Hi,

I used dynamic panel estimation model - GMM to examine the relationship between institutions and carbon emissions. I have used a quadratic term and interaction term in the same equation. I want to know how to find the turning point when there is a linear, quadratic term and interaction term in an equation.

I ran the following command:

Variables of interest:

frleg: Institutions
i.inclevel: categorical variable with 4 different income levels. High-income level countries is the base category.

Code:
 xtabond2 co2int L.co2int i.inclevel frleg frleg2 c.frleg#i.inclevel lpopden indus  period3-period31, gmm(L.(co2int frleg frleg2 c.frleg#i.inclevel  lpopden  indus), collapse lag(1 4) ortho) iv(period3-period31  c.frleg#i.inclevel frleg2 frleg, equation(level)) robust twostep small ortho
In the output below, I have removed periods to condense the table.
Code:
.  

Dynamic panel-data estimation, two-step system GMM
------------------------------------------------------------------------------
Group variable: cty                             Number of obs      =      2937
Time variable : year                            Number of groups   =       121
Number of instruments = 75                      Obs per group: min =         3
F(42, 120)    =     32.71                                      avg =     24.27
Prob > F      =     0.000                                      max =        30
--------------------------------------------------------------------------------------
                     |              Corrected
              co2int |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
---------------------+----------------------------------------------------------------
              co2int |
                 L1. |   .6831759   .0431564    15.83   0.000     .5977293    .7686226
                     |
            inclevel |
  high-income, OECD  |          0  (empty)
         low-income  |   2.143515   .9536475     2.25   0.026     .2553593    4.031671
lower-middle-income  |   1.462294   .7817783     1.87   0.064    -.0855731     3.01016
upper-middle-income  |    1.53233   .6849764     2.24   0.027     .1761243    2.888536
                     |
               frleg |     7.7978   3.146071     2.48   0.015     1.568798     14.0268
              frleg2 |  -4.561939      1.862    -2.45   0.016    -8.248569   -.8753081
                     |
    inclevel#c.frleg |
  high-income, OECD  |          0  (empty)
         low-income  |  -3.563873   1.382582    -2.58   0.011    -6.301289   -.8264566
lower-middle-income  |  -1.873472    1.08048    -1.73   0.085    -4.012747    .2658019
upper-middle-income  |  -1.923528   .9477975    -2.03   0.045    -3.800101   -.0469549
                     |
             lpopden |   .0289588   .0539836     0.54   0.593     -.077925    .1358426
               indus |   .0023753   .0034381     0.69   0.491    -.0044318    .0091824
             
               _cons |  -2.699571   1.347477    -2.00   0.047    -5.367482   -.0316595
--------------------------------------------------------------------------------------
Instruments for orthogonal deviations equation
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    BOD.L(1/4).(L.co2int L.frleg L.frleg2 1bL.inclevel#coL.frleg
    3L.inclevel#cL.frleg 4L.inclevel#cL.frleg 5L.inclevel#cL.frleg L.lpopden
    L.indus) collapsed
Instruments for levels equation
  Standard
    period3 period4 period5 period6 period7 period8 period9 period10 period11
    period12 period13 period14 period15 period16 period17 period18 period19
    period20 period21 period22 period23 period24 period25 period26 period27
    period28 period29 period30 period31 1b.inclevel#co.frleg
    3.inclevel#c.frleg 4.inclevel#c.frleg 5.inclevel#c.frleg frleg2 frleg
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    D.(L.co2int L.frleg L.frleg2 1bL.inclevel#coL.frleg 3L.inclevel#cL.frleg
    4L.inclevel#cL.frleg 5L.inclevel#cL.frleg L.lpopden L.indus) collapsed
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -4.76  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =  -0.32  Pr > z =  0.750
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(32)   =  79.01  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(32)   =  40.32  Prob > chi2 =  0.148
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  GMM instruments for levels
    Hansen test excluding group:     chi2(24)   =  36.88  Prob > chi2 =  0.045
    Difference (null H = exogenous): chi2(8)    =   3.44  Prob > chi2 =  0.904
My question is:
1# How to estimate the turning point for institutions (frleg) with respect to each income-level of countries? i.e. turning point for low, middle or upper-middle income countries.

I know if there were no interaction term in this regression, then using the formula b/|2a| would have worked. But, how do I estimate with respect to each income group category?

Can someone please help me with me?

Thank you.

Ritika