AR(2) in System GMM xtabond2

[Meisye Alfian]

Hi everyone, I've gone through Roodman's 2009 work and a few discussions on this platform, yet I remain uncertain about whether I understand things correctly.

I am running a one-step system GMM on HDI as dependent var and L.HDI, IDI_1, IDG as independent variables
with the following output:

 xtabond2 HDI L.HDI IDI_1 IDG , gmm ( L.HDI IDI_1 , lag ( 3 4 ) ) iv ( IDG ) small robust artest(3)

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: id                              Number of obs      =       237
Time variable : year                            Number of groups   =        34
Number of instruments = 33                      Obs per group: min =         6
F(3, 33)      =  2.83e+06                                      avg =      6.97
Prob > F      =     0.000                                      max =         7
------------------------------------------------------------------------------
             |               Robust
         HDI | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         HDI |
         L1. |    .968693   .0107476    90.13   0.000     .9468269    .9905591
             |
       IDI_1 |  -.0138277   .0066768    -2.07   0.046    -.0274118   -.0002437
         IDG |    .003112     .00298     1.04   0.304    -.0029509    .0091749
       _cons |   3.617811   .5071562     7.13   0.000     2.585994    4.649628
------------------------------------------------------------------------------
Instruments for first differences equation
  Standard
    D.IDG
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(2/3).(L.HDI IDI_1)
Instruments for levels equation
  Standard
    IDG
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    DL.(L.HDI IDI_1)
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -2.81  Pr > z =  0.005
Arellano-Bond test for AR(2) in first differences: z =  -1.66  Pr > z =  0.097
Arellano-Bond test for AR(3) in first differences: z =  -1.22  Pr > z =  0.224
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(29)   = 172.40  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(29)   =  33.39  Prob > chi2 =  0.262
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  GMM instruments for levels
    Hansen test excluding group:     chi2(18)   =  31.23  Prob > chi2 =  0.027
    Difference (null H = exogenous): chi2(11)   =   2.16  Prob > chi2 =  0.998
  iv(IDG)
    Hansen test excluding group:     chi2(28)   =  33.21  Prob > chi2 =  0.228
    Difference (null H = exogenous): chi2(1)    =   0.19  Prob > chi2 =  0.665

My questions are:
1. AR(2) hypothesis seems to be rejected at 10% level. Would it pose any problem? Because other literature seems to take confidence only when it is not rejected at 10%.
2. The coefficient of HDI.L1 (first lag of HDI) seems to be near 1, does it signify the existence of a unit root? would this be a problem?
3. Is there anything from the result that I should be aware of?

[Meisye Alfian]

I'm sorry, this is the correct code that I use, not using lag 3 and 4 for instrument but lag 2 and 3:

  
 xtabond2 HDI L.HDI IDI_1 IDG , gmm ( L.HDI IDI_1 , lag ( 2 3 ) ) iv ( IDG ) small robust artest(3)

[Sebastian Kripfganz]

1. Whether a p-value of 0.097 provides sufficient confidence against serially correlated errors is up to your judgement. You would need to decide about what probability for a type-I error you are willing to accept; i.e., the significance level. There might be a commonly accepted standard in your literature. This p-value certainly does not provide much comfort. When interpreting those regression results, a word of caution might be in order.

2. Essentially, your dependent variable is almost entirely explained by its own past. This is not necessarily a problem, depending on the research question.

3. 33 groups is very small for this type of estimation. The results may not be very reliable.

[Meisye Alfian]

Thank you so much Prof. Kripfganz , that's really helpful. I'll try to fix my model specifications and add more groups.

Meanwhile, some other questions regarding the old specification to help me understand better the system GMM if you don't mind:

1. Since I was still worried about the unit root problem, I tried to do first-differencing to my dependent variable

 xtabond2 D.HDI L.HDI IDI_1 IDG, gmm ( L.HDI IDI_1 , lag ( 2 3 ) ) iv(IDG) small robust artest(3)

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: id                              Number of obs      =       237
Time variable : year                            Number of groups   =        34
Number of instruments = 33                      Obs per group: min =         6
F(3, 33)      =    194.29                                      avg =      6.97
Prob > F      =     0.000                                      max =         7
------------------------------------------------------------------------------
             |               Robust
       D.HDI | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         HDI |
         L1. |   -.031307   .0107476    -2.91   0.006    -.0531731   -.0094409
             |
       IDI_1 |  -.0138277   .0066768    -2.07   0.046    -.0274118   -.0002437
         IDG |    .003112     .00298     1.04   0.304    -.0029509    .0091749
       _cons |   3.617811   .5071562     7.13   0.000     2.585994    4.649628
------------------------------------------------------------------------------
Instruments for first differences equation
  Standard
    D.IDG
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(2/3).(L.HDI IDI_1)
Instruments for levels equation
  Standard
    IDG
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    DL.(L.HDI IDI_1)
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -2.81  Pr > z =  0.005
Arellano-Bond test for AR(2) in first differences: z =  -1.66  Pr > z =  0.097
Arellano-Bond test for AR(3) in first differences: z =  -1.22  Pr > z =  0.224
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(29)   = 172.40  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(29)   =  33.39  Prob > chi2 =  0.262
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  GMM instruments for levels
    Hansen test excluding group:     chi2(18)   =  31.23  Prob > chi2 =  0.027
    Difference (null H = exogenous): chi2(11)   =   2.16  Prob > chi2 =  0.998
  iv(IDG)
    Hansen test excluding group:     chi2(28)   =  33.21  Prob > chi2 =  0.228
    Difference (null H = exogenous): chi2(1)    =   0.19  Prob > chi2 =  0.665

but I want to make sure if combining D.HDI as dependent variable and lag levels of HDI as independent variable is possible in system GMM? or is it only possible to use the L.D.HDI as the independent variable when I am using D.HDI?

2. Assuming I chose to believe that the errors are serially correlated in second order AR(2) with the significance level of 10%, would then using the lag 3 as instrument be a problem? since there's no serial correlation in third order.