System GMM xtabond2

[Fillipe Soares]

Dear all,

I'm currently investigating the relationship between carbon emissions and socioeconomic variables for Brazilian states. My data has 27 individuals (states) and 6 years (5-year interval 1990-2015). According to the literature review my regressors are likely to be endogenous then I've decided to run the system GMM estimator by using the program xtabond2. However, as a new Stata user, I'm not convinced if I'm doing it correctly due to the unusual output I have.

My variables are: I = carbon emissions, P = population, Arpc = per capita GDP and T = technology. I've also included year fixed effects.

xtabond2 L(0/1).I P Arpc T i.AnoStata, gmmstyle( L.(I P Arpc T), laglimits(1 2) collapse equation(diff)) gmmstyle( L.(I P Arpc T
> ), laglimits(0 0) collapse eq(level)) ivstyle(i.AnoStata, eq(level)) twostep robust

Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.
Warning: Two-step estimated covariance matrix of moments is singular.
  Using a generalized inverse to calculate optimal weighting matrix for two-step estimation.
  Difference-in-Sargan/Hansen statistics may be negative.

Dynamic panel-data estimation, two-step system GMM

Group variable: Ufs                             Number of obs      =       135
Time variable : AnoStata                        Number of groups   =        27
Number of instruments = 17                      Obs per group: min =         5
Wald chi2(10) =  2.00e+06                                      avg =      5.00
Prob > chi2   =     0.000                                      max =         5

Corrected
I       Coef.   Std. Err.      z    P>z     [95% Conf. Interval]

I
L1.    .7259731    .206855     3.51   0.000     .3205448    1.131401
            
P    .2356326   .2079675     1.13   0.257    -.1719761    .6432413
Arpc      .08098   .2533465     0.32   0.749    -.4155699      .57753
T     .051556   .2607521     0.20   0.843    -.4595087    .5626208
            
Ano
1990            0  (empty)
1995      1.93051   1.654896     1.17   0.243    -1.313027    5.174047
2000     2.019052   1.712828     1.18   0.238    -1.338029    5.376133
2005      1.83546   1.831226     1.00   0.316    -1.753678    5.424598
2010     1.981132   1.858962     1.07   0.287    -1.662367    5.624631
2015     1.997801   1.917377     1.04   0.297    -1.760188     5.75579
            
_cons           0  (omitted)

Instruments for first differences equation
GMM-type (missing=0, separate instruments for each period unless collapsed)
L(1/2).(L.I L.P L.Arpc L.T) collapsed
Instruments for levels equation
Standard
1990b.Ano 1995.Ano 2000.Ano 2005.Ano 2010.Ano 2015.Ano
_cons
GMM-type (missing=0, separate instruments for each period unless collapsed)
D.(L.I L.P L.Arpc L.T) collapsed

Arellano-Bond test for AR(1) in first differences: z =  -2.23  Pr > z =  0.026
Arellano-Bond test for AR(2) in first differences: z =   1.97  Pr > z =  0.049

Sargan test of overid. restrictions: chi2(6)    =  23.71  Prob > chi2 =  0.001
(Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(6)    =   7.12  Prob > chi2 =  0.310
(Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
GMM instruments for levels
Hansen test excluding group:     chi2(2)    =   4.11  Prob > chi2 =  0.128
Difference (null H = exogenous): chi2(4)    =   3.01  Prob > chi2 =  0.557
gmm(L.I L.P L.Arpc L.T, collapse eq(level) lag(0 0))
Hansen test excluding group:     chi2(2)    =   4.11  Prob > chi2 =  0.128
Difference (null H = exogenous): chi2(4)    =   3.01  Prob > chi2 =  0.557
iv(1990b.Ano 1995.Ano 2000.Ano 2005.Ano 2010.Ano 2015.Ano, eq(level))
Hansen test excluding group:     chi2(2)    =   5.94  Prob > chi2 =  0.051
Difference (null H = exogenous): chi2(4)    =   1.18  Prob > chi2 =  0.882

According to the above output, we can't reject the null hypothesis of the validity of the overidentifying restrictions, the values reported for the Diff-in-Hansen test are the p-values for the validity of the additional moment restrictions necessary for system GMM. Again, we do not reject the null that the additional moment conditions are valid. On the other hand, there is evidence for second-order autocorrelation and my lagged dependent variable is the only regressor which has statistical significance.


All comments and suggestions are very welcome and valuable.


Best regards,

[Sebastian Kripfganz]

xtabond2 computes the degrees of freedom for the overidentification tests incorrectly if you have omitted (or empty) coefficients in the regression output. This usually happens when you specify time effects with the factor variable notation. xtabond2 reports 6 degrees of freedom while it should be 8 (= 17 instruments - 9 coefficients). This obviously affects the p-values.

To solve the problem, you should create manual time dummies first and then run the regression with these separate time dummies for the years 2000 to 2015. Note that you need to remove one additional dummy besides the base year 1990 because of the dynamic nature of the model. This is why your constant term is omitted.