Estimation of turning points--EKC

[Yolanda Lee]

Hello all,

I'm writing a professional paper on carbon dioxide emissions. I want to integrate a section on the Environmental Kuznets Curve but I need some help with the math. I have a variable 'gdp per capita'. I want to use it as a log(gdp per capita) but I also want to find the turning point. I know I need to square the term but my question is do I take the log of it first then square it or square it then take the log.

I also then need help on how to transform the squared log form back into levels to find the dollar amount where gdp per capita squared reaches a maximum. I would really appreciate any insight into my problem.

Thanks in advance!

[Carlo Lazzaro]

Yolanda:
provided that I do not know anything about EKC, I would suggest you:
- ln(gdp per capita) before squaring (it would be probably wise to center the variable around a meaningful value beforehand);
- the formula for calculating the turning point via regression coefficient is: (-coefficient of the linear term/(2*coefficient of the squared term)). After this calculation has been performed, I would check whether the resulting value falls within the range of -gdp per capita- or not;
- you can exp(ln(gdp per capita)) to go back to the original metric (this procedure may at best approximate the value of the pdp per capita in its original metric, though).

[Anna Schmid]

Dear Carlo Lazzaro

I am actually doing exactly the same thing for my bachelors thesis and a am trying to find out the turning points of sulfur dioxid and other environmental indicators as dependent variables and gdp as an independent variable.
Now you say, that i can calculate the turning points of these indicators with: (-coefficient of the linear term/(2*coefficient of the squared term). First, thank you. Second, can you maybe give a reference to that and an explanation why it is working like that? Because it is a paper and i have to justify every move

Kind Regards,

Anna

[Clyde Schechter]

Second, can you maybe give a reference to that and an explanation why it is working like that?

You can find this in any high school algebra textbook that goes as far as quadratic functions. Find the section on the axis of symmetry of a parabola. I don't think you really need to give a reference for this.

[Jorrit Gosens]

You should not square before logging as log(x-squared)=2*log(x), and including both in a regression will give perfect collinearity.
On Carlo’s remark to check identified maximum with original data: I agree, but would add that the optimum falling outside original data is not strange with the EKC. The EKC hypothesis works, sometimes. Some pollutants do not very well follow the relationship, and such a finding should not lead you to conclude you might have done something wrong, but rather that the EKC didn’t work in this case. You might check to see if the original data at least starts to plateau at higher values of income.
A good reference, which has both good and weak points of the EKC is following one. Good read when you are going to do this type of analysis anyway: http://steadystate.org/wp-content/up...znetsCurve.pdf

[Clyde Schechter]

On Carlo’s remark to check identified maximum with original data: I agree, but would add that the optimum falling outside original data is not strange with the EKC.

That may well be, but if the turning point falls outside the data, then it isn't a real turning point, and, arguably, you may not even really have a quadratic model for the data. Extrapolating regression models beyond the range of the predictor variables is notoriously unreliable.

[Carlo Lazzaro]

I agree with Clyde about the risk of extrapolating regression models beyond the range of the independent variables.
My favourite reference on this topic is: http://www.bmj.com/content/317/7155/409

[Jorrit Gosens]

I believe we're arguing the same point, Clyde. The EKC hypothesizes a quadratic relation between pollutants and income. What I meant to say is that if the analysis done by Yolanda would not reveal such a pattern, than that is not necessarily surprising or wrong, as this has been the case in many earlier instances of comparable analyses.

[Carlo Lazzaro]

Anna wrote:

Now you say, that i can calculate the turning points of these indicators with: (-coefficient of the linear term/(2*coefficient of the squared term). First, thank you. Second, can you maybe give a reference to that and an explanation why it is working like that? Because it is a paper and i have to justify every move

Here is a possible reference: http://www.stata.com/bookstore/micro...ata/index.html, page 250.

[Clyde Schechter]

Jorrit Gosens . OK, thanks. I have never heard of EKC before, but in light of your explanation it does indeed seem we're saying the same thing in different ways. Thank you.

[Marco Giansoldati]

Dear Members,

I wish to kindly ask a clarfication which I hope it is not too trivial.

I wish to look at the relationship between GDP and GDP squared on the CO2 emissions in the realm of the Environmental Kuznets Curve. The simple relationship I would lilke to test is as follows:

CO2 = a +bGDP+ cGDP^2.

The problem arises when I wish use logs to estimate the relationship between the dependent and the explanatories and look for a turning point.

The two options I should use are either one of the following two:

1) ln(CO2) = a +bln(GDP)+ cln(GDP^2) OR

2) ln(CO2) = a +bln(GDP)+ c(ln(GDP))^2

I looked at the paper indicated by Jorrit Gosens, who suggests I should go for the second option:

http://www.steadystate.org/wp-content/uploads/Stern_KuznetsCurve.pdf

as can be seen at page 1422, Equation 1.

Yet, in a number of articles I found support for the first argument. Please have a look at the following one

https://www.sciencedirect.com/science/article/pii/S0140988317303328

where Equation 2 at page 551 reads as follows

Here it seems that the Authors go for the first option, unless they have just opted for a notational shortcut.

I would be extremely thankful if you could kindly provide me with your kind suggestions.

Thanks.

Marco

[Jorrit Gosens]

It's fairly inconsequential.

e.g.:

clear 
set obs 1000
gen x = 100*_n
gen lnx2 = ln(x*x)
gen lnxlnx = ln(x)*ln(x)
line  lnx2 lnxlnx

Theoretically, though, you are trying to estimate the relation:

X = a*Y + b*Y^2 + c
where Y=ln(z)
Which would suggest it is more correct to use X = a*ln(z) + b*ln(z)^2 + c

[Jorrit Gosens]

Actually, see my earlier point here:

You should not square before logging as log(x-squared)=2*log(x), and including both in a regression will give perfect collinearity.

So somewhat consequential

clear 
set obs 1000
gen y = runiform()
gen x = 100*_n
gen lnx =ln(x)
gen lnx2 = ln(x*x)
gen lnxlnx = ln(x)*ln(x)
reg y lnx lnx2 lnxlnx

reg y lnx lnx2 lnxlnx
note: lnx omitted because of collinearity

      Source |       SS           df       MS      Number of obs   =     1,000
-------------+----------------------------------   F(2, 997)       =      2.35
       Model |  .397322852         2  .198661426   Prob > F        =    0.0960
    Residual |  84.3277287       997  .084581473   R-squared       =    0.0047
-------------+----------------------------------   Adj R-squared   =    0.0027
       Total |  84.7250515       999  .084809861   Root MSE        =    .29083

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         lnx |          0  (omitted)
        lnx2 |   .0946561   .0520523     1.82   0.069    -.0074885    .1968007
      lnxlnx |  -.0103471   .0053978    -1.92   0.056    -.0209395    .0002452
       _cons |  -.3414879   .4980443    -0.69   0.493    -1.318823    .6358474
------------------------------------------------------------------------------

[Marco Giansoldati]

Thank you very much Jorrit Gosens . I will keep you posted.

[Marco Giansoldati]

Thank you all.