Dynamic panel data model with xtabond2/ivreg2/Difference-in-Hansen test

[Ferdinand Springer]

Dear all,
I am using an unbalanced panel data with T=11 and N=200,000 to estimate a dynamic panel data model with the xtabond2 command (with Stata 14.1).

So far, I have assumed that the lag of my dependent variable (L.y) is endogenous and all of my other control variables are exogenous. When I estimate the model using xtabond2 and a two-step system GMM with the underlying assumptions, the coefficients are alright but the AR(2)-test is pretty weak and the Hansen-test doesn’t show the desired result. I dropped the first two years manually and I used the suboption orthogonal since I have gaps in my data.
Do you think that my specification in xtabond2 is correct or did I make some mistakes that I didnt’t recognize so far?

xtabond2 y L.y x1 x2 x3 x4 x5 x6 x7 yr3-yr11, twostep robust orthogonal ///
> gmm(L.y, lag(2 6) equation(both)) ///
> iv(x1 x2 x2 x4 x5 x6 x7, equation(both)) ///
> iv(yr3-yr11, equation(level))
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 ------------------------------------------------------------------------------
             |              Corrected
           y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           y |
         L1. |   .9530541   .0045813   208.03   0.000     .9440749    .9620332
             |
          x1 |   .0009905   .0030752     0.32   0.747    -.0050369    .0070178
          x2 |   -.001352   .0025792    -0.52   0.600    -.0064072    .0037032
          x3 |   .0043922   .0084974     0.52   0.605    -.0122625    .0210468
          x4 |  -.0003227   .0000352    -9.18   0.000    -.0003917   -.0002538
          x5 |  -.0012368   .0007791    -1.59   0.112    -.0027639    .0002903
          x6 |   .0751949   .0200101     3.76   0.000     .0359758    .1144139
          x7 |  -.0035094   .0006449    -5.44   0.000    -.0047734   -.0022453
         yr3 |   .3537909   .0098877    35.78   0.000     .3344114    .3731705
         yr4 |   .3586882   .0052821    67.91   0.000     .3483354     .369041
         yr5 |   .1584298   .0051346    30.86   0.000     .1483661    .1684935
         yr6 |   .1583574   .0041391    38.26   0.000     .1502449    .1664698
         yr7 |   .1061737   .0038656    27.47   0.000     .0985972    .1137502
         yr8 |   .1504396   .0037814    39.78   0.000     .1430281     .157851
         yr9 |   .1798849   .0034038    52.85   0.000     .1732137    .1865561
       yr10 |   .1689228   .0034978    48.29   0.000     .1620672    .1757784
       yr11 |    .0115624   .0032083     3.60   0.000     .0052743    .0178506
    _cons |    .1823155   .0733113     2.49   0.013      .038628    .3260029
 ------------------------------------------------------------------------------
Instruments for orthogonal deviations equation
  Standard
    FOD.(x1 x2 x2 x4 x5 x6 x7)
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(2/6).L.y
Instruments for levels equation
  Standard
    yr3 yr4 yr5 yr6 yr7 yr8 yr9 yr10 y11
    x1 x2 x2 x4 x5 x6 x7
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    DL.L.y
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z = -58.80  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =  -1.57  Pr > z =  0.117
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(36)   = 313.40  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(36)   = 292.69  Prob > chi2 =  0.000
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  GMM instruments for levels
    Hansen test excluding group:     chi2(28)   = 277.21  Prob > chi2 =  0.000
    Difference (null H = exogenous): chi2(8)    =  15.48  Prob > chi2 =  0.050
  iv(x1 x2 x2 x4 x5 x6 x7)
    Hansen test excluding group:     chi2(30)   = 275.27  Prob > chi2 =  0.000
    Difference (null H = exogenous): chi2(6)    =  17.42  Prob > chi2 =  0.008
  iv(yr3 yr4 yr5 yr6 yr7 yr8 yr9 yr10 yr11, eq(level))
    Hansen test excluding group:     chi2(27)   = 188.63  Prob > chi2 =  0.000
    Difference (null H = exogenous): chi2(9)    = 104.06  Prob > chi2 =  0.000

When I specify my iv’s separately, I get similar results for the coefficients but different results for the Difference-in-Hansen test:

Difference-in-Hansen tests of exogeneity of instrument subsets:
  GMM instruments for levels
    Hansen test excluding group:     chi2(29)   = 279.04  Prob > chi2 =  0.000
    Difference (null H = exogenous): chi2(8)    =  14.31  Prob > chi2 =  0.074
  iv(x1)
    Hansen test excluding group:     chi2(36)   = 292.79  Prob > chi2 =  0.000
    Difference (null H = exogenous): chi2(1)    =   0.56  Prob > chi2 =  0.454
  iv(x2)
    Hansen test excluding group:     chi2(36)   = 292.54  Prob > chi2 =  0.000
    Difference (null H = exogenous): chi2(1)    =   0.81  Prob > chi2 =  0.368
  iv(x3)
    Hansen test excluding group:     chi2(36)   = 292.69  Prob > chi2 =  0.000
    Difference (null H = exogenous): chi2(1)    =   0.66  Prob > chi2 =  0.417
  iv(x4)
    Hansen test excluding group:     chi2(36)   = 293.14  Prob > chi2 =  0.000
    Difference (null H = exogenous): chi2(1)    =   0.21  Prob > chi2 =  0.643
  iv(x5)
    Hansen test excluding group:     chi2(36)   = 293.26  Prob > chi2 =  0.000
    Difference (null H = exogenous): chi2(1)    =   0.09  Prob > chi2 =  0.766
  iv(x6)
    Hansen test excluding group:     chi2(36)   = 293.26  Prob > chi2 =  0.000
    Difference (null H = exogenous): chi2(1)    =   0.10  Prob > chi2 =  0.757
  iv(x7)
    Hansen test excluding group:     chi2(36)   = 293.35  Prob > chi2 =  0.000
    Difference (null H = exogenous): chi2(1)    =   0.00  Prob > chi2 =  0.982
  iv(yr3 yr4 yr5 yr6 yr7 yr8 yr9 yr10 yr11, eq(level))
    Hansen test excluding group:     chi2(28)   = 188.71  Prob > chi2 =  0.000
    Difference (null H = exogenous): chi2(9)    = 104.64  Prob > chi2 =  0.000

I am struggling with the correct interpretation of the Difference-in-Hansen test since the 'Hansen test excluding group' is always highly rejected whereas the 'Difference' test is not.

I am also not sure if all of my controls are really exogenous or if they're predetermined or even endogenous.
Can I use the ivreg2 command to test if my controls are exogenous? So for example:

xi: ivreg2 y L.y x2 x3 x4 x5 x6 x7 (x1=L.x1) i.yr, gmm2s robust cluster(ID) endogtest(x1)

Do you have any suggestions how I could fix my regression command so that the AR(2) and Hansen test show the desired results?
Thanks a lot in advance for your help, any help is highly appreciated.

Kind regards,
Ferdi

[Sebastian Kripfganz]

I have ongoing debates with other people whether iv(x1 x2 x2 x4 x5 x6 x7, equation(both)) is a reasonable specification. You will notice that this is not the same as iv(x1 x2 x2 x4 x5 x6 x7, equation(diff)) iv(x1 x2 x2 x4 x5 x6 x7, equation(level)). If you know what the first version is doing and this is what you want it to do, then there is nothing to worry about. If you are suprised that the two specifications differ, then probably the second version is what you actually want to get. This might also explain some of the differences you observe for the Difference-in-Hansen test. (Apparently, your second model has one additional instrument for whatever reason.)

Regarding the Hansen test, I would worry primarily whether the level instruments are valid. Notice that you are assuming that all of your variables x1 x2 x2 x4 x5 x6 x7 are not just strictly exogenous with respect to the idiosyncratic error term but also are uncorrelated with the unobserved unit-specific effects. This is a very strong assumption (equivalent to a random-effects assumption). In addition, the instruments for the lagged dependent variable used for the level model might be invalid if mean stationarity is not satisfied. In that case, a difference-GMM estimator might be preferable and help to obtain a better outcome of the Hansen test.

To deal with the low p-value of the Arellano-Bond AR(2) test, it might help to add additional lags of the dependent variable as regressors as a means to capture any remaining serial correlation.