ECON 5120: Panel data econometrics
Seminar 3: October 2., 2007
Problem 3.1: Error autotocorrelation and heteroskedasticity
Standard variance components model:
(0.1)
Rewriting the model in matrix form gives:
,,
are uncorrelated and have zero expectation, with covariance matrix
. Model (0.1) can thus be written compactly as:
(0.2) ,
where
(0.3)
Models with autocorrelation:
Model A1:
Assume that the AR(1) process started infinetly long back in time. Then .
We derive the variance of :
From Lillard and Willis (1978) p.989[1], we know that
The first line is the variance, the second the covariance between inventions in different periods, for the same individual. Thus, the covariance of is:
We can write the residual covariance matrix of the genuine disturbance, :
and the covariance matrix of as:
In model A1, there is homoschedasiticity and autocorrelation, but not equicorrelation unless . This can be seen from the fact that[2]:
The correlation is independent of i but varies with for .
Model A2:
We go through the same steps as for model A1, the only difference being that is now individual specific (has subscript i). The aggregate covariance matrix is thus:
In this model, is both heteroskedastic and autocorrelated. We now have:
which varies with both i and for .
Model A3:
We go through the same steps as for models A1 and A2, the only difference being that now both and are individual specific (have subscript i). The aggregate covariance matrix is thus:
In this model, as in A2, is both heteroskedastic and autocorrelated. We now have:
which varies with both i and for .
To sum up, the models A1-A3 all have autocorrelation, and A2-A3 also have heteroskedasticity. However, none of them have equicorrelation, as the correlation of varies with for .
Models with error heteroskedasticity:
Model H1:
The covariance matrix for this model is: which can be written out just as (0.3), only with individual . We thus have:
in this model is heteroskedastic, and it is equicorrelated for individual i.
Model H2:
The covariance matrix for this model is: which can be written out just as (0.3), only with individual . We thus have:
in this model is heteroskedastic, and it is equicorrelated for individual i.
Model H3:
The covariance matrix for this model is: which can be written out just as (0.3), only with individualand . We thus have:
in this model is heteroskedastic, and it is equicorrelated for individual i.
All of the models H1-H3 have heteroskedastic disturbances, ’s, which are equicorrelated for individual i.
Would you consider any of these extensions as improvements of the model? That depends on the real structure of the data. For instance, there is no point in using an autocorrelation covariance matrix for estimation unless there actually is autocovariance in the disturbances. We lose more periods when we have to account for autocorrelation.
Problem 3.2 A: Instrument variables
If we treat all variables as exogenous, we can use the one-stage within estimator. Xtreg
We assume that the model can be written
=
= ~IID()
. xtreg ln_wage age* ten not_s uni so, fe i(idcode)
Fixed-effects (within) regression Number of obs = 19007
Group variable (i): idcode Number of groups = 4134
R-sq: within = 0.1333 Obs per group: min = 1
between = 0.2375 avg = 4.6
overall = 0.2031 max = 12
F(6,14867) = 381.19
corr(u_i, Xb) = 0.2074 Prob > F = 0.0000
------
ln_wage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
------+------
age | .0311984 .0033902 9.20 0.000 .0245533 .0378436
age2 | -.0003457 .0000543 -6.37 0.000 -.0004522 -.0002393
tenure | .0176205 .0008099 21.76 0.000 .0160331 .0192079
not_smsa | -.0972535 .0125377 -7.76 0.000 -.1218289 -.072678
union | .0975672 .0069844 13.97 0.000 .0838769 .1112576
south | -.0620932 .013327 -4.66 0.000 -.0882158 -.0359706
_cons | 1.091612 .0523126 20.87 0.000 .9890729 1.194151
------+------
sigma_u | .3910683
sigma_e | .25545969
rho | .70091004 (fraction of variance due to u_i)
------
F test that all u_i=0: F(4133, 14867) = 8.31 Prob > F = 0.0000
(We note that only 19007 obs are used in the regression, due to missing variables in UNION)
The F test is a test for absence of fixed effects. We can assume fixed effects.
We can make a plot to look at the residuals. First we predict Xb, We have named it “yhatt”
We then compute the residuals: y- yhatt: We have named it “res”, or we can type:
predict <varname>, ue
Either way we obtain , the combined residual.
. predict yhatt
. gen res= ln_wage- yhatt
(or predict res_ ,ue)
. hist res
. hist res, normal
. twoway (scatter res yhatt) (mspline res yhatt)
. twoway (scatter res year) (mspline res year)
This gives the plots below. By looking at the plot it seems that our model seems to fit assumptions on ~IID()
We can also predict the first differenced overall component =
, by typing:
predict res2,e
We then obtain these plots.
IV-modell 2:
xtivreg ln_wage age* not_s (tenure = south union), fe
If we believe that tenure is an endogenous variable, we can try to handle this with instruments. It is suggested that we use union and south as instruments for tenure.
We then need another specification of the model.
1) = = ~IID()
X is still a vector of exogenous variables; Y is a vector of observations of endogenous variables, that are allowed to correlate with. N is the number of observations, and n is the number of girls.(groups)
We then use xtivreg which is a twostage estimator.
First we estimate
It is important to construct instruments that are strongly correlated with the endogenous variable, but not . We find that the correlation between and not_smsa is -0.1451 and between and union is 0,0792. The correlations between the endogenous variable and the instruments are -0.0266 and -0.1321. So they are not very good instruments.
. corr tenure south union
(obs=19007)
| tenure south union
------+------
tenure | 1.0000
south | -0.0266 1.0000
union | 0.1600 -0.1321 1.0000
. corr res3 not_smsa union
(obs=19007)
| res3 not_smsa union
------+------
res3 | 1.0000
not_smsa | -0.1451 1.0000
union | 0.0792 -0.0693 1.0000
IV-modell 3:
xtivreg ln_wage age* not_s (tenure = south union), fe
Fixed-effects (within) IV regression Number of obs = 19007
Group variable: idcode Number of groups = 4134
R-sq: within = . Obs per group: min = 1
between = 0.1304 avg = 4.6
overall = 0.0897 max = 12
F(4138,14869) = 74.14
corr(u_i, Xb) = -0.6843 Prob > F = 0.0000
------
ln_wage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
------+------
tenure | .2403531 .0373419 6.44 0.000 .1671583 .3135478
age | .0118437 .0090032 1.32 0.188 -.0058037 .0294912
age2 | -.0012145 .0001968 -6.17 0.000 -.0016003 -.0008286
not_smsa | -.0167178 .0339236 -0.49 0.622 -.0832123 .0497767
_cons | 1.678287 .1626657 10.32 0.000 1.359442 1.997132
------+------
sigma_u | .70661941
sigma_e | .63029359
rho | .55690561 (fraction of variance due to u_i)
------
F test that all u_i=0: F(4133,14869) = 1.44 Prob > F = 0.0000
------
Instrumented: tenure
Instruments: age age2 not_smsa south union
. correlate, _coef
| tenure age age2 not_smsa _cons
------+------
tenure | 1.0000
age | -0.3709 1.0000
age2 | -0.7337 -0.3543 1.0000
not_smsa | 0.4131 -0.1526 -0.3054 1.0000
_cons | 0.6172 -0.9515 0.0637 0.2079 1.0000
Note: corr(u_i, Xb) = -0.6843 is high, the model seems to be even worse than before
(note: this does not happen if we use another instrument than union, for instance hours)
Looking at the residuals we see that they do not seem to fit assumptions on IID (0,σ2). Our model specification with instrument variables does not improve our estimation.
By using tenure as a endogenous variable, using south and union as instruments, we find that age and not_smsa are no longer significant. If we believe for instance from other studies that these should be significant, we should use a different model specification.
IV-modell 4:
We are asked to use a between estimation.
After passing 1) trough the between estimator we are left with
Where for
We similarly define as the matrix of instruments after they have passed trough the between transformation. These instruments are used to correct the biases on the coefficients. We do not succeed, we find that sd(u_i + avg(e_i.))= 0,4445007. The residual plots also show that the coefficients are not constant for different values of X.
xtivreg ln_wage age* not_smsa (tenure= union south), be i(idcode)
Between-effects IV regression: Number of obs = 19007
Group variable: idcode Number of groups = 4134
R-sq: within = 0.0881 Obs per group: min = 1
between = . avg = 4.6
overall = 0.1483 max = 12
chi2(4) = 512.38
sd(u_i + avg(e_i.))= .4445007 Prob > chi2 = 0.0000
------
ln_wage | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------+------
tenure | .1349486 .0102693 13.14 0.000 .1148212 .155076
age | .0424055 .0135488 3.13 0.002 .0158503 .0689607
age2 | -.0008035 .0002143 -3.75 0.000 -.0012235 -.0003835
not_smsa | -.2619405 .0162784 -16.09 0.000 -.2938457 -.2300354
_cons | .8430686 .2029267 4.15 0.000 .4453395 1.240798
------+------
Instrumented: tenure
Instruments: age age2 not_smsa union south
. correlate, _coef
| tenure age age2 not_smsa _cons
------+------
tenure | 1.0000
age | -0.1843 1.0000
age2 | 0.0832 -0.9898 1.0000
not_smsa | 0.0096 -0.0202 0.0156 1.0000
_cons | 0.1310 -0.9921 0.9767 0.0016 1.0000
IV-modell 5:
If we believe, or are willing to assume, that all’s are uncorrelated with the other covariates, we can fit the random-effects model. There are two variance components to estimate, the variance of and . Since the variance components are unknown, the consistent estimates are required to implement feasible GLS.
A consistent estimator is obtained by
The residuals in estimating are first obtained form OLS regression.
The estimates and their standard errors are calculated using .
(note: We are not quite sure about this, and hope that this can be commented on at the seminar.)
xtivreg ln_wage age* not_s (tenure = south union), re i(idcode)
. xtivreg ln_wage age* not_s (tenure = south union), re i(idcode)
G2SLS random-effects IV regression Number of obs = 19007
Group variable: idcode Number of groups = 4134
R-sq: within = 0.0620 Obs per group: min = 1
between = 0.1745 avg = 4.6
overall = 0.1206 max = 12
Wald chi2(4) = 941.52
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
------
ln_wage | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------+------
tenure | .1772948 .0111724 15.87 0.000 .1553972 .1991924
age | .0191674 .0066388 2.89 0.004 .0061555 .0321792
age2 | -.0008496 .0001057 -8.04 0.000 -.0010567 -.0006425
not_smsa | -.2119932 .0130456 -16.25 0.000 -.2375622 -.1864243
_cons | 1.42761 .1037797 13.76 0.000 1.224205 1.631014
------+------
sigma_u | .33156584
sigma_e | .63029359
rho | .21674808 (fraction of variance due to u_i)
------
Instrumented: tenure
Instruments: age age2 not_smsa south union
. correlate, _coef
| tenure age age2 not_smsa _cons
------+------
tenure | 1.0000
age | -0.2370 1.0000
age2 | -0.2199 -0.8895 1.0000
not_smsa | 0.0847 -0.0247 -0.0171 1.0000
_cons | 0.3147 -0.9874 0.8287 -0.0007 1.0000
3.2.B
The STATA output from running the regressions can be found on the pages below. We have given the models numbers, and comment on all the models first.
Model 3.2.B1. xtabond n w k ys yr1980-yr1984, lags(1)
Model 3.2.B2 xtabond n w k ys yr1980-yr1984, lags(1) robust
Model 3.2.-B3 xtabond n w k ys yr1980-yr1984, lags(1) twostep
Model 3.2.-B4 xtabond n w k ys yr1980-yr1984, lags(2)
Model 3.2.-B5 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1)
Model 3.2.-B6 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) robust
Model 3.2.-B7 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) twostep
Model 3.2.-B8 xtabond n l(0/2).w l(0/2).k l(0/2).ys yr1980-yr1984, lags(2)
Model 3.2.-B9 xtabond n l(0/2).w l(0/2).k l(0/2).ys yr1980-yr1984, lags(2) robust
Model 3.2.-B10 xtabond n l(0/2).w l(0/2).k l(0/2).ys yr1980-yr1984, lags(2) twostep
Dynamic panel data models allow past realisations of the dependent variable to affect its current level.
1)
~IID(o,)
and are assumed to be independent for each i over all t.
is a vector of parameters to be estimated
Var(
Arellano and Bond derive a generalized method-of-moments estimator for , using lagged levels of the dependent variable as instruments.
This method assumes that there is no second-order autocorrelation in the .
xtabond includes the test for autocorrelation and the Sargan test of over-identifying restrictions for this model.
We do not know the AR structure but it can be different for each individual () and the variances () may be different for different individuals. (se Model A1-A3)
First differencing of the equation removes theand produces an equation that can be estimated using instrumental variables.
In all the models we use the lagged dependent variable as an instrument variable. We have then lost the three first observations, to lags and differencing. Since contains only strictly exogenous covariates, will serve as its own instrument. The instrument matrix has one row for each time period we are instrumenting.
The difficult part is to define and implement this kind of instrument matrix for each i. We have tried different methods of this in model B1-B10, without much success. It might be that we have omitted variables, and that our attempts will be no use with this model.
We have an unbalanced panel. This makes the algebra more difficult as we can not use kroneker products. But stata handles this. Missing observations are handled by dropping the rows for which there are no data, and filling inn zeroes in columns where missing data would be required.
It=Index set of individuals which are observed in period t . t =1,….T
Pi=Index set of periods where individual i is observed i =1,…N
T the number of periods when at least one individual is observed.
N is the number of individuals which are observed at least one period.
Vi define D as a (NxT) matrix whose element (i,t) is
Dit =
In model Model 3.2.B1 the genuine disturbance follows an AR(1) prosess.
Sargan test of over-identifying restrictions is rejected. Possibly due to heteroskedasticity. The presence of second order autocorrelation would imply that the estimates are inconsistent.
Model 3.2.B2 is similar to B1 but we now have computed robust standard errors, taken into account that we suspect heteroskedasticity .We see that the coefficients are the same, as they should be, and the (robust) standard errors are larger. But we still suspect that the estimates are inconsistent, because of the presence of second order autocorr.
Model 3.2.B3.
Areallo Bond recommends one step, but we see that Sagran test is not rejected and the autocorrelation test says there is no first order autocorrelation. But the estimates may still be inconsistent, because of the presence of second order autocorrelation.
We also note that several of the coeff. have changed, one has even switched sign.
Model 3.2.-B4
We use two lags of the dependent variable, but it is not significant for lag 2. The other results do not differ much.
Model 3.2.-B5 and B6
Here we use both the 1 difference and the lagged variable. The results do not differ much.
Model 3.2.-B7-10
We now use two lags of the dependent variable. But the estimates may still be inconsistent, because of the presence of second order autocorrelation.
xtabond n w k ys yr1980-yr1984, lags(1)
Arellano-Bond dynamic panel-data estimation Number of obs = 751
Group variable (i): id Number of groups = 140
Wald chi2(9) = 645.91
Time variable (t): year Obs per group: min = 5
avg = 5.364286
max = 7
One-step results
------
D.n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------+------
n |
LD. | .3566042 .0761519 4.68 0.000 .2073492 .5058592
w |
D1. | -.5114253 .0526485 -9.71 0.000 -.6146145 -.4082361
k |
D1. | .3086461 .0282417 10.93 0.000 .2532934 .3639988
ys |
D1. | .5032803 .0958316 5.25 0.000 .3154537 .6911069
yr1980 |
D1. | .0195602 .0143097 1.37 0.172 -.0084863 .0476067
yr1981 |
D1. | .0205486 .0226508 0.91 0.364 -.0238461 .0649432
yr1982 |
D1. | .0432438 .0296175 1.46 0.144 -.0148054 .1012929
yr1983 |
D1. | .0742359 .0370875 2.00 0.045 .0015457 .1469261
yr1984 |
D1. | .0918581 .0444807 2.07 0.039 .0046775 .1790387
_cons | -.0148382 .0056797 -2.61 0.009 -.0259702 -.0037062
------
Sargan test of over-identifying restrictions:
chi2(27) = 83.97 Prob > chi2 = 0.0000
Arellano-Bond test that average autocovariance in residuals of order 1 is 0:
H0: no autocorrelation z = -2.79 Pr > z = 0.0052
Arellano-Bond test that average autocovariance in residuals of order 2 is 0:
H0: no autocorrelation z = -0.72 Pr > z = 0.4745
Model 3.2.B2 xtabond n w k ys yr1980-yr1984, lags(1) robust
Arellano-Bond dynamic panel-data estimation Number of obs = 751
Group variable (i): id Number of groups = 140
Wald chi2(9) = 433.33
Time variable (t): year Obs per group: min = 5
avg = 5.364286
max = 7
One-step results
------
| Robust
D.n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------+------
n |
LD. | .3566042 .1371188 2.60 0.009 .0878562 .6253522
w |
D1. | -.5114253 .1701517 -3.01 0.003 -.8449164 -.1779342
k |
D1. | .3086461 .0534522 5.77 0.000 .2038817 .4134105
ys |
D1. | .5032803 .1513647 3.32 0.001 .2066109 .7999496
yr1980 |
D1. | .0195602 .013986 1.40 0.162 -.0078518 .0469722
yr1981 |
D1. | .0205486 .0303305 0.68 0.498 -.038898 .0799952
yr1982 |
D1. | .0432438 .0395268 1.09 0.274 -.0342273 .1207148
yr1983 |
D1. | .0742359 .0459919 1.61 0.107 -.0159065 .1643784
yr1984 |
D1. | .0918581 .0573505 1.60 0.109 -.0205468 .204263
_cons | -.0148382 .0061046 -2.43 0.015 -.0268031 -.0028734
------
Arellano-Bond test that average autocovariance in residuals of order 1 is 0:
H0: no autocorrelation z = -2.22 Pr > z = 0.0263
Arellano-Bond test that average autocovariance in residuals of order 2 is 0:
H0: no autocorrelation z = -0.61 Pr > z = 0.5443
Model 3.2.-B3 xtabond n w k ys yr1980-yr1984, lags(1) twostep
Arellano-Bond dynamic panel-data estimation Number of obs = 751
Group variable (i): id Number of groups = 140
Wald chi2(9) = 618.30
Time variable (t): year Obs per group: min = 5
avg = 5.364286
max = 7
Two-step results
------
D.n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------+------
n |
LD. | .2651432 .0559895 4.74 0.000 .1554058 .3748807
w |
D1. | -.4103142 .0430384 -9.53 0.000 -.494668 -.3259604
k |
D1. | .2563969 .0351334 7.30 0.000 .1875368 .3252571
ys |
D1. | .5436233 .0916815 5.93 0.000 .3639307 .7233158
yr1980 |
D1. | .0203073 .0099064 2.05 0.040 .0008911 .0397236
yr1981 |
D1. | .003441 .0192103 0.18 0.858 -.0342105 .0410925
yr1982 |
D1. | .0051906 .0264286 0.20 0.844 -.0466085 .0569898
yr1983 |
D1. | .0205109 .0318032 0.64 0.519 -.0418222 .082844
yr1984 |
D1. | .0190986 .0360729 0.53 0.596 -.0516029 .0898001
_cons | -.0119679 .004482 -2.67 0.008 -.0207523 -.0031834
------
Warning: Arellano and Bond recommend using one-step results for
inference on coefficients
Sargan test of over-identifying restrictions:
chi2(27) = 32.22 Prob > chi2 = 0.2242
Arellano-Bond test that average autocovariance in residuals of order 1 is 0:
H0: no autocorrelation z = -1.24 Pr > z = 0.2165
Arellano-Bond test that average autocovariance in residuals of order 2 is 0:
H0: no autocorrelation z = -0.32 Pr > z = 0.7473
Model 3.2.-B4 xtabond n w k ys yr1980-yr1984, lags(2)
Arellano-Bond dynamic panel-data estimation Number of obs = 611
Group variable (i): id Number of groups = 140
Wald chi2(10) = 429.41
Time variable (t): year Obs per group: min = 4
avg = 4.364286
max = 6
One-step results
------
D.n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------+------
n |
LD. | .3809966 .0913604 4.17 0.000 .2019335 .5600597
L2D. | -.0314535 .0372183 -0.85 0.398 -.1044 .041493
w |
D1. | -.5582806 .0595507 -9.37 0.000 -.674998 -.4415633
k |
D1. | .3604439 .0334723 10.77 0.000 .2948394 .4260483
ys |
D1. | .506865 .1101652 4.60 0.000 .2909451 .7227848
yr1980 |
D1. | .0058845 .0194738 0.30 0.763 -.0322833 .0440524
yr1981 |
D1. | -.0010127 .032771 -0.03 0.975 -.0652427 .0632172
yr1982 |
D1. | .0158584 .0452833 0.35 0.726 -.0728953 .1046121
yr1983 |
D1. | .0370505 .0581743 0.64 0.524 -.0769689 .15107
yr1984 |
D1. | .0427605 .071393 0.60 0.549 -.0971672 .1826881
_cons | .0009947 .0124716 0.08 0.936 -.0234491 .0254385
------
Sargan test of over-identifying restrictions:
chi2(25) = 74.97 Prob > chi2 = 0.0000
Arellano-Bond test that average autocovariance in residuals of order 1 is 0:
H0: no autocorrelation z = -3.13 Pr > z = 0.0017
Arellano-Bond test that average autocovariance in residuals of order 2 is 0:
H0: no autocorrelation z = -0.39 Pr > z = 0.6973
Model 3.2.-B5 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1)
Arellano-Bond dynamic panel-data estimation Number of obs = 751
Group variable (i): id Number of groups = 140
Wald chi2(12) = 813.95
Time variable (t): year Obs per group: min = 5
avg = 5.364286
max = 7
One-step results
------
D.n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------+------
n |
LD. | .5630709 .1094238 5.15 0.000 .3486042 .7775376
w |
D1. | -.5534161 .0561889 -9.85 0.000 -.6635442 -.4432879
LD. | .3098602 .0751487 4.12 0.000 .1625714 .4571489
k |
D1. | .3063797 .0297639 10.29 0.000 .2480435 .3647159
LD. | -.0522857 .0503968 -1.04 0.300 -.1510616 .0464901
ys |
D1. | .6228309 .1195031 5.21 0.000 .3886092 .8570526
LD. | -.597117 .143242 -4.17 0.000 -.8778661 -.3163679
yr1980 |
D1. | .0044292 .0156642 0.28 0.777 -.026272 .0351304
yr1981 |
D1. | -.0377724 .0244028 -1.55 0.122 -.085601 .0100561
yr1982 |
D1. | -.0710787 .032883 -2.16 0.031 -.1355282 -.0066292
yr1983 |
D1. | -.0812401 .0425751 -1.91 0.056 -.1646857 .0022055
yr1984 |
D1. | -.080054 .0513176 -1.56 0.119 -.1806347 .0205267
_cons | .0050601 .0078047 0.65 0.517 -.0102369 .020357
------
Sargan test of over-identifying restrictions:
chi2(27) = 77.00 Prob > chi2 = 0.0000
Arellano-Bond test that average autocovariance in residuals of order 1 is 0:
H0: no autocorrelation z = -3.39 Pr > z = 0.0007
Arellano-Bond test that average autocovariance in residuals of order 2 is 0:
H0: no autocorrelation z = -1.23 Pr > z = 0.2203
.
Model 3.2.-B6 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) robust
Arellano-Bond dynamic panel-data estimation Number of obs = 751
Group variable (i): id Number of groups = 140
Wald chi2(12) = 624.34
Time variable (t): year Obs per group: min = 5
avg = 5.364286
max = 7
One-step results
------
| Robust
D.n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------+------
n |
LD. | .5630709 .1197828 4.70 0.000 .3283009 .7978409
w |
D1. | -.5534161 .1772358 -3.12 0.002 -.9007918 -.2060403
LD. | .3098602 .1202698 2.58 0.010 .0741357 .5455846
k |
D1. | .3063797 .0547069 5.60 0.000 .1991562 .4136032
LD. | -.0522857 .0679217 -0.77 0.441 -.1854099 .0808384
ys |
D1. | .6228309 .1694083 3.68 0.000 .2907968 .954865
LD. | -.597117 .1872489 -3.19 0.001 -.9641182 -.2301159
yr1980 |
D1. | .0044292 .0144535 0.31 0.759 -.023899 .0327575
yr1981 |
D1. | -.0377724 .0260604 -1.45 0.147 -.0888499 .013305
yr1982 |
D1. | -.0710787 .0357855 -1.99 0.047 -.141217 -.0009405
yr1983 |
D1. | -.0812401 .0470945 -1.73 0.085 -.1735437 .0110635
yr1984 |
D1. | -.080054 .0564116 -1.42 0.156 -.1906187 .0305108
_cons | .0050601 .0094224 0.54 0.591 -.0134075 .0235276
------
Arellano-Bond test that average autocovariance in residuals of order 1 is 0:
H0: no autocorrelation z = -3.23 Pr > z = 0.0012
Arellano-Bond test that average autocovariance in residuals of order 2 is 0:
H0: no autocorrelation z = -1.25 Pr > z = 0.2099
Model 3.2.-B7 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) twostep
Arellano-Bond dynamic panel-data estimation Number of obs = 751
Group variable (i): id Number of groups = 140
Wald chi2(12) = 1060.48
Time variable (t): year Obs per group: min = 5
avg = 5.364286
max = 7
Two-step results
------
D.n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------+------
n |
LD. | .4670255 .0745046 6.27 0.000 .3209992 .6130518
w |
D1. | -.4870518 .0470883 -10.34 0.000 -.5793432 -.3947605
LD. | .2396211 .0611649 3.92 0.000 .1197401 .3595021
k |
D1. | .2229986 .0403502 5.53 0.000 .1439136 .3020836
LD. | .0524942 .0500748 1.05 0.294 -.0456507 .1506391
ys |
D1. | .600489 .1031834 5.82 0.000 .3982532 .8027247
LD. | -.4223655 .1189128 -3.55 0.000 -.6554304 -.1893007
yr1980 |
D1. | .0028122 .0107633 0.26 0.794 -.0182834 .0239079
yr1981 |
D1. | -.0430203 .0201913 -2.13 0.033 -.0825945 -.0034462
yr1982 |
D1. | -.0651432 .0270214 -2.41 0.016 -.1181041 -.0121823
yr1983 |
D1. | -.0671289 .0310614 -2.16 0.031 -.1280081 -.0062497
yr1984 |
D1. | -.0738373 .0354618 -2.08 0.037 -.1433411 -.0043335
_cons | .0007818 .0053805 0.15 0.884 -.0097637 .0113274
------
Warning: Arellano and Bond recommend using one-step results for
inference on coefficients
Sargan test of over-identifying restrictions:
chi2(27) = 37.14 Prob > chi2 = 0.0925
Arellano-Bond test that average autocovariance in residuals of order 1 is 0:
H0: no autocorrelation z = -2.55 Pr > z = 0.0108
Arellano-Bond test that average autocovariance in residuals of order 2 is 0:
H0: no autocorrelation z = -1.02 Pr > z = 0.3076
Model 3.2.-B8 xtabond n l(0/2).w l(0/2).k l(0/2).ys yr1980-yr1984, lags(2)
Arellano-Bond dynamic panel-data estimation Number of obs = 611
Group variable (i): id Number of groups = 140
Wald chi2(16) = 549.88
Time variable (t): year Obs per group: min = 4
avg = 4.364286
max = 6
One-step results
------
D.n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------+------
n |
LD. | .7590458 .1534595 4.95 0.000 .4582706 1.059821
L2D. | -.1182499 .0491858 -2.40 0.016 -.2146523 -.0218474
w |
D1. | -.6264705 .0683354 -9.17 0.000 -.7604053 -.4925357
LD. | .4450418 .1093473 4.07 0.000 .2307251 .6593584
L2D. | -.1459958 .0759505 -1.92 0.055 -.294856 .0028644
k |
D1. | .3552865 .0379609 9.36 0.000 .2808846 .4296884
LD. | -.0810551 .0601376 -1.35 0.178 -.1989226 .0368124
L2D. | -.0184798 .0422759 -0.44 0.662 -.101339 .0643794
ys |
D1. | .6353047 .1386783 4.58 0.000 .3635001 .9071092
LD. | -.8009587 .1938173 -4.13 0.000 -1.180834 -.4210837
L2D. | .2040576 .1563103 1.31 0.192 -.102305 .5104202
yr1980 |
D1. | .0108957 .0221529 0.49 0.623 -.0325231 .0543146
yr1981 |
D1. | -.0227497 .0370657 -0.61 0.539 -.0953972 .0498978
yr1982 |
D1. | -.0338001 .0509725 -0.66 0.507 -.1337044 .0661041
yr1983 |
D1. | -.0194175 .0673381 -0.29 0.773 -.1513978 .1125628
yr1984 |
D1. | -.0011615 .084187 -0.01 0.989 -.166165 .1638419
_cons | -.0004955 .0150878 -0.03 0.974 -.0300669 .029076
------
Sargan test of over-identifying restrictions:
chi2(25) = 59.25 Prob > chi2 = 0.0001
Arellano-Bond test that average autocovariance in residuals of order 1 is 0:
H0: no autocorrelation z = -4.26 Pr > z = 0.0000
Arellano-Bond test that average autocovariance in residuals of order 2 is 0:
H0: no autocorrelation z = -0.11 Pr > z = 0.9096
Model 3.2.-B9 xtabond n l(0/2).w l(0/2).k l(0/2).ys yr1980-yr1984, lags(2) robust
Arellano-Bond dynamic panel-data estimation Number of obs = 611
Group variable (i): id Number of groups = 140
Wald chi2(16) = 647.69
Time variable (t): year Obs per group: min = 4
avg = 4.364286
max = 6
One-step results
------
| Robust
D.n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------+------
n |
LD. | .7590458 .1341298 5.66 0.000 .4961561 1.021935
L2D. | -.1182499 .0457147 -2.59 0.010 -.2078491 -.0286506
w |
D1. | -.6264705 .1906682 -3.29 0.001 -1.000173 -.2527678
LD. | .4450418 .1795079 2.48 0.013 .0932128 .7968707
L2D. | -.1459958 .0873153 -1.67 0.095 -.3171306 .0251389
k |
D1. | .3552865 .0601116 5.91 0.000 .2374698 .4731031
LD. | -.0810551 .0744821 -1.09 0.276 -.2270374 .0649272
L2D. | -.0184798 .032538 -0.57 0.570 -.0822531 .0452934
ys |
D1. | .6353047 .1773702 3.58 0.000 .2876654 .9829439
LD. | -.8009587 .262686 -3.05 0.002 -1.315814 -.2861035
L2D. | .2040576 .1642452 1.24 0.214 -.117857 .5259722
yr1980 |
D1. | .0108957 .0175574 0.62 0.535 -.0235161 .0453075
yr1981 |
D1. | -.0227497 .0312617 -0.73 0.467 -.0840216 .0385222
yr1982 |
D1. | -.0338001 .041608 -0.81 0.417 -.1153503 .0477501
yr1983 |
D1. | -.0194175 .0558735 -0.35 0.728 -.1289274 .0900925
yr1984 |
D1. | -.0011615 .073711 -0.02 0.987 -.1456325 .1433095
_cons | -.0004955 .0126406 -0.04 0.969 -.0252707 .0242797
------
Arellano-Bond test that average autocovariance in residuals of order 1 is 0:
H0: no autocorrelation z = -3.95 Pr > z = 0.0001
Arellano-Bond test that average autocovariance in residuals of order 2 is 0:
H0: no autocorrelation z = -0.10 Pr > z = 0.9206
Model 3.2.-B10 xtabond n l(0/2).w l(0/2).k l(0/2).ys yr1980-yr1984, lags(2) twostep
Arellano-Bond dynamic panel-data estimation Number of obs = 611
Group variable (i): id Number of groups = 140
Wald chi2(16) = 1059.42
Time variable (t): year Obs per group: min = 4
avg = 4.364286
max = 6
Two-step results
------
D.n | Coef. Std. Err. z P>|z| [95% Conf. Interval]
------+------
n |
LD. | .7219585 .0872442 8.28 0.000 .550963 .892954
L2D. | -.0968684 .0277448 -3.49 0.000 -.1512472 -.0424896
w |
D1. | -.5542483 .0568186 -9.75 0.000 -.6656107 -.4428859
LD. | .4028884 .0935179 4.31 0.000 .2195968 .5861801
L2D. | -.1332653 .053101 -2.51 0.012 -.2373413 -.0291892
k |
D1. | .2791604 .0455152 6.13 0.000 .1899522 .3683685
LD. | -.0196619 .0552974 -0.36 0.722 -.1280427 .0887189
L2D. | -.0470922 .0263051 -1.79 0.073 -.0986492 .0044649
ys |
D1. | .5826981 .1170406 4.98 0.000 .3533028 .8120935
LD. | -.6633483 .143779 -4.61 0.000 -.94515 -.3815466
L2D. | .2129541 .119221 1.79 0.074 -.0207148 .4466229
yr1980 |
D1. | .004616 .0132031 0.35 0.727 -.0212616 .0304936
yr1981 |
D1. | -.0434272 .0247054 -1.76 0.079 -.091849 .0049945
yr1982 |
D1. | -.0524147 .0323349 -1.62 0.105 -.1157899 .0109604
yr1983 |
D1. | -.0320466 .0419099 -0.76 0.444 -.1141886 .0500954
yr1984 |
D1. | -.0347231 .0531522 -0.65 0.514 -.1388994 .0694532
_cons | .0022866 .0090722 0.25 0.801 -.0154946 .0200678
------
Warning: Arellano and Bond recommend using one-step results for
inference on coefficients
Sargan test of over-identifying restrictions:
chi2(25) = 31.68 Prob > chi2 = 0.1673
Arellano-Bond test that average autocovariance in residuals of order 1 is 0:
H0: no autocorrelation z = -3.48 Pr > z = 0.0005
Arellano-Bond test that average autocovariance in residuals of order 2 is 0:
H0: no autocorrelation z = -0.25 Pr > z = 0.8048
Looking at the residuals=predicted_y - y, it does not seem like we can assume standard assumptions of normality and constant variance.
. correlate, _coef
| LD. L2D. D. LD. L2D. D. LD. L2D.
| n n w w w k k k
------+------
n |
LD. | 1.0000
L2D. | -0.4966 1.0000
w |
D1. | 0.0185 -0.2676 1.0000
LD. | 0.4882 -0.0882 -0.7629 1.0000
L2D. | -0.1588 -0.0786 0.6073 -0.5442 1.0000
k |
D1. | -0.0416 -0.0503 -0.0785 0.1200 0.0040 1.0000
LD. | -0.5603 0.2779 0.0322 -0.3463 0.0884 -0.6392 1.0000
L2D. | -0.2603 -0.2597 0.0791 -0.2135 -0.0362 0.2655 -0.1719 1.0000
ys |
D1. | 0.0966 -0.0180 -0.5238 0.5203 -0.1663 0.0837 -0.1507 -0.0715
LD. | -0.3557 -0.0026 0.7719 -0.8535 0.5145 -0.0846 0.2683 0.2198
L2D. | 0.1330 0.0324 -0.5510 0.5372 -0.5240 0.0052 -0.1195 -0.2073
yr1980 |
D1. | -0.0800 0.2176 -0.3169 0.1438 -0.3610 0.0955 -0.0237 0.0962
yr1981 |
D1. | -0.0505 0.1136 -0.3418 0.2121 -0.4099 0.1523 -0.0725 0.1953
yr1982 |
D1. | -0.0930 0.1934 -0.2635 0.0739 -0.4432 0.0434 0.0826 0.0797
yr1983 |
D1. | -0.0177 0.1196 -0.1840 -0.0008 -0.4918 -0.0791 0.0910 0.0821
yr1984 |
D1. | -0.0732 0.1882 -0.3305 0.0932 -0.5857 -0.0790 0.1422 0.0951
_cons | 0.1179 -0.2728 0.2353 -0.0558 0.5535 0.2314 -0.2538 -0.0425
| D. LD. L2D. D. D. D. D. D.
| ys ys ys yr1980 yr1981 yr1982 yr1983 yr1984
------+------
ys |
D1. | 1.0000
LD. | -0.6909 1.0000
L2D. | 0.1428 -0.6316 1.0000
yr1980 |
D1. | 0.3594 -0.2632 0.1602 1.0000
yr1981 |
D1. | 0.3682 -0.1718 0.0343 0.8633 1.0000
yr1982 |
D1. | 0.0774 -0.0219 0.1796 0.8038 0.8670 1.0000
yr1983 |
D1. | -0.1406 0.0445 0.2605 0.6932 0.7232 0.9192 1.0000
yr1984 |
D1. | -0.1235 -0.0656 0.3313 0.6807 0.6959 0.8886 0.9577 1.0000
_cons | 0.2124 0.0330 -0.2920 -0.6376 -0.6224 -0.8055 -0.8860 -0.9200
|
| _cons
------+------
_cons | 1.0000.
Side 1 av 21
[1] Lillard, Lee A. and Robert J. Willis (1978). “Dynamic Aspects of Earning Mobility”. Econometrica, Vol. 46, No. 5, pp. 985-1012.
[2] Note: