System GMM and AB test for AR(2)

[Connie Gao]

Hi All:

I am using system GMM method to estimate my panel data (T=6). I am new to this method. I know that the assumption of system GMM is the changes in the instrumenting variables are uncorrelated with fixed effect. But I do not know whether system GMM should rely on the AB test for AR(2). My results for AR(2) is significant. If it relies on AR(2), how should I correct it?

The following is my code:

xtabond2 mcs_abs L.mcs_abs i.RWL1 i.RWL1#i.b i.housect age i.Rmarital i.Rchild i.cfreq i.Ghealth1 i.Xoccup i.healthstress i.moneystress i.disease i.Rstate yeardum1-yeardum6, gmmstyle(mcs_abs i.RWL1, laglimits(2 4) collapse eq(level)) gmmstyle(i.RWL1#i.b i.Ghealth1 i.housect i.disease,lag(1 2) eq(level)) ivstyle(age i.Rchild i.Rmarital i.cfreq i.Xoccup i.healthstress i.moneystress i.Rstate yeardum1-yeardum6, equation(level)) twostep robust small orthogonal

The test is following:

Arellano-Bond test for AR(1) in first differences: z =  -9.51  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =   3.54  Pr > z =  0.000
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(121)  = 145.59  Prob > chi2 =  0.063
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(121)  = 124.96  Prob > chi2 =  0.384
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  gmm(lfstfy 1b.RWL1 2.RWL1 3.RWL1 4.RWL1 5.RWL1 6.RWL1 7.RWL1 8.RWL1, collapse eq(level) 
> lag(2 4))
    Hansen test excluding group:     chi2(97)   = 107.38  Prob > chi2 =  0.221
    Difference (null H = exogenous): chi2(24)   =  17.58  Prob > chi2 =  0.823
  iv(age 0b.Rchild 1.Rchild 0b.Rmarital 1.Rmarital 0b.disease 1.disease 2.disease 0b.cfreq
>  1.cfreq 2.cfreq 3.cfreq 1b.Xoccup 2.Xoccup 3.Xoccup 1b.healthstress 2.healthstress 3.he
> althstress 4.healthstress 5.healthstress 1b.moneystress 2.moneystress 3.moneystress 4.mo
> neystress 5.moneystress 1b.Rstate 2.Rstate 3.Rstate 4.Rstate 5.Rstate 6.Rstate yeardum1 
> yeardum2 yeardum3 yeardum4 yeardum5 yeardum6, eq(level))
    Hansen test excluding group:     chi2(93)   = 109.94  Prob > chi2 =  0.111
    Difference (null H = exogenous): chi2(28)   =  15.03  Prob > chi2 =  0.978

Thank you,
Connie

[Sebastian Kripfganz]

You do not seem to be using a system GMM estimator. All of your instruments are specified for the level model, i.e. you are essentially using a level GMM estimator. As you are using lagged differenced of the dependent variable as instruments, you still need absence of serial error correlation for the validity of these instruments. In other words, the rejection of the AR(2) test tends to be a problem. One way to deal with it might be to add further lags of the dependent variable as regressors to proxy for the remaining serial correlation.

As an aside, many of your instruments are specified untransformed for the level model. If you are aware that this requires the assumption that those variables are uncorrelated with the individual-specific "fixed effects", then this does not have to be a problem. This is similar to a classical random-effects assumption. I believe that some of these variables are time-invariant which indeed requires strong assumptions to identify their effects.

More on GMM estimation of dynamic panel models:
XTDPDGMM: new Stata command for efficient GMM estimation of linear (dynamic) panel models with nonlinear moment conditions

More on time-invariant variables in linear dynamic panel models:

• Kripfganz, S. and C. Schwarz (forthcoming). Estimation of Linear Dynamic Panel Data Models with Time-Invariant Regressors. Journal of Applied Econometrics, DOI: 10.1002/jae.2681

[Connie Gao]

Thank you for the comments and references, Sebastian. Actually, I was confused with equation (level) and equation (difference), but I think I know it now.
I have tried serval models based on difference-GMM and system-GMM, could you please help me to take a look at which is better?
1) The first is difference-GMM

xtabond2 lfstfy L.lfstfy i.RWL1 i.RWL1#i.b i.housect age i.Rmarital i.Rchild i.cfreq i.Ghealth1 i.Xoccup i.disease i.healthstress i.moneystress  i.Rstate yeardum2-yeardum6, gmmstyle(lfstfy i.RWL1, laglimits(2 4) collapse) gmmstyle(i.RWL1#i.b i.housect age i.Rmarital i.Rchild i.cfreq i.Ghealth1 i.Xoccup i.disease i.healthstress i.moneystress,lag(1 .)) ivstyle(i.Rstate yeardum2-yeardum6, equation(level)) robust small orthogonal noleveleq

The test results

------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z = -10.82  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =   0.80  Pr > z =  0.425
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(492)  = 530.84  Prob > chi2 =  0.110
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(492)  = 489.33  Prob > chi2 =  0.525
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  gmm(lfstfy 1b.RWL1 2.RWL1 3.RWL1 4.RWL1 5.RWL1 6.RWL1 7.RWL1 8.RWL1, collapse lag(2
> 4))
    Hansen test excluding group:     chi2(468)  = 469.12  Prob > chi2 =  0.477
    Difference (null H = exogenous): chi2(24)   =  20.21  Prob > chi2 =  0.685

2) The second is system-GMM with equation (diff)

xtabond2 lfstfy L.lfstfy i.RWL1 i.RWL1#i.b i.housect age i.Rmarital i.Rchild i.cfreq i.Ghealth1 i.Xoccup i.disease i.healthstress i.moneystress  i.Rstate yeardum2-yeardum6, gmmstyle(lfstfy i.RWL1, laglimits(2 4) collapse eq(diff)) gmmstyle(i.RWL1#i.b i.housect age i.Rmarital i.Rchild i.cfreq i.Ghealth1 i.Xoccup i.disease i.healthstress i.moneystress,lag(1 .) eq(diff)) ivstyle(i.Rstate yeardum2-yeardum6, equation(level)) twostep robust small orthogonal

The test results

Arellano-Bond test for AR(1) in first differences: z =  -8.81  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =   0.70  Pr > z =  0.484
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(501)  = 550.50  Prob > chi2 =  0.062
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(501)  = 499.75  Prob > chi2 =  0.507
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  gmm(lfstfy 1b.RWL1 2.RWL1 3.RWL1 4.RWL1 5.RWL1 6.RWL1 7.RWL1 8.RWL1, collapse eq(dif
> f) lag(2 4))
    Hansen test excluding group:     chi2(477)  = 477.30  Prob > chi2 =  0.487
    Difference (null H = exogenous): chi2(24)   =  22.44  Prob > chi2 =  0.553
  iv(1b.Rstate 2.Rstate 3.Rstate 4.Rstate 5.Rstate 6.Rstate yeardum2 yeardum3 yeardum4
>  yeardum5 yeardum6, eq(level))
    Hansen test excluding group:     chi2(492)  = 492.11  Prob > chi2 =  0.490
    Difference (null H = exogenous): chi2(9)    =   7.63  Prob > chi2 =  0.572

3) The third one system GMM without equation option in pre-determined gmmstyle

xtabond2 lfstfy L.lfstfy i.RWL1 i.RWL1#i.b i.housect age i.Rmarital i.Rchild i.cfreq i.Ghealth1 i.Xoccup i.disease i.Rstate yeardum2-yeardum6, gmmstyle(lfstfy i.RWL1, laglimits(2 5) collapse eq(diff)) gmmstyle(i.RWL1#i.b i.housect age i.Rmarital i.Rchild i.cfreq i.Ghealth1 i.Xoccup i.disease,lag(1 .)) ivstyle(i.Rstate yeardum2-yeardum6, equation(level)) twostep robust small orthogonal

Arellano-Bond test for AR(1) in first differences: z =  -9.83  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =   2.33  Pr > z =  0.020
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(550)  = 800.54  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(550)  = 499.36  Prob > chi2 =  0.940
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  GMM instruments for levels
    Hansen test excluding group:     chi2(400)  = 353.14  Prob > chi2 =  0.956
    Difference (null H = exogenous): chi2(150)  = 146.22  Prob > chi2 =  0.572
  gmm(lfstfy 1b.RWL1 2.RWL1 3.RWL1 4.RWL1 5.RWL1 6.RWL1 7.RWL1 8.RWL1, collapse eq(dif
> f) lag(2 5))
    Hansen test excluding group:     chi2(520)  = 479.66  Prob > chi2 =  0.897
    Difference (null H = exogenous): chi2(30)   =  19.69  Prob > chi2 =  0.924
  iv(1b.Rstate 2.Rstate 3.Rstate 4.Rstate 5.Rstate 6.Rstate yeardum2 yeardum3 yeardum4
>  yeardum5 yeardum6, eq(level))
    Hansen test excluding group:     chi2(542)  = 492.04  Prob > chi2 =  0.939
    Difference (null H = exogenous): chi2(8)    =   7.32  Prob > chi2 =  0.503

I also tried not specifying equation option in both gmmstyle, but AR(2) fails in this contest.

I prefer model 2), but I don't why AR(2) is sensitive to whether or not adding equation (diff) in gmmstyle. If I add eq(diff) in gmmstyle, does that mean model 2) is the same as model 1)?

Thank you,

Connie

[Sebastian Kripfganz]

In the first model, you are specifying the option noleveleq but also the suboption equation(level) for the last set of instruments. This combination does not seem to make sense.

In all specifications, the number of instruments is quite high. This might potentially yield a too-many-instruments problem.

Given the output you show us, it is hardly possible to make any judgement about which model to prefer.

I do not have a straightforward answer regarding the sensitivity of the AR(2) test.

[Connie Gao]

Thank you, Sebastian! I will narrow down the number of IVs.