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.
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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)
Arellano–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 | |||||||
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 | ||||||
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 | |||||
| Order z Prob > z | ||
| 1 -4.6414 0.0000 | ||
| 2 -1.0572 0.2904 | ||
| 3 -.19492 0.8455 | ||
| 4 .04472 0.9643 | ||
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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