Undefined turning points

[Emil Hermansson]

Hello!

I am writing my thesis about the environmental Kuznets curve (EKC). The analysis compares turning points of the EKC between country groups. However, when I run the regressions (estimated by pooled OLS) for the EKC I do not get results that allow me to extract any turning points since the turning points are not defined (without the use of complex numbers). Why do I get these results, and how should I interpret them? A screenshot of my regression output, estimation regression and the formula for the turning points should be is attached below:

[Nick Cox]

I guess I would try a plot of predicted ln CO_2 against ln GDP PC. Perhaps your cubic is doing more than you intended.

Picky point. Subscripts are impossible in variable names, but Co is cobalt, and CO is carbon and oxygen.

[Emil Hermansson]

Hello Nick!

Thanks for your respons and input, you are right about the Co/CO point will change that! I get the following scatterplot out of it:

Cant really get why the turning points are not defined from this..

[Nick Cox]

You would get a more precise view from

twoway function _b[ln_GDPPC] * ln_GDPPC + _b[ln_GDPPC_2] * ln_GDPPC_2 + _b[ln_GDPPC_3] * ln_GDPPC_3, range(ln_GDPPC)

but my reading is that the fitted curve is monotonic with an inflexion but no turning points. That's entirely possible with a cubic and indeed exactly true of x^3.

Indeed. plotting the first derivative makes everything clear. I get

twoway function 13.04218 -1.344725 * 2 * x + 0.0463432 * x^2 * 3, range(7 14) yli(0)

showing that there is almost a turning point, but the curve is still monotonic.

[Emil Hermansson]

Hello again Nick,

Thank you very much for this, you are right, that explains a lot.

Have a good day.

Best regards
Emil.

[Nick Cox]

I should have said -- almost an inflexion. Fortunately none of my former calculus teachers are in a position to read this thread.

[Emil Hermansson]

I should have said -- almost an inflexion. Fortunately none of my former calculus teachers are in a position to read this thread.

Yes, it does not have inflection point either right? Cause from my calculations I cannot define X for f''(x)=0 either.

[Nick Cox]

Correct. It is sufficient to note that the first derivative is always positive over the data range, so the fitted curve is monotonic without an inflexion even.

[Nick Cox]

In fact “over the data range” is over-cautious. The first derivative is a a quadratic so its properties are clear.

[Daniel Loaiza]

Hi Nick, one question, how can I calculate the inflection point when GDP has first difference applied? I mean, I have a significant coefficient value of 0.65 for GDP and -6.4 for GDP2, if I calculate -b/2a it gives me approximately 2, then if I apply exponential it gives close to 8. However, the GDP per capita of my panel is It ranges between 1,000 and 18,000. In several studies I have seen that they calculate it but I don't know how they apply the inverse transformation of the first difference. I hope you can help me. greetings