st: Failing to reject Arellano-Bond test for AR(1) in first differences imply a random walk?

[Evan Markel <markelev@gmail.com>]

Dear listservers,

I am estimating an xtabond2 model using a panel where N=434
microfinance institutions (MFI's) and where T=5. After executing
xtabond2 system GMM this reduces to N=233 and T=3.

My primary concern right now is the implication of failing to reject
the null hypothesis of no autocorrelation in the Arellano-Bond test
for AR(1). I have read Roodman 2006 and understand that negative first
order serial correlation is to be expected in AR(1) because of the
mathematical relation between the first difference and the first lag
of difference.

Specifically (eit)-(eit-1) is related to (eit-1)-(eit-2). But what
does this mean when I fail to reject the null. Does it give evidence
of a random walk? I have found several examples and discussions
stating why first order serial correlation is expected in AR(1), but
have not found a discussion on the implications of the contrary.

 Below is an extract from my xtabond2 output.

Instruments for orthogonal deviations equation
  Standard
    FOD.(iyear L2.pm)
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(1/5).(oss lindi lsolid lvillage)
Instruments for levels equation
  Standard
    iyear L2.pm
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    D.(oss lindi lsolid lvillage)
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -1.00  Pr > z =  0.315
Arellano-Bond test for AR(2) in first differences: z =   0.10  Pr > z =  0.917
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(31)   =  87.67  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(31)   =  19.86  Prob > chi2 =  0.939
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  GMM instruments for levels
    Hansen test excluding group:     chi2(19)   =  12.54  Prob > chi2 =  0.861
    Difference (null H = exogenous): chi2(12)   =   7.32  Prob > chi2 =  0.836
  iv(iyear L2.pm)
    Hansen test excluding group:     chi2(29)   =  16.57  Prob > chi2 =  0.968
    Difference (null H = exogenous): chi2(2)    =   3.29  Prob > chi2 =  0.193

My variable of interest is Operational Self-sufficiency (OSS).
Following Roodman 2009 I regress OSS on the lagged level of OSS which
yields a low R^2 of .117 possibly signally lag 1.oss is a weak
instrument. Maybe this is a source of my failure to reject AR(1)?

regress oss l.oss

      Source |       SS                df            MS
                  Number of obs =     784
-------------+------------------------------
                           F(  1,   782) =  103.65
       Model |    19.0811599        1      19.0811599
           Prob > F      =  0.0000
    Residual |   143.966445       782   .184100313
        R-squared     =  0.1170
-------------+------------------------------
                           Adj R-squared =  0.1159
       Total |      163.047605       783     .208234489
           Root MSE      =  .42907

------------------------------------------------------------------------------
         oss |      Coef.           Std. Err.      t    P>|t|     [95%
Conf. Interval]
-------------+----------------------------------------------------------------
         oss |
         L1. |     .3328396   .0326934    10.18   0.000     .2686624    .3970168
             |
       _cons |   .7341655   .0399425    18.38   0.000     .6557582    .8125727


Thank you in advance for your help,

Evan M.
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