System GMM- Autoregressive Coefficient Greater Than One

[vincent fernandes]

I am using System GMM to examine if an Kuznets curve exists in Non-Communicable Diseases (NCD), I am using the following code:

xtabond2 ln_NCD L.ln_NCD ln_GDP sq_ln_GDP year i.year, gmm(L.ln_NCD ln_GDP sq_ln_GDP, collapse) iv(year i.year, equation(level)) twostep robust artests(4) orthogonal small

The results show that the coefficient of the lagged dependant variable is 1.012***, I am unsure if this result is valid? If I estimate the model using the first difference of ln_NCD all the variables become insignificant and the AR(2) and Hansen nulls can be rejected even after adjusting lag limits.

Some journal articles still report models with the autoregressive coefficient of one, but I am unsure whether the model should be changed and if so how?

Results:

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: country_Num                     Number of obs      =  2272
Time variable : year                            Number of groups   = 142
Number of instruments = 66                      Obs per group: min =16
F(21, 141)    =   3748.85                                      avg = 16.00
Prob > F      =     0.000                                      max = 16
------------------------------------------------------------------------------
             |              Corrected
      ln_NCD |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      ln_NCD |
         L1. |   1.015486   .0083677   121.36   0.000     .9989437   1.032028
             |
      ln_GDP |  -.1577481   .0430502    -3.66   0.000    -.2428555  -.0726407
   sq_ln_GDP |   .0088189   .0024055     3.67   0.000     .0040634   .0135743
        year |  -.0078024    .003255    -2.40   0.018    -.0142372  -.0013675
             |
        year |
       2000  |          0  (empty)
       2001  |  -.1132571   .0488529    -2.32   0.022     -.209836   -.0166782
       2002  |   -.105692   .0455994    -2.32   0.022    -.1958389   -.0155451
       2003  |  -.0982847   .0423466    -2.32   0.022     -.182001   -.0145683
       2004  |  -.0908042   .0390907    -2.32   0.022    -.1680838   -.0135246
       2005  |  -.0830651   .0358306    -2.32   0.022    -.1538998   -.0122304
       2006  |   -.075597   .0325721    -2.32   0.022    -.1399898   -.0112042
       2007  |  -.0682752   .0293124    -2.33   0.021    -.1262238   -.0103267
       2008  |  -.0605534   .0260546    -2.32   0.022    -.1120614   -.0090453
       2009  |  -.0524794   .0228044    -2.30   0.023    -.0975622   -.0073966
       2010  |  -.0450336    .019547    -2.30   0.023    -.0836767   -.0063904
       2011  |  -.0373474   .0162897    -2.29   0.023     -.069551   -.0051438
       2012  |  -.0291999   .0130376    -2.24   0.027    -.0549743   -.0034256
       2013  |  -.0216554   .0097808    -2.21   0.028    -.0409913   -.0023194
       2014  |  -.0145751   .0065174    -2.24   0.027    -.0274597   -.0016906
       2015  |  -.0074117   .0032589    -2.27   0.024    -.0138544   -.0009691
       2016  |          0  (omitted)
             |
       _cons |   16.18347   6.557335     2.47   0.015     3.220066    29.14687
------------------------------------------------------------------------------
Instruments for orthogonal deviations equation
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(1/16).(L.ln_NCD ln_GDP sq_ln_GDP) collapsed
Instruments for levels equation
  Standard
    year 2000b.year 2001.year 2002.year 2003.year 2004.year 2005.year
    2006.year 2007.year 2008.year 2009.year 2010.year 2011.year 2012.year
    2013.year 2014.year 2015.year 2016.year
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    D.(L.ln_NCD ln_GDP sq_ln_GDP) collapsed
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =   2.24  Pr > z =  0.025
Arellano-Bond test for AR(2) in first differences: z =  -1.85  Pr > z =  0.065
Arellano-Bond test for AR(3) in first differences: z =  -1.42  Pr > z =  0.155
Arellano-Bond test for AR(4) in first differences: z =   1.38  Pr > z =  0.168
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(44)   = 559.13  Prob > chi2 = 0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(44)   =  43.95  Prob > chi2 = 0.474
  (Robust, but weakened by many instruments.)

The problem persists in the extended model:

xtabond2 ln_NCD L.ln_NCD ln_GDP sq_ln_GDP ln_the_per_cap_mean D_ln_PollutionPM25 ln_PhysicalActivity ln_HospitalBedsper1000 ln_Pop_over_65 ln_urban_pop ln_trade ln_BMI_WB Human_Capital year i.year, gmm(L.ln_NCD ln_GDP sq_ln_GDP ln_the_per_cap_mean ln_PhysicalActivity ln_HospitalBedsper1000 ln_Pop_over_65 ln_BMI_WB Human_Capital, laglimit(10 15) collapse) iv(ln_PollutionPM25 ln_urban_pop ln_trade year i.year, equation(level)) twostep robust small orthogonal artests(4)

Results:

 Dynamic panel-data estimation, two-step system GMM
------------------------------------------------------------------------------
Group variable: country_Num                     Number of obs      =      2272
Time variable : year                            Number of groups   =       142
Number of instruments = 82                      Obs per group: min =        16
F(30, 141)    =  6.78e+07                                      avg =     16.00
Prob > F      =     0.000                                      max =        16
----------------------------------------------------------------------------------------
                       |              Corrected
                ln_NCD |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-----------------------+----------------------------------------------------------------
                ln_NCD |
                   L1. |   1.000572   .0012721   786.57   0.000     .9980569    1.003086
                       |
                ln_GDP |  -.0581613   .0205383    -2.83   0.005    -.0987641   -.0175584
             sq_ln_GDP |   .0035004   .0011806     2.97   0.004     .0011665    .0058343
   ln_the_per_cap_mean |  -.0010093   .0020293    -0.50   0.620    -.0050211    .0030024
    D_ln_PollutionPM25 |  -.0050512   .0041004    -1.23   0.220    -.0131574     .003055
   ln_PhysicalActivity |   .0018993   .0081889     0.23   0.817    -.0142896    .0180883
ln_HospitalBedsper1000 |    .000579    .005011     0.12   0.908    -.0093273    .0104853
        ln_Pop_over_65 |  -.0193053   .0042337    -4.56   0.000    -.0276751   -.0109355
          ln_urban_pop |   .0056669   .0050194     1.13   0.261    -.0042561    .0155899
              ln_trade |  -.0003482   .0022376    -0.16   0.877    -.0047717    .0040754
             ln_BMI_WB |  -.0288796   .0719827    -0.40   0.689    -.1711844    .1134252
         Human_Capital |   .0000605   .0009432     0.06   0.949    -.0018041    .0019252
                  year |    .000123   .0001329     0.93   0.356    -.0001397    .0003857
                       |
                  year |
                 2000  |          0  (empty)
                 2001  |  -.0002431   .0020365    -0.12   0.905    -.0042691    .0037828
                 2002  |  -.0001769   .0018859    -0.09   0.925    -.0039052    .0035515
                 2003  |  -.0003033    .001754    -0.17   0.863    -.0037708    .0031642
                 2004  |  -.0003709   .0016119    -0.23   0.818    -.0035576    .0028157
                 2005  |  -.0001796   .0014797    -0.12   0.904    -.0031049    .0027457
                 2006  |  -.0001765   .0013895    -0.13   0.899    -.0029234    .0025705
                 2007  |  -.0003638   .0012888    -0.28   0.778    -.0029117     .002184
                 2008  |  -.0001412   .0012165    -0.12   0.908    -.0025461    .0022637
                 2009  |   .0002143   .0009889     0.22   0.829    -.0017407    .0021694
                 2010  |   .0002028   .0008768     0.23   0.817    -.0015306    .0019363
                 2011  |   .0004399    .000819     0.54   0.592    -.0011793    .0020591
                 2012  |    .000613   .0006914     0.89   0.377    -.0007538    .0019798
                 2013  |   .0005916   .0005102     1.16   0.248     -.000417    .0016003
                 2014  |   .0001537   .0003613     0.43   0.671    -.0005605    .0008679
                 2015  |   .0004369   .0004986     0.88   0.382    -.0005488    .0014225
                 2016  |          0  (omitted)
                       |
                 _cons |          0  (omitted)
----------------------------------------------------------------------------------------
Instruments for orthogonal deviations equation
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(10/15).(L.ln_NCD ln_GDP sq_ln_GDP ln_the_per_cap_mean
    ln_PhysicalActivity ln_HospitalBedsper1000 ln_Pop_over_65 ln_BMI_WB
    Human_Capital) collapsed
Instruments for levels equation
  Standard
    ln_PollutionPM25 ln_urban_pop ln_trade year 2000b.year 2001.year 2002.year
    2003.year 2004.year 2005.year 2006.year 2007.year 2008.year 2009.year
    2010.year 2011.year 2012.year 2013.year 2014.year 2015.year 2016.year
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    DL9.(L.ln_NCD ln_GDP sq_ln_GDP ln_the_per_cap_mean ln_PhysicalActivity
    ln_HospitalBedsper1000 ln_Pop_over_65 ln_BMI_WB Human_Capital) collapsed
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =   2.06  Pr > z =  0.040
Arellano-Bond test for AR(2) in first differences: z =  -1.88  Pr > z =  0.060
Arellano-Bond test for AR(3) in first differences: z =  -1.45  Pr > z =  0.147
Arellano-Bond test for AR(4) in first differences: z =   1.33  Pr > z =  0.182
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(51)   =  72.17  Prob > chi2 =  0.027
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(51)   =  50.24  Prob > chi2 =  0.504
  (Robust, but weakened by many instruments.)

[Sebastian Kripfganz]

1. A coefficient of the lagged dependent variable slightly above 1, but not significantly different from 1, can happen in empirical work. It could be a characteristic of the data and does not necessarily imply that there is anything wrong.
2. You cannot separately identify a linear time trend from the time dummies. You should either remove the linear time trend or the time dummies.
3. There is a bug in xtabond2 that produces incorrect degrees of freedom (and therefore incorrect p-values) when (time) dummies are specified with the factor variable notation and some dummies are omitted.
4. There is a bug in xtabond2 that produces incorrect coefficient estimates when the orthogonal option is specified.

To deal with the third and fourth issue, I recommend to use the xtdpdgmm command instead:
XTDPDGMM: new Stata command for GMM estimation of linear dynamic panel models

[amira elshal]

Sebastian Kripfganz Thanks for the useful points. One quick question: What about the long-run effect? If the coefficient of the lagged dependent variable is greater than one, the long-run effect of an explanatory variable of interest will be negative (while it should be positive). Please see below. For example, the long-run effect of a disaster will be calculated as (0.004+0.008)/(1-1.013).

[Sebastian Kripfganz]

A coefficient estimate for the lagged dependent variable that is (significantly) larger than 1 raises serious doubts about the correct specification of the model or estimator. Calculating long-run effects in this situation does not make much sense. In your case, I would suggest to substantially reduce the number of instruments (by restricting lags of the instruments and/or by using collapsed instruments).

[Raymond Zhang]

Hi,@Sebastian,I want to ask a simple question.If the coefficient estimate for the lagged dependent variable is not significant,I mean the P-value is larger than 0.1.Does it indicate that xtabond2 is not appropriate？

Raymond

[Sebastian Kripfganz]

If the coefficient of the lagged dependent variable is not statistically significant, you could estimate a static model instead of a dynamic model. xtabond2/xtdpdgmm can still potentially be used in this case. See: https://www.statalist.org/forums/for...-with-xtabond2