Dear Statalist users, I hope this thread finds you well.

I’m working with xtabond2 command in Stata. I read David Roodman’s papers, Mr. Sebastian Kripfganz’s presentation notes and I also read almost all threads about system GMM in the forum. However, I suspect that my coding is wrong and I’m doing something wrong. So, could you please help?

Here is my study: I’m working on carbon emissions convergence based on Solow-Swan model. I’m trying to estimate convergence equation with 29 countries and 11 data points (5 year averages between 1965 and 2014). However, I cannot find any significant GDP coefficient for conditional convergence. FE and OLS gives meaningful results and I know that that the system GMM coefficient should be lying between them.

My variables are carbon emissions per capita, gdp per capita and population growth and gross savings. Population growth and gross domestic savings come from theoretical derivations and I treat them as exogenous variable.

Specifically, my xtabond2 code as follows:

Code:
xtabond2 lncopc l.lncopc lngdppc lngdsav lnpopgrate t1-t11, gmm(l.lncopc l.lngdppc, lag (2 8) equation(diff) collapse) iv(lngdsav lnpopgrate, equation(level)) iv(t1-t11, equation(level)) robust small
The output that Stata provides:

Code:
 xtabond2 lncopc l.lncopc lngdppc lngdsav lnpopgrate t1-t11, gmm(l.lncopc l.lngdppc, lag (2 8) equation(diff) collapse) iv(lngdsav lnpopgrate
> , equation(level)) iv(t1-t11, equation(level)) robust small
Favoring speed over space. To switch, type or click on mata: mata set matafavor space, perm.
Warning: Two-step estimated covariance matrix of moments is singular.
  Using a generalized inverse to calculate robust weighting matrix for Hansen test.
  Difference-in-Sargan/Hansen statistics may be negative.

Dynamic panel-data estimation, one-step system GMM
------------------------------------------------------------------------------
Group variable: country1                        Number of obs      =       219
Time variable : year2                           Number of groups   =        27
Number of instruments = 26                      Obs per group: min =         1
F(15, 26)     =     34.63                                      avg =      8.11
Prob > F      =     0.000                                      max =        10
------------------------------------------------------------------------------
             |               Robust
      lncopc |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      lncopc |
         L1. |   .8322342   .0626964    13.27   0.000     .7033598    .9611085
             |
     lngdppc |  -.0855172   .1160706    -0.74   0.468    -.3241037    .1530694
     lngdsav |   .1367784   .1342324     1.02   0.318    -.1391403     .412697
  lnpopgrate |   .0084767   .0185688     0.46   0.652     -.029692    .0466454
          t1 |          0  (omitted)
          t2 |   .1008801   .1818593     0.55   0.584     -.272937    .4746972
          t3 |   .2263383   .0878163     2.58   0.016     .0458293    .4068472
          t4 |   .1056935   .0771406     1.37   0.182    -.0528712    .2642582
          t5 |   .0392374   .0825815     0.48   0.639    -.1305113    .2089861
          t6 |   .1132047   .0556357     2.03   0.052    -.0011561    .2275655
          t7 |   .1107995   .0393627     2.81   0.009     .0298883    .1917106
          t8 |   .1224347   .0500722     2.45   0.022     .0195098    .2253596
          t9 |   .1421915   .0231602     6.14   0.000     .0945851    .1897979
         t10 |   .0998298   .0176985     5.64   0.000       .06345    .1362096
         t11 |          0  (omitted)
       _cons |   .7242149   .8193276     0.88   0.385     -.959937    2.408367
------------------------------------------------------------------------------
Instruments for first differences equation
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(2/8).(L.lncopc L.lngdppc) collapsed
Instruments for levels equation
  Standard
    t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11
    lngdsav lnpopgrate
    _cons
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -2.37  Pr > z =  0.018
Arellano-Bond test for AR(2) in first differences: z =  -1.09  Pr > z =  0.276
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(10)   =  14.00  Prob > chi2 =  0.173
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(10)   =  11.88  Prob > chi2 =  0.293
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  iv(lngdsav lnpopgrate, eq(level))
    Hansen test excluding group:     chi2(8)    =  10.76  Prob > chi2 =  0.216
    Difference (null H = exogenous): chi2(2)    =   1.12  Prob > chi2 =  0.572
  iv(t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11, eq(level))
    Hansen test excluding group:     chi2(1)    =   3.36  Prob > chi2 =  0.067
    Difference (null H = exogenous): chi2(9)    =   8.52  Prob > chi2 =  0.483


.
In results, number of instruments are barely less than number of groups? Can it be the reason of the problem? I also see that I have omitted time dummy and constant. I tried to overcome this issue by using time dummies beginning with t2 (dropping t1), even though the coefficient of gdp per capita still insignificant, Hansen and Sargan test results were improved.

Code:
xtabond2 lncopc l.lncopc lngdppc lngdsav lnpopgrate t2-t11, gmm(l.lncopc l.lngdppc, lag (2 8) equation(diff) collapse) iv(lngdsav lnpopgrate
> , equation(level)) iv(t1-t11, equation(level)) robust small
Favoring speed over space. To switch, type or click on mata: mata set matafavor space, perm.
t9 dropped due to collinearity
Warning: Two-step estimated covariance matrix of moments is singular.
  Using a generalized inverse to calculate robust weighting matrix for Hansen test.
  Difference-in-Sargan/Hansen statistics may be negative.

Dynamic panel-data estimation, one-step system GMM
------------------------------------------------------------------------------
Group variable: country1                        Number of obs      =       219
Time variable : year2                           Number of groups   =        27
Number of instruments = 26                      Obs per group: min =         1
F(13, 26)     =     40.35                                      avg =      8.11
Prob > F      =     0.000                                      max =        10
------------------------------------------------------------------------------
             |               Robust
      lncopc |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      lncopc |
         L1. |   .8322342   .0623899    13.34   0.000       .70399    .9604784
             |
     lngdppc |  -.0855172    .115503    -0.74   0.466     -.322937    .1519027
     lngdsav |   .1367784    .133576     1.02   0.315     -.137791    .4113478
  lnpopgrate |   .0084767    .018478     0.46   0.650    -.0295054    .0464588
          t2 |  -.0413114   .1707277    -0.24   0.811    -.3922472    .3096244
          t3 |   .0841467   .0787222     1.07   0.295    -.0776691    .2459626
          t4 |   -.036498   .0672219    -0.54   0.592    -.1746747    .1016786
          t5 |  -.1029541   .0717872    -1.43   0.163    -.2505147    .0446065
          t6 |  -.0289868   .0469637    -0.62   0.542     -.125522    .0675484
          t7 |   -.031392   .0274301    -1.14   0.263    -.0877754    .0249913
          t8 |  -.0197568   .0392182    -0.50   0.619     -.100371    .0608573
         t10 |  -.0423617   .0166714    -2.54   0.017    -.0766303   -.0080932
         t11 |  -.1421915   .0230469    -6.17   0.000    -.1895651   -.0948179
       _cons |   .8664065   .8084108     1.07   0.294    -.7953058    2.528119
------------------------------------------------------------------------------
Instruments for first differences equation
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(2/8).(L.lncopc L.lngdppc) collapsed
Instruments for levels equation
  Standard
    t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11
    lngdsav lnpopgrate
    _cons
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -2.37  Pr > z =  0.018
Arellano-Bond test for AR(2) in first differences: z =  -1.09  Pr > z =  0.276
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(12)   =  14.00  Prob > chi2 =  0.300
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(12)   =  11.88  Prob > chi2 =  0.455
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  iv(lngdsav lnpopgrate, eq(level))
    Hansen test excluding group:     chi2(10)   =  10.76  Prob > chi2 =  0.376
    Difference (null H = exogenous): chi2(2)    =   1.12  Prob > chi2 =  0.572
  iv(t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11, eq(level))
    Hansen test excluding group:     chi2(3)    =   3.36  Prob > chi2 =  0.340
    Difference (null H = exogenous): chi2(9)    =   8.52  Prob > chi2 =  0.483


.
Lastly, in some topics I saw that some statalisters use equation() suboption in gmmstyle two different ways. They sometimes use in that way: gmm(x y, eq(level)) gmm(x y, eq(diff)). Why one should use variables in level and diff equation both? How can I distinguish between them?

I would be appreciated in you answer my questions.

Thank you in advance!

Esra.