xtabond2, differenced dependent variable

[Bjoern Brey]

Hello all,

I wanted to estimate a model with a differenced dependent variable in STATA using the xtabond2 command, as the model is dynamic I would include a first differenced lag of the dependent variable on the right hand side.

The standard specification and explanations for xtabond only use non-differenced dependent variables, so is it possible to have a differenced dependent variable (plus lags) in xtabond2?
If yes what would be the correct specification for the gmm?
gmmstyle(LD.dependent_variable, laglimits (2 .) collapse),
gmmstyle(L.dependent_variable, laglimits (2 .),
gmmstyle(L2.dependent_variable, laglimits (2 .)
or something completely different?

My dataset is N=30, T=16; which I know is fairly short N & long T for arellano bond estimation but it is the only data available for analysing the question.

Thank you very much for any comment and help.

[Sebastian Kripfganz]

Let me first ask: What is your reason for specifying a dynamic model completely in first differences?

Note the following equivalence:
\[ y_{it} = \lambda y_{i,t-1} + \beta x_{it} + \alpha_i + u_{it} \]
\[ \Delta y_{it} = (\lambda - 1) y_{i,t-1} + \beta x_{it} + \alpha_i + u_{it} \]

The underlying model is the same in both specifications. You can interpret the coefficients \( \beta \) as marginal effects of \( x_{it} \) on the differenced dependent variable \( \Delta y_{it} \) no matter whether you actually estimate the first or the second equation. The valid instruments for the right-hand side variables are the same. You will even get the identical coefficient estimates for \( \beta \).

[Andrew Musau]

Bjoern: xtabond2 takes care of the fixed effect if you enter your variables at level (I assume that the reason you want to difference prior to estimation is to eliminate the fixed effect). Consider the dynamic panel model

y_it = α + γy_it-1+ βx_it + η_i +u_it

Eliminating the fixed effect η_i by differencing

Δy_i2 = y_{i2 -} y_i1= γ(y_i1 - y_i0)+ β (x_{i2 -} x_i1)+ (u_i2 - u_i1)

reduces to

Δy_i2 = γΔy_i1 + βΔx_i2 + Δu_i2 (Eq. 1)
__________________________________________________

In Eq. 1, Δy_i1 depends on Δu_i1 and therefore we cannot condition on the lagged dependent variable when the T dimension is small. Intuitively, the problem arises because of the difficulty in distinguishing between fixed effects and lagged dependence - an individual can have a high value of y either because he has a high fixed effect η or because his values of y were high in the past. With only a few time series observations, one cannot distinguish between these two effects (a large N only increases the number of problematic cases).

Now, if we were to advance and use your logic of first differencing, then estimating the equation - we can just consider Δy_i1 as endogenous and use an estimator such as 2SLS (the example below uses abdata.dta available online)

webuse abdata
ivregress 2sls D.n (D.nL1= nL2) D.(nL2 w wL1 k kL1 kL2 ys ysL1 ysL2 yr1979 emp yr1980 yr1981 yr1982 yr1983)

Instrumental variables (2SLS) regression               Number of obs =     611
                                                       Wald chi2(16) =  539.01
                                                       Prob > chi2   =  0.0000
                                                       R-squared     =  0.4571
                                                       Root MSE      =  .10581

------------------------------------------------------------------------------
         D.n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         nL1 |
         D1. |   .3498306   .9820426     0.36   0.722    -1.574938    2.274599
             |
         nL2 |
         D1. |  -.0637868   .0866018    -0.74   0.461    -.2335232    .1059496
             |
           w |
         D1. |  -.5473248   .1294125    -4.23   0.000    -.8009686    -.293681
             |
         wL1 |
         D1. |   .2562104   .5863569     0.44   0.662     -.893028    1.405449
             |
           k |
         D1. |    .340475   .0668565     5.09   0.000     .2094387    .4715113
             |
         kL1 |
         D1. |   .0568763   .3060622     0.19   0.853    -.5429946    .6567472
             |
         kL2 |
         D1. |   .0165387   .1181213     0.14   0.889    -.2149748    .2480521
             |
          ys |
         D1. |   .5695901   .2240617     2.54   0.011     .1304373    1.008743
             |
        ysL1 |
         D1. |   -.506269   .7151837    -0.71   0.479    -1.908003    .8954653
             |
        ysL2 |
         D1. |   .0277203   .2462848     0.11   0.910    -.4549891    .5104297
             |
      yr1979 |
         D1. |   .0090343   .0213699     0.42   0.672      -.03285    .0509186
             |
         emp |
         D1. |   .0160809   .0026665     6.03   0.000     .0108546    .0213072
             |
      yr1980 |
         D1. |   .0233671   .0297529     0.79   0.432    -.0349476    .0816817
             |
      yr1981 |
         D1. |    .004924   .0242929     0.20   0.839    -.0426893    .0525372
             |
      yr1982 |
         D1. |  -.0074509   .0241174    -0.31   0.757    -.0547203    .0398184
             |
      yr1983 |
         D1. |  -.0099749    .020234    -0.49   0.622    -.0496327    .0296829
             |
       _cons |  -.0050355   .0129166    -0.39   0.697    -.0303515    .0202806
------------------------------------------------------------------------------
Instrumented:  D.nL1
Instruments:   D.nL2 D.w D.wL1 D.k D.kL1 D.kL2 D.ys D.ysL1 D.ysL2 D.yr1979
               D.emp D.yr1980 D.yr1981 D.yr1982 D.yr1983 nL2

Notice that the first lag of the variable "n" is treated as endogenous and the instrument set is listed at the bottom (all differenced before estimation). The problem with this approach is that once you difference the variables, Δu_it may be correlated with Δu_it-1 and 2SLS is no longer efficient. Generalized Method of Moments (GMM) estimation addresses this problem and is the basis of the Arellano-Bond estimator. Using xtabond2 and estimating the above model, we need not difference the variables before hand:

xtabond2 n L.n L2.n w L.w L(0/2).(k ys) yr*, gmm(L.n) iv(w L.w L(0/2).(k ys) yr*) nolevel robust

Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.
yr1976 dropped due to collinearity
yr1977 dropped due to collinearity
yr1978 dropped due to collinearity
Warning: Two-step estimated covariance matrix of moments is singular.
  Using a generalized inverse to calculate robust weighting matrix for Hansen test.
  Difference-in-Sargan/Hansen statistics may be negative.

Dynamic panel-data estimation, one-step difference GMM
------------------------------------------------------------------------------
Group variable: id                              Number of obs      =       611
Time variable : year                            Number of groups   =       140
Number of instruments = 41                      Obs per group: min =         4
Wald chi2(16) =   1727.45                                      avg =      4.36
Prob > chi2   =     0.000                                      max =         6
------------------------------------------------------------------------------
             |               Robust
           n |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           n |
         L1. |   .6862261   .1445943     4.75   0.000     .4028266    .9696257
         L2. |  -.0853582   .0560155    -1.52   0.128    -.1951467    .0244302
             |
           w |
         --. |  -.6078208   .1782055    -3.41   0.001    -.9570972   -.2585445
         L1. |   .3926237   .1679931     2.34   0.019     .0633632    .7218842
             |
           k |
         --. |   .3568456   .0590203     6.05   0.000      .241168    .4725233
         L1. |  -.0580012   .0731797    -0.79   0.428    -.2014308    .0854284
         L2. |  -.0199475   .0327126    -0.61   0.542    -.0840631    .0441681
             |
          ys |
         --. |   .6085073   .1725313     3.53   0.000     .2703522    .9466624
         L1. |  -.7111651   .2317163    -3.07   0.002    -1.165321   -.2570095
         L2. |   .1057969   .1412021     0.75   0.454    -.1709542     .382548
             |
      yr1979 |   .0095545   .0102896     0.93   0.353    -.0106127    .0297217
      yr1980 |   .0220152   .0177104     1.24   0.214    -.0126966     .056727
      yr1981 |  -.0117743   .0295079    -0.40   0.690    -.0696086      .04606
      yr1982 |  -.0270588   .0292751    -0.92   0.355    -.0844369    .0303193
      yr1983 |  -.0213204   .0304599    -0.70   0.484    -.0810207    .0383798
      yr1984 |  -.0077033   .0314106    -0.25   0.806     -.069267    .0538604
------------------------------------------------------------------------------
Instruments for first differences equation
  Standard
    D.(w L.w k L.k L2.k ys L.ys L2.ys yr1976 yr1977 yr1978 yr1979 yr1980
    yr1981 yr1982 yr1983 yr1984)
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(1/8).L.n
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -3.60  Pr > z =  0.000
Arellano-Bond test for AR(2) in first differences: z =  -0.52  Pr > z =  0.606
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(25)   =  67.59  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(25)   =  31.38  Prob > chi2 =  0.177
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  iv(w L.w k L.k L2.k ys L.ys L2.ys yr1976 yr1977 yr1978 yr1979 yr1980 yr1981 yr1982 yr1983 yr1984)
    Hansen test excluding group:     chi2(11)   =  12.01  Prob > chi2 =  0.363
    Difference (null H = exogenous): chi2(14)   =  19.37  Prob > chi2 =  0.151

It may be useful to follow the implementation in the original paper by David Roodman that appears in the Stata Journal - see link below

http://www.stata-journal.com/sjpdf.h...iclenum=st0159

[Bjoern Brey]

Thank you both very much. This made xtabond2 much clearer to me. I think the specification outlined by Andrew Musau is the one that I should use for estimation.