My Fifth Letter to You (Merry Christmas!)

By Graham

Dear Daniel,

It’s a few days after Christmas and I’m here at home thinking once again about you. My Christmas has been very merry indeed – with lots of family and friends and games and fun and good food (more about that in a second). But of course, everything is tinged with sadness because you are not here with us, and I wish you were.

Our Christmas tree is magnificent – two metres tall and bushy and perfectly formed. We have loaded it with ornaments and lights and it is in the corner of the dining room, with plenty of space for Christmas presents. Next to the tree, I have set up the Nativity scene on the sill of the dining room window. It is the set of figurines I inherited from your Irish grandmother, who is with the angels in heaven now. I’m sure she is very pleased to look down and see that I still set it up every Christmas.

It was also very popular this Christmas morning. Our visitors included a little girl your age, who duly turned over the baby Jesus when she came out of her room and sang ‘Away in the Manger’, which is a family tradition.

Then there were presents! We unwrapped tonnes of gifts (I got a lovely coat from Jo, which I have been wearing every day since). The children were all delighted with the gifts Santa had brought (they even caught a glimpse of him out the window on Christmas Eve!) and ran around the house like mad people, playing with everything and eating far too many sugary things. Then, later on, we ate a delicious meal of turkey, chicken, ham, stuffing, brussels sprouts, broccoli, turnip, mashed potato, roast potato, turnip, gravy, cranberry sauce and bread sauce. And for dessert we had Yule Log and chocolate cake and chocolate pralines. It was amazing!

What about you? Did Santa come to you? Did you get nice gifts? Did you have good food? I hope so, my dear son. I hope you had an excellent, merry Christmas and are doing well.

I miss you so much. My heart is as always full of love for you, and for Anna and Daphne. I hope 2018 is the year when I get to see you again.

Take care and Happy New Year,





 Category: Dear Daniel

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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: