberon’ in the mornings, the endless diaper changes (!), but also ‘flying’ you around the living room Superman-style. I hope that somewhere, in your subconscious, those memories will remain embedded.

There has been more legal stuff, but I won’t bore you with that. Suffice it to say that I am still doing everything I can to remain in your life and be your dad. I will never give up on you.

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:

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.

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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:

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:

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