ECON 5120: Panel data econometrics
Seminar 3: October 2., 2007
Problem 3.1: Error autotocorrelation and heteroskedasticity
Standard variance components model:
yit  k  xit   i  uit ,
(0.1)
uit
IID(0,  ),
2
 it  i  uit ,
i
IID(0,   ),
i  1,..., N ; t  1,..., T .
2
xit  uit   it ,
Rewriting the model in matrix form gives:
yi  eT k  X i   εi ,
εi  eT i  ui , i  1,..., N
ε1 ,..., εN are uncorrelated and have zero expectation, with covariance matrix
V (εi )  E(εi εi )  E (eT i  ui )(eT i  ui )   2 (eT eT  )   2 IT . Model (0.1) can thus be
written compactly as:
(0.2)
yi  eT k  X i    i ,
 i  eT  i  ui
IID(0T ,1 , T ) ,
i  1,..., N ,
where
(0.3)
  2   2
 2

 2
 2   2
T  T ( 2 ,  2 )   2 (eT eT  )   2 IT  


2
 2
 
Models with autocorrelation:
Model A1:
 2
 2





 2   2 
 it   i  uit ,
i
IID(0,  2 ),
uit   ui ,t 1  it ,
  1, it
 i  it  xit
IID(0,  2 ),
i, t.
Assume that the AR(1) process started infinetly long back in time. Then ut   r 0  t r .

r
We derive the variance of  it :
Side 1 av 21

var( it )  var( i  uit )  var  i   r 0  rt r
  2  var


r 0



 rt r   2   r 0 (  r )2 var(t r )   2 

1
 2
1  2
From Lillard and Willis (1978) p.9891, we know that

 2
2
2
2
   u   
  2 i  j , t  s

2
(1


)


 2   2   
 2
 2
 2
2

2 
E( it  js )       u     
    2  1  2    i  j , t  s    0
(1   2 )
     


0 i j


The first line is the variance, the second the covariance between inventions in different
periods, for the same individual. Thus, the covariance of  it is:
cov( it ,  js )   2   
2
, i  j, t  s    0
(1   2 )
We can write the residual covariance matrix of the genuine disturbance, uit  (ui1 ,..., uiT )´ :

 1


2
  2
*
 
 
1  2 

  T 1

1
2


1
.
.
 T 1 

. 
. 


1 
and the covariance matrix of  it  ( i1 ,...,  iT )´ (i ,..., i )´(ui1,..., uiT )´ as:

 2
2
  
1  2


 2
2
   
1  2

2
T  *   2 eT eT    2
    2 
 
1  2



2
  2   T 1 

1  2

 2
 2
2
2
  
  
1  2
1  2
2
 2
 
1  2
2
 2  
.
 2
1  2
  
2
T 1
 2
  
1  2
2
 2 
.
 2
1  2
.
 2 

1  2 
.
 2 
 2
1  2












Lillard, Lee A. and Robert J. Willis (1978). “Dynamic Aspects of Earning Mobility”. Econometrica, Vol. 46,
No. 5, pp. 985-1012.
1
Side 2 av 21
In model A1, there is homoschedasiticity and autocorrelation, but not equicorrelation unless
  0 . This can be seen from the fact that2:
corr ( it ,  is ) 
 2
1  2
 2
2
 
1  2
 2   t  s
The correlation is independent of i but varies with t  s for   0 .
Model A2:
 it   i  uit ,
i
IID(0,  2 ),
uit   ui ,t 1  it ,
  1, it
 i  it  xit
(0,  2i ),
i, t.
We go through the same steps as for model A1, the only difference being that  2i is now
individual specific (has subscript i). The aggregate covariance matrix is thus:

 2i
 2i
 2i
 2i 
2
2
2
2
2
T 1

















1  2
1  2
1  2
1  2 



2
 2i
 2i
2
2
  2   i 2

 
  
.
1 
1  2
1  2


*
2


2
2
2
Ti  i    eT eT  
 i
 i
2
2
  2   2  i

  
 
.
 

1  2
1  2
1  2




2
2



 i
2
  2   T 1 i

.
.
 
2
2


1


1




In this model,  it is both heteroskedastic and autocorrelated. We now have:
 2i
  
1  2
corr ( it ,  is ) 
2
 2  i 2
1 
which varies with both i and t  s for   0 .
2
2
Note: corr ( it ,  is ) 
cov( it ,  is )

st.dev( it ) * st.dev( is )
 2   t  s
t s
 2
1  2
2
2
    2  2   2
1 
1 
2

 2
1  2
2
 2   2
1 
 2   t  s
Side 3 av 21
Model A3:
 it   i  uit ,
i
IID(0,  2 ),
uit  i ui ,t 1  it ,
i  1, it
(0,  2i ),
 i  it  xit i, t.
We go through the same steps as for models A1 and A2, the only difference being that now
both  2i and i are individual specific (have subscript i). The aggregate covariance matrix is
thus:

 2i
2
  
1  i 2


2
  2  i i

1  i 2

Ti  i*   2 eT eT    2
 2i
2
    i 1   2
i



2
  2   T 1  i
i
 
1  i 2

 2i
 2i
2
2
   i
  
1  i 2
1  2
2
 2 
 2i
1  i 2
 2i
   i
1  i 2
2
 2  i
   i
2
 2i
1  i 2
T 1
 2i 

1  i 2 
.
 2i
 
1  i 2
.
.
 2i
 
1  i 2
2
.
2
In this model, as in A2,  it is both heteroskedastic and autocorrelated. We now have:
corr ( it ,  is ) 
2i
1  i 2
2
 2  i 2
1  i
 2  i t  s
which varies with both i and t  s for   0 .
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  it
varies with t  s for   0 .
Models with error heteroskedasticity:
Model H1:
i
 it   i  uit ,
(0,  2i ), i  1,..., N
uit
IID(0,  2 )
 i  it  xit
i, t.
Side 4 av 21












The covariance matrix for this model is: V ( i )   2i (eT eT  )   2 IT which can be written out
just as (0.3), only with individual  2i . We thus have:
 2i
corr ( it ,  is )  2
 i   2
 it in this model is heteroskedastic, and it is equicorrelated for individual i.
Model H2:
 it   i  uit ,
i
uit
IID(0,  2 ),
(0,  i2 ) i  1,..., N
 i  it  xit
i, t.
The covariance matrix for this model is: V ( i )   2 (eT eT  )   i2 IT which can be written out
just as (0.3), only with individual  i2 . We thus have:
corr ( it ,  is ) 
 2
 2   i2
 it in this model is heteroskedastic, and it is equicorrelated for individual i.
Model H3:
 it   i  uit ,
i
(0,  2i ), i  1,..., N
uit
(0,  i2 ), i  1,..., N
 i  it  xit
i, t.
The covariance matrix for this model is: V ( i )   2i (eT eT  )   i2 IT which can be written out
just as (0.3), only with individual  2i and  i2 . We thus have:
corr ( it ,  is ) 
 2i
 2i   i2
 it in this model is heteroskedastic, and it is equicorrelated for individual i.
All of the models H1-H3 have heteroskedastic disturbances,  it ’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.
Side 5 av 21
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
yit  Yit   X it    i   it = Z it  u it
i  it = u it u it ~IID( 0,  2 )
. xtreg ln_wage age* ten not_s uni so, fe i(idcode)
Fixed-effects (within) regression
Group variable (i): idcode
Number of obs
Number of groups
=
=
19007
4134
R-sq:
Obs per group: min =
avg =
max =
1
4.6
12
within = 0.1333
between = 0.2375
overall = 0.2031
corr(u_i, Xb)
= 0.2074
F(6,14867)
Prob > F
=
=
381.19
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  i  vit , 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
u it ~IID( 0,  2 )
Side 6 av 21
We can also predict the first differenced overall component  = uit  uit 1
, 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) yit  Yit   X it    i   it
= Z it  i  it
u it = i  it u it ~IID( 0,  2 )
X is still a vector of exogenous variables; Y is a vector of observations of endogenous
variables, that are allowed to correlate with it . 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 u it . We find that the correlation between u it and not_smsa is -0.1451 and
between u it 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.
Side 7 av 21
. 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
Group variable: idcode
Number of obs
Number of groups
=
=
19007
4134
R-sq:
Obs per group: min =
avg =
max =
1
4.6
12
within =
.
between = 0.1304
overall = 0.0897
corr(u_i, Xb)
= -0.6843
F(4138,14869)
Prob > F
=
=
74.14
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)
Side 8 av 21


Looking at the residuals y  y  Z  y 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 y i    Z i  i   i
 1  Ti
Where wi    wit for w  y, Z , v
 Ti  t 1
We similarly define X i as the matrix of instruments X it 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:
Group variable: idcode
Number of obs
Number of groups
=
=
19007
4134
R-sq:
Obs per group: min =
avg =
max =
1
4.6
12
within = 0.0881
between =
.
overall = 0.1483
sd(u_i + avg(e_i.))=
.4445007
chi2(4)
Prob > chi2
=
=
512.38
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
Side 9 av 21
. 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  i ’s are uncorrelated with the other covariates,
we can fit the random-effects model. There are two variance components to estimate, the
variance of  i and  i . Since the variance components are unknown, the consistent estimates
are required to implement feasible GLS.
A consistent estimator is obtained by ˆ gls  ( X '  1 X ) 1 ( X '  1 y)
 
ui ' u j

The residuals in estimating  i , j 
are first obtained form OLS regression.
T

The estimates and their standard errors are calculated using  1 .
(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
Group variable: idcode
Number of obs
Number of groups
=
=
19007
4134
R-sq:
Obs per group: min =
avg =
max =
1
4.6
12
within = 0.0620
between = 0.1745
overall = 0.1206
corr(u_i, X)
= 0 (assumed)
Wald chi2(4)
Prob > chi2
=
=
941.52
0.0000
Side 10 av 21
-----------------------------------------------------------------------------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
0
.5
Density
1
1.5
|
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
-2
0
2
4
res2
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) ............................................................................. 13
Model 3.2.B2 xtabond n w k ys yr1980-yr1984, lags(1) robust .................................................................. 14
Model 3.2.-B3 xtabond n w k ys yr1980-yr1984, lags(1) twostep............................................................... 14
Model 3.2.-B4 xtabond n w k ys yr1980-yr1984, lags(2) ............................................................................ 15
Model 3.2.-B5 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) ............................................... 16
Model 3.2.-B6 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) robust .................................... 17
Model 3.2.-B7 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) twostep .................................. 17
Model 3.2.-B8 xtabond n l(0/2).w l(0/2).k l(0/2).ys yr1980-yr1984, lags(2) ............................................... 18
Model 3.2.-B9 xtabond n l(0/2).w l(0/2).k l(0/2).ys yr1980-yr1984, lags(2) robust .................................... 19
Model 3.2.-B10 xtabond n l(0/2).w l(0/2).k l(0/2).ys yr1980-yr1984, lags(2) twostep ................................ 19
Dynamic panel data models allow past realisations of the dependent variable to affect its
current level.
1) y it  y it 1 i  X it    i   it
 it   i ,t 1   it ,
it ~ (0, 2 )
Side 11 av 21
 i ~IID(o,  2i )
 i and  it are assumed to be independent for each i over all t.
 is a vector of parameters to be estimated
Var(  it  
 2i
1   i2
Arellano and Bond derive a generalized method-of-moments estimator for  i ,  i using
lagged levels of the dependent variable as instruments.
This method assumes that there is no second-order autocorrelation in the  it .
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 ( it ~ (0,  i2 ) )
and the variances (  i ~ IID (0,  i2 ) ) may be different for different individuals. (se Model A1A3)
First differencing of the equation removes the  i and 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 x it contains only strictly
exogenous covariates, xit will serve as its own instrument. The instrument matrix has one
row for each time period we are instrumenting.
 yi 2

 0
Zi  
.

 0

0
0
yi3 
0
.
.
0
 y i ,T  2
x i 4 

x i 5 
. 

xiT 
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.
Side 12 av 21
Vi define D as a (NxT) matrix whose element (i,t) is
Dit =
i  1,..N
1 if individual i is observed in period t

 0 if individual i is not observed in period t t  1,...T



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
Group variable (i): id
Time variable (t): year
Number of obs
Number of groups
=
=
751
140
Wald chi2(9)
=
645.91
Obs per group: min =
avg =
max =
5
5.364286
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 |
Side 13 av 21
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
Group variable (i): id
Time variable (t): year
Number of obs
Number of groups
=
=
751
140
Wald chi2(9)
=
433.33
Obs per group: min =
avg =
max =
5
5.364286
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
Group variable (i): id
Number of obs
Number of groups
=
=
751
140
Wald chi2(9)
=
618.30
Side 14 av 21
Time variable (t): year
Obs per group: min =
avg =
max =
5
5.364286
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
Group variable (i): id
Time variable (t): year
Number of obs
Number of groups
=
=
611
140
Wald chi2(10)
=
429.41
Obs per group: min =
avg =
max =
4
4.364286
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
Side 15 av 21
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
Group variable (i): id
Time variable (t): year
Number of obs
Number of groups
=
=
751
140
Wald chi2(12)
=
813.95
Obs per group: min =
avg =
max =
5
5.364286
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
.
Side 16 av 21
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
Group variable (i): id
Time variable (t): year
Number of obs
Number of groups
=
=
751
140
Wald chi2(12)
=
624.34
Obs per group: min =
avg =
max =
5
5.364286
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
Group variable (i): id
Time variable (t): year
Number of obs
Number of groups
=
=
751
140
Wald chi2(12)
=
1060.48
Obs per group: min =
avg =
max =
5
5.364286
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
Side 17 av 21
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
Group variable (i): id
Time variable (t): year
Number of obs
Number of groups
=
=
611
140
Wald chi2(16)
=
549.88
Obs per group: min =
avg =
max =
4
4.364286
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 |
Side 18 av 21
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
Group variable (i): id
Time variable (t): year
Number of obs
Number of groups
=
=
611
140
Wald chi2(16)
=
647.69
Obs per group: min =
avg =
max =
4
4.364286
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
Group variable (i): id
Number of obs
Number of groups
=
=
611
140
Wald chi2(16)
=
1059.42
Side 19 av 21
Time variable (t): year
Obs per group: min =
avg =
max =
4
4.364286
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
-2
-2
.3
-4
.2
-4
.1
1976
1978
1980
year
1982
1984
-1
0
Density
0
0
.4
2
2
.5
Looking at the residuals=predicted_y - y, it does not seem like we can assume
standard assumptions of normality and constant variance.
-4
-2
0
res
2
Median spline
-.5
0
Linear prediction
res
res
Median spline
. correlate, _coef
|
|
LD.
n
L2D.
n
D.
w
LD.
w
L2D.
w
D.
k
LD.
k
L2D.
k
Side 20 av 21
.5
-------------+-----------------------------------------------------------------------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 21 av 21