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## Dynamic panel-data (DPD) analysis

Stata has suite of tools for dynamic panel-data analysis:

• xtabond implements the Arellano and Bond estimator, which uses moment conditions in which lags of the dependent variable and first differences of the exogenous variables are instruments for the first-differenced equation.
• xtdpdsys implements the Arellano and Bover/Blundell and Bond system estimator, which uses the xtabond moment conditions and moment conditions in which lagged first differences of the dependent variable are instruments for the level equation.
• xtdpd, for advanced users, is a more flexible alternative that can fit models with low-order moving-average correlations in the idiosyncratic errors and predetermined variables with a more complicated structure than allowed with xtabond and xtdpdsys.
• Postestimation tools allow you to test for serial correlation in the first-differenced residuals and test the validity of the overidentifying restrictions.

### Example

Building on the work of Layard and Nickell (1986), Arellano and Bond (1991) fit a dynamic model of labor demand to an unbalanced panel of firms located in the United Kingdom. First we model employment on wages, capital stock, industry output, year dummies, and a time trend, including one lag of employment and two lags of wages and capital stock. We will use the one-step Arellano–Bond estimator and request their robust VCE:

. webuse abdata

. xtabond n L(0/2).(w k) yr1980-yr1984 year, vce(robust)

ArellanoÂ–Bond dynamic panel-data estimation     Number of obs     =        611
Group variable: id                              Number of groups  =        140
Time variable: year
Obs per group:
min =          4
avg =   4.364286
max =          6

Number of instruments =     40                  Wald chi2(13)     =    1318.68
Prob > chi2       =     0.0000
One-step results
(Std. err. adjusted for clustering on id)

Robust
n     Coefficient  std. err.      z    P>|z|     [95% conf. interval]

n
L1.       .6286618   .1161942     5.41   0.000     .4009254    .8563983

w
--.      -.5104249   .1904292    -2.68   0.007    -.8836592   -.1371906
L1.       .2891446    .140946     2.05   0.040     .0128954    .5653937
L2.      -.0443653   .0768135    -0.58   0.564     -.194917    .1061865

k
--.       .3556923   .0603274     5.90   0.000     .2374528    .4739318
L1.      -.0457102   .0699732    -0.65   0.514    -.1828552    .0914348
L2.      -.0619721   .0328589    -1.89   0.059    -.1263743    .0024301

yr1980      -.0282422   .0166363    -1.70   0.090    -.0608488    .0043643
yr1981      -.0694052    .028961    -2.40   0.017    -.1261677   -.0126426
yr1982      -.0523678   .0423433    -1.24   0.216    -.1353591    .0306235
yr1983      -.0256599   .0533747    -0.48   0.631    -.1302723    .0789525
yr1984      -.0093229   .0696241    -0.13   0.893    -.1457837    .1271379
year       .0019575   .0119481     0.16   0.870    -.0214604    .0253754
_cons      -2.543221   23.97919    -0.11   0.916    -49.54158    44.45514

Instruments for differenced equation
GMM-type: L(2/.).n
Standard: D.w LD.w L2D.w D.k LD.k L2D.k D.yr1980 D.yr1981 D.yr1982
D.yr1983 D.yr1984 D.year
Instruments for level equation
Standard: _cons


Because we included one lag of n in our regression model, xtabond used lags 2 and back as instruments. Differences of the exogenous variables also serve as instruments.

Here we refit our model, using xtdpdsys instead so that we can obtain the Arellano–Bover/Blundell–Bond estimates:

. xtdpdsys n L(0/2).(w k) yr1980-yr1984 year, vce(robust)

System dynamic panel-data estimation            Number of obs     =        751
Group variable: id                              Number of groups  =        140
Time variable: year
Obs per group:
min =          5
avg =   5.364286
max =          7

Number of instruments =     47                  Wald chi2(13)     =    2579.96
Prob > chi2       =     0.0000
One-step results

Robust
n   Coefficient  std. err.      z    P>|z|     [95% conf. interval]

n
L1.     .8221535    .093387     8.80   0.000     .6391184    1.005189

w
--.    -.5427935   .1881721    -2.88   0.004     -.911604   -.1739831
L1.     .3703602   .1656364     2.24   0.025     .0457189    .6950015
L2.    -.0726314   .0907148    -0.80   0.423    -.2504292    .1051664

k
--.     .3638069   .0657524     5.53   0.000     .2349346    .4926792
L1.    -.1222996   .0701521    -1.74   0.081    -.2597951     .015196
L2.    -.0901355   .0344142    -2.62   0.009    -.1575862   -.0226849

yr1980    -.0308622    .016946    -1.82   0.069    -.0640757    .0023512
yr1981    -.0718417   .0293223    -2.45   0.014    -.1293123    -.014371
yr1982    -.0384806   .0373631    -1.03   0.303    -.1117111    .0347498
yr1983    -.0121768   .0498519    -0.24   0.807    -.1098847    .0855311
yr1984    -.0050903   .0655011    -0.08   0.938    -.1334701    .1232895
year     .0058631   .0119867     0.49   0.625    -.0176304    .0293566
_cons    -10.59198   23.92087    -0.44   0.658    -57.47602    36.29207

Instruments for differenced equation
GMM-type: L(2/.).n
Standard: D.w LD.w L2D.w D.k LD.k L2D.k D.yr1980 D.yr1981 D.yr1982
D.yr1983 D.yr1984 D.year
Instruments for level equation
GMM-type: LD.n
Standard: _cons


Comparing the footers of the two commands’ output illustrates the key difference between the two estimators. xtdpdsys included the lagged differences of n as instruments in the level equation; xtabond did not.

The moment conditions of these GMM estimators are valid only if there is no serial correlation in the idiosyncratic errors. Because the first difference of white noise is necessarily autocorrelated, we need only concern ourselves with second and higher autocorrelation. We can use estat abond to test for autocorrelation:

. estat abond, artests(4)

Dynamic panel-data estimation                   Number of obs     =        751
Group variable: id                              Number of groups  =        140
Time variable: year
Obs per group:
min =          5
avg =   5.364286
max =          7

Number of instruments =     47                  Wald chi2(13)     =    2579.96
Prob > chi2       =     0.0000
One-step results
(Std. err. adjusted for clustering on id)

Robust
n   Coefficient  std. err.      z    P>|z|     [95% conf. interval]

n
L1.     .8221535    .093387     8.80   0.000     .6391184    1.005189

w
--.    -.5427935   .1881721    -2.88   0.004     -.911604   -.1739831
L1.     .3703602   .1656364     2.24   0.025     .0457189    .6950015
L2.    -.0726314   .0907148    -0.80   0.423    -.2504292    .1051664

k
--.     .3638069   .0657524     5.53   0.000     .2349346    .4926792
L1.    -.1222996   .0701521    -1.74   0.081    -.2597951     .015196
L2.    -.0901355   .0344142    -2.62   0.009    -.1575862   -.0226849

yr1980    -.0308622    .016946    -1.82   0.069    -.0640757    .0023512
yr1981    -.0718417   .0293223    -2.45   0.014    -.1293123    -.014371
yr1982    -.0384806   .0373631    -1.03   0.303    -.1117111    .0347498
yr1983    -.0121768   .0498519    -0.24   0.807    -.1098847    .0855311
yr1984    -.0050903   .0655011    -0.08   0.938    -.1334701    .1232895
year     .0058631   .0119867     0.49   0.625    -.0176304    .0293566
_cons    -10.59198   23.92087    -0.44   0.658    -57.47602    36.29207

Instruments for differenced equation
GMM-type: L(2/.).n
Standard: D.w LD.w L2D.w D.k LD.k L2D.k D.yr1980 D.yr1981 D.yr1982
D.yr1983 D.yr1984 D.year
Instruments for level equation
GMM-type: LD.n
Standard: _cons

Arellano–Bond test for zero autocorrelation in first-differenced errors
H0: No autocorrelation

Order         z   Prob > z

1   -4.6414     0.0000
2   -1.0572     0.2904
3   -.19492     0.8455
4   .04472      0.9643



### References

Arellano, M., and S. Bond. 1991.
Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. The Review of Econometric Studies 58: 277–297.
Layard, R., and S. J. Nickell. 1986.
Unemployment in Britain. Economica 53: 5121–5169.

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