2-step weighting matrix vs 3-step weighting matrix from xtdpdgmm command

[Abraham Risyad]

Dear Statalist,

Currently, I am researching the impact of branding on the return of capital employed (ROCE). I suspect a dynamic nature in my unbalanced panel data and an endogeneity issue in the branding variable. Therefore, I use a two-step system GMM for my analysis. I used the xtdpdgmm command to run my regression, and the post-estimation result showed two different Hansen test results that confused me. The number of moment conditions from my regression result is 98, and I suspect this number also indicates the number of instruments used. This number is still below the number of banks in my dataset, which is 114 banks. The Hansen test result for 2-step moment functions with 2-step weighting matrix is insignificant. However, if I use the Hansen test result with 3-step weighting matrix, it is significant.

Can anyone please enlighten me on the difference between the 2-step and 3-step weighting matrix from the xtdpdgmm post-estimation result? Also, for my case here, should I rely on the result from the 2-step or 3-step weighting matrix, or should I rely simultaneously on both of them?

. xtdpdgmm ROCE L_ROCE $control, gmm(L_ROCE, lag(2 3)) gmm(DB_Branding, lag(1 2)) two vce(r) teffect
note: 254.Date omitted because of collinearity.

Generalized method of moments estimation

Fitting full model:
Step 1         f(b) =   77.04881
Step 2         f(b) =  .53330455

Group variable: Sandi                        Number of obs         =      2615
Time variable: Date                          Number of groups      =       114

Moment conditions:     linear =      98      Obs per group:    min =         4
                    nonlinear =       0                        avg =   22.9386
                        total =      98                        max =        24

                                (Std. err. adjusted for 114 clusters in Sandi)
------------------------------------------------------------------------------
             |              WC-Robust
        ROCE | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
      L_ROCE |   .4208216   .1976822     2.13   0.033     .0333715    .8082716
 DB_Branding |  -11.13101   3.905712    -2.85   0.004    -18.78607   -3.475956
        Size |  -.0309021    1.78705    -0.02   0.986    -3.533456    3.471651
         CAR |  -.0035476   .0093652    -0.38   0.705    -.0219029    .0148078
         NPL |   -2.08351   .8617056    -2.42   0.016    -3.772422   -.3945978
         LDR |  -.0022799   .0265356    -0.09   0.932    -.0542888    .0497289
         NIM |   1.385492    .510181     2.72   0.007      .385556    2.385429
Business_Mix |  -.1251238   .1944438    -0.64   0.520    -.5062266     .255979
   Inflation |   2.219998   .9313944     2.38   0.017     .3944988    4.045498
  GDP_Growth |  -.2106766   .1346259    -1.56   0.118    -.4745386    .0531854
             |
        Date |
        233  |  -1.350172   .9389638    -1.44   0.150    -3.190508    .4901628
        234  |  -.4392329   1.004881    -0.44   0.662    -2.408763    1.530297
        235  |  -2.500202   1.082535    -2.31   0.021    -4.621932   -.3784727
        236  |   .2346466   1.073901     0.22   0.827     -1.87016    2.339453
        237  |  -2.125878   1.044461    -2.04   0.042    -4.172983   -.0787731
        238  |  -1.862617   .8534568    -2.18   0.029    -3.535361   -.1898722
        239  |  -.9545664   .5586313    -1.71   0.087    -2.049464    .1403309
        240  |   .1372608   1.360941     0.10   0.920    -2.530135    2.804657
        241  |          0  (empty)
        242  |   1.213023   .6496135     1.87   0.062    -.0601956    2.486243
        243  |   .3086538   .7763399     0.40   0.691    -1.212944    1.830252
        244  |   2.849401   .9740243     2.93   0.003      .940348    4.758453
        245  |    3.97898   1.916145     2.08   0.038     .2234046    7.734556
        246  |    2.64422   .9373806     2.82   0.005     .8069881    4.481453
        247  |   1.275293   1.017346     1.25   0.210    -.7186695    3.269255
        248  |   1.940281   .9799561     1.98   0.048     .0196022    3.860959
        249  |  -3.595698   1.815245    -1.98   0.048    -7.153512   -.0378827
        250  |  -7.201618   3.259837    -2.21   0.027    -13.59078   -.8124556
        251  |  -7.300011   2.911381    -2.51   0.012    -13.00621   -1.593809
        252  |  -4.884409   2.438686    -2.00   0.045    -9.664147   -.1046724
        253  |  -2.552789   1.104793    -2.31   0.021    -4.718143   -.3874359
        254  |          0  (empty)
        255  |  -1.380236    .455906    -3.03   0.002    -2.273795   -.4866764
             |
       _cons |   .8589907   20.95298     0.04   0.967     -40.2081    41.92608
------------------------------------------------------------------------------
Instruments corresponding to the linear moment conditions:
 1, model(level):
   234:L2.L_ROCE 235:L2.L_ROCE 236:L2.L_ROCE 237:L2.L_ROCE 238:L2.L_ROCE
   239:L2.L_ROCE 240:L2.L_ROCE 241:L2.L_ROCE 242:L2.L_ROCE 243:L2.L_ROCE
   244:L2.L_ROCE 245:L2.L_ROCE 246:L2.L_ROCE 247:L2.L_ROCE 248:L2.L_ROCE
   249:L2.L_ROCE 250:L2.L_ROCE 251:L2.L_ROCE 252:L2.L_ROCE 253:L2.L_ROCE
   254:L2.L_ROCE 255:L2.L_ROCE 235:L3.L_ROCE 236:L3.L_ROCE 237:L3.L_ROCE
   238:L3.L_ROCE 239:L3.L_ROCE 240:L3.L_ROCE 241:L3.L_ROCE 242:L3.L_ROCE
   243:L3.L_ROCE 244:L3.L_ROCE 245:L3.L_ROCE 246:L3.L_ROCE 247:L3.L_ROCE
   248:L3.L_ROCE 249:L3.L_ROCE 250:L3.L_ROCE 251:L3.L_ROCE 252:L3.L_ROCE
   253:L3.L_ROCE 254:L3.L_ROCE 255:L3.L_ROCE
 2, model(level):
   233:L1.DB_Branding 234:L1.DB_Branding 235:L1.DB_Branding 236:L1.DB_Branding
   237:L1.DB_Branding 238:L1.DB_Branding 239:L1.DB_Branding 240:L1.DB_Branding
   241:L1.DB_Branding 242:L1.DB_Branding 243:L1.DB_Branding 244:L1.DB_Branding
   245:L1.DB_Branding 246:L1.DB_Branding 247:L1.DB_Branding 248:L1.DB_Branding
   249:L1.DB_Branding 250:L1.DB_Branding 251:L1.DB_Branding 252:L1.DB_Branding
   253:L1.DB_Branding 254:L1.DB_Branding 255:L1.DB_Branding 242:L2.DB_Branding
   243:L2.DB_Branding 245:L2.DB_Branding 246:L2.DB_Branding 247:L2.DB_Branding
   248:L2.DB_Branding 253:L2.DB_Branding 254:L2.DB_Branding
 3, model(level):
   233bn.Date 234.Date 235.Date 236.Date 237.Date 238.Date 239.Date 240.Date
   241.Date 242.Date 243.Date 244.Date 245.Date 246.Date 247.Date 248.Date
   249.Date 250.Date 251.Date 252.Date 253.Date 254.Date 255.Date
 4, model(level):
   _cons

. estat overid

Sargan-Hansen test of the overidentifying restrictions
H0: overidentifying restrictions are valid

2-step moment functions, 2-step weighting matrix       chi2(66)    =   60.7967
                                                       Prob > chi2 =    0.6580

2-step moment functions, 3-step weighting matrix       chi2(66)    =   94.3566
                                                       Prob > chi2 =    0.0126

. estat serial

Arellano-Bond test for autocorrelation of the first-differenced residuals
H0: no autocorrelation of order 1      z =   -3.2123   Prob > |z|  =    0.0013
H0: no autocorrelation of order 2      z =   -0.1084   Prob > |z|  =    0.9137

Sincerely,
Abraham

[Sebastian Kripfganz]

While the number of instruments (moment conditions) is smaller than the number of groups, it is still relatively high. (There is unfortunately no general threshold for deciding about how many instruments are too many.) With this many instruments, estimating the optimal weighting matrix can be difficult. I believe, this is what happened here. The weighting matrix is imprecisely estimated and therefore differs substantially when computed in the second step (based on first-step residuals) or the third step (based on second-step residuals). Put differently, this difference in the overidentification tests is simply an indication about the problems with estimating the optimal weighting matrix. If the weighting matrix was estimated with sufficient precision, there should not be any relevant difference in these two tests.

My recommendation would be to use the collapse option, which should bring down the number of instruments substantially.

[Abraham Risyad]

Dear Sebastian,

Thank you so much for your insight. I have tried to use the collapse option, and the number of instruments has reduced significantly.
However, another problem emerges is the estat overid post-estimation command somehow cannot calculate the Hansen test.

. estat overid

Sargan-Hansen test of the overidentifying restrictions
H0: overidentifying restrictions are valid

2-step moment functions, 2-step weighting matrix       chi2(0)     =    0.0000
note: coefficients are just-identified                 Prob > chi2 =         .

2-step moment functions, 3-step weighting matrix       chi2(0)     =    0.0000
note: coefficients are just-identified                 Prob > chi2 =         .

[Sebastian Kripfganz]

Now you do not have enough instruments. You should also choose appropriate instruments for your $control variables. In your initial example, you only included instruments for DB_Branding, but none of the others. This also applies if those variables are exogenous. You might want to have a look at the xtdpdgmmfe command, which is also part of the xtdpdgmm package and provides a potentially simpler syntax:

help xtdpdgmmfe