Interpreting Coefficients of Quadratic Terms in Regression

[Patricia Nicole]

Hi! I'm quite confused on how to interpret the coefficients of quadratic terms in my regression, these are my coefficients, where CO2 is my dependent variable:

lnco2 | Coefficient
-------------+-----------------
lngdp | .5752118
lngdpsq | -.0305936
lncoal | .1103289
lnelec | .2510876
lnrenew | -.0859841
lnurban | -.9068971
lntrade | .038965
_cons | .8618177
-------------------------------

I would like to interpret the effect of a 1% increase in GDP per capita to CO2 emissions. The presence of the quadratic term (GDP per capita squared) makes it complicated. According to my prof, the increase in CO2 is not constant across values of GDP per capita since I have a quadratic term.

Would appreciate any help, thanks!

[Felix Bittmann]

This usually means that you have an inverted u shape. I would suggest not creating quadratic by hand and rather rely on a graphic interpretation. Like

reg lnco2 c.lngdp##c.lngdp othervars
margins, at(lngdp=(1(1)10))
marginsplot

You only need to set the range, here I did from 1 to 10. This depends on the scaling of the lngdp variable and must be set by you.

[Carlo Lazzaro]

Patricia:
as an aside to Felix's excellent advice, you should also check by the usual formula:

-b/2a

whether the value of your turning point:

. di -.5752118/(2*-.0305936)
9.4008518

is actually included in the range of your -lngdp- variable (otherwise the existence of a turning point is an algebric artifact).
By the way, following the FAQ you should have shared the whole outcome table of -regress-, so that interested listers can check, among ather results, the statistical significance of your squared term (that, as Felix wisely pointed out, should be coded relying on the wonderful capabilities of -fvvarlist- notation).
Eventually, your teacher/preofessor/supervisor is probably advising you to study the sign of the first derivative of the quadratic function. As you have an inverted U-shape (aka: the sad parabola ), to the left of the maximum the funtion is increasing (as the slope of the first derivative is positive), whereas the opposite happens to the right of the maximum.

[Patricia Nicole]

Thanks for your suggestions! Although i'm aware that it may be an inverted u-shape, I would like to know how I can interpret it in sentence format.

My previous interpretation is as ff: A 1% increase in GDP per capita results in a 0.5752118% increase in the mean of CO2 emissions.

Although it's wrong.

[Felix Bittmann]

I think you are missing the point with such a short summary. When there is a U shape it is not possible to "compress" the information like so because the effect of GDP is not constant. Sure, you can simply compute the "average" effect but then why bother and include the quadratic term? I think the important message is that the effect of GDP is not constant. I would use the graph and then simply describe what you see. Like "for lower values of GDP the relationship is linear and then reaches a plateau (or inverts) which might be due to..."

[Carlo Lazzaro]

Patricia:
if your turning point does exists in the light of previous replies, your last statement it is true only to the left of the maximum (where the first derivative, that is -lngdp- of the quadratic function has a positive slope); conversely, to the right of the maximum, the first derivative of the quadratic function has a negative slope, hence a 1% increase in GDP per capita causes a 0.5752118% reduction in the mean of CO2 emissions.
What above is proved via a toy-example (let's assume that data are cross-sectional and not panel):

use "https://www.stata-press.com/data/r17/nlswork.dta"
. reg ln_wage c.age##c.age

      Source |       SS           df       MS      Number of obs   =    28,510
-------------+----------------------------------   F(2, 28507)     =   1379.28
       Model |  574.966399         2    287.4832   Prob > F        =    0.0000
    Residual |  5941.72375    28,507  .208430342   R-squared       =    0.0882
-------------+----------------------------------   Adj R-squared   =    0.0882
       Total |  6516.69015    28,509  .228583611   Root MSE        =    .45654

------------------------------------------------------------------------------
     ln_wage | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         age |   .0855891   .0035923    23.83   0.000     .0785481    .0926302
             |
 c.age#c.age |  -.0010982   .0000596   -18.42   0.000    -.0012151   -.0009814
             |
       _cons |   .1647917   .0521021     3.16   0.002      .062669    .2669143
------------------------------------------------------------------------------

*Turning point (maximum) at age=39 years
. di -.0855891/(2* -.0010982 )
38.967902

. tabulate age fitted if age <=40 & age>=35

    Age in |
   current |                         Linear prediction
      year |  1.815079   1.822694   1.828112   1.831188   1.831334   1.832359 |     Total
-----------+------------------------------------------------------------------+----------
        35 |     1,264          0          0          0          0          0 |     1,264
        36 |         0      1,111          0          0          0          0 |     1,111
        37 |         0          0        956          0          0          0 |       956
        38 |         0          0          0          0        868          0 |       868
        39 |         0          0          0          0          0        770 |       770
        40 |         0          0          0        577          0          0 |       577
-----------+------------------------------------------------------------------+----------
     Total |     1,264      1,111        956        577        868        770 |     5,546


.

As you can see, the linear prediction reaches a maximum at age=39, then preogressively declines.