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8/10/2019 Panel Data Notes
1/9
PANEL DATA
FIXED EFFECTS MODEL
RHO = COEFFICIENT OF VARIATION
WE CAN USE LEAST SQUARES WITH TIME DATA BECAUSE Ui F test is rejected. Since there is no Ui, we no
longer have to do transformation.
1% increase in enrollment results to a 0.26% increase in rent
delta: 1 unit
time variable: year, 80 to 90, but with gaps
panel variable: city (strongly balanced)
. xtset city year
F test that all u_i=0: F(63, 60) = 6.95 Prob > F = 0.0000
rho .84025883 (fraction of variance due to u_i)
sigma_e .06418028
sigma_u .14719727
_cons 2.037717 1.139795 1.79 0.079 -.2422116 4.317646
y90 .3773375 .0378882 9.96 0.000 .3015498 .4531252
lavginc .3196525 .0672163 4.76 0.000 .1851999 .4541051
lenroll .2621794 .1035767 2.53 0.014 .0549952 .4693636
lpop -.1937295 .1256064 -1.54 0.128 -.4449796 .0575206
lrent Coef. Std. Err. t P>|t| [95 Conf. Interval]
corr(u_i, Xb) = -0.0452 Prob > F = 0.0000
F(4,60) = 615.18
overall = 0.7886 max = 2
between = 0.2692 avg = 2.0
R-sq: within = 0.9762 Obs per group: min = 2
Group variable: city Number of groups = 64
Fixed-effects (within) regression Number of obs = 128
. xtreg lrent lpop lenroll lavginc y90,fe
8/10/2019 Panel Data Notes
2/9
LS estimate is underestimated because of absence of Ui in LS model
If there is actually a correlation in covariance of x and e, RE estimate will be biased.
_cons -.1170694 .5202727 -0.23 0.822 -1.146917 .9127785
y90 .2679622 .0355981 7.53 0.000 .197498 .3384264
lavginc .554786 .0540609 10.26 0.000 .4477758 .6617962
lenroll .1316076 .0316488 4.16 0.000 .0689607 .1942545
lpop -.0880099 .0256218 -3.43 0.001 -.1387267 -.0372932
lrent Coef. Std. Err. t P>|t| [95 Conf. Interval]
Total 14.0581346 127 .110693974 Root MSE = .1291
Adj R-squared = 0.8494
Residual 2.05016337 123 .016667995 R-squared = 0.8542
Model 12.0079713 4 3.00199282 Prob > F = 0.0000
F( 4, 123) = 180.11
Source SS df MS Number of obs = 128
. regress lrent lpop lenroll lavginc y90
rho .75021227 (fraction of variance due to u_i)
sigma_e .06418028
sigma_u .11122648
_cons .8181798 .542852 1.51 0.132 -.2457905 1.88215
y90 .3247298 .0290265 11.19 0.000 .267839 .3816206
lavginc .4377074 .0521273 8.40 0.000 .3355398 .5398749
lenroll .1404553 .039585 3.55 0.000 .0628702 .2180405
lpop -.0793277 .0327066 -2.43 0.015 -.1434314 -.015224
lrent Coef. Std. Err. z P>|z| [95 Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
Random effects u_i ~ Gaussian Wald chi2(4) = 2389.99
overall = 0.8486 max = 2
between = 0.5134 avg = 2.0
R-sq: within = 0.9743 Obs per group: min = 2
Group variable: city Number of groups = 64
Random-effects GLS regression Number of obs = 128
. xtreg lrent lpop lenroll lavginc y90,re
8/10/2019 Panel Data Notes
3/9
. estimates store RE
rho .75021227 (fraction of variance due to u_i)
sigma_e .06418028
sigma_u .11122648
_cons .8181798 .542852 1.51 0.132 -.2457905 1.88215
y90 .3247298 .0290265 11.19 0.000 .267839 .3816206
lavginc .4377074 .0521273 8.40 0.000 .3355398 .5398749
lenroll .1404553 .039585 3.55 0.000 .0628702 .2180405
lpop -.0793277 .0327066 -2.43 0.015 -.1434314 -.015224
lrent Coef. Std. Err. z P>|z| [95 Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
Random effects u_i ~ Gaussian Wald chi2(4) = 2389.99
overall = 0.8486 max = 2
between = 0.5134 avg = 2.0
R-sq: within = 0.9743 Obs per group: min = 2
Group variable: city Number of groups = 64
Random-effects GLS regression Number of obs = 128
. xtreg lrent lpop lenroll lavginc y90,re
. estimates store FE
F test that all u_i=0: F(63, 60) = 6.95 Prob > F = 0.0000
rho .84025883 (fraction of variance due to u_i)
sigma_e .06418028
sigma_u .14719727
_cons 2.037717 1.139795 1.79 0.079 -.2422116 4.317646
y90 .3773375 .0378882 9.96 0.000 .3015498 .4531252
lavginc .3196525 .0672163 4.76 0.000 .1851999 .4541051
lenroll .2621794 .1035767 2.53 0.014 .0549952 .4693636
lpop -.1937295 .1256064 -1.54 0.128 -.4449796 .0575206
lrent Coef. Std. Err. t P>|t| [95 Conf. Interval]
corr(u_i, Xb) = -0.0452 Prob > F = 0.0000
F(4,60) = 615.18
overall = 0.7886 max = 2
between = 0.2692 avg = 2.0
R-sq: within = 0.9762 Obs per group: min = 2
Group variable: city Number of groups = 64
Fixed-effects (within) regression Number of obs = 128
. xtreg lrent lpop lenroll lavginc y90,fe
(V_b-V_B is not positive definite)
Prob>chi2 = 0.0368
= 10.23
chi2(4) = (b-B)'[(V_b-V_B)^(-1)](b-B)
Test: Ho: difference in coefficients not systematic
B = inconsistent under Ha, efficient under Ho; obtained from xtreg
b = consistent under Ho and Ha; obtained from xtreg
y90 .3773375 .3247298 .0526077 .0243512
lavginc .3196525 .4377074 -.1180548 .0424356
lenroll .2621794 .1404553 .1217241 .0957139
lpop -.1937295 -.0793277 -.1144018 .1212734
FE RE Difference S.E.
(b) (B) (b-B) sqrt(diag(V_b-V_B))
Coefficients
. hausman FE RE
8/10/2019 Panel Data Notes
4/9
WAGE_PANEL 2 DATA
WE CAN DETECT TIME INVARIANT VARIABLES THROUGH STANDARD DEVIATION UNDER T.
Log linear model. Average wage of union members is 17% higher than non-union members. This can be
too high. It is possible that there are other factors that can also affect wage that are also correlated with
union membership. These are some possible U.
delta: 1 unit
time variable: year, 1980 to 1987
panel variable: crossid (strongly balanced)
. xtset crossid year
within .3236137 -.4360092 1.313991 T = 8
between .3766116 0 1 n = 545
married overall .4389908 .4963208 0 1 N = 4360
within 31.1431 -44.07523 158.9248 T = 8
between 26.35134 17.5 215.5 n = 545
expersq overall 50.42477 40.78199 0 324 N = 4360
within 2.291551 3.014679 10.01468 T = 8
between 1.654918 3.5 14.5 n = 545
exper overall 6.514679 2.825873 0 18 N = 4360
within 0 11.76697 11.76697 T = 8
between 1.747585 3 16 n = 545
educ overall 11.76697 1.746181 3 16 N = 4360
within .3622636 -2.467201 3.204687 T = 8
between .3907468 .3333435 3.174173 n = 545
lwage overall 1.649147 .5326094 -3.579079 4.05186 N = 4360
Variable Mean Std. Dev. Min Max Observations
. xtsum lwage educ exper expersq married
_cons -.0343057 .0632559 -0.54 0.588 -.1583195 .0897081
union .1685243 .0170652 9.88 0.000 .1350679 .2019808
married .1230112 .0155714 7.90 0.000 .0924833 .1535392
expersq -.0027349 .0007099 -3.85 0.000 -.0041267 -.0013432
exper .0861696 .0101415 8.50 0.000 .0662871 .1060521
educ .0989945 .0046227 21.41 0.000 .0899316 .1080574
lwage Coef. Std. Err. t P>|t| [95 Conf. Interval]
Total 1236.52965 4359 .283672779 Root MSE = .48287
Adj R-squared = 0.1781
Residual 1015.19113 4354 .233162869 R-squared = 0.1790
Model 221.338512 5 44.2677025 Prob > F = 0.0000
F( 5, 4354) = 189.86
Source SS df MS Number of obs = 4360
. regress lwage educ exper expersq married union
8/10/2019 Panel Data Notes
5/9
Note that educ is omitted, because it is already previously shown that it is time invariant. Notice that
union estimate now is smaller in FE model. Ui is also non-zero.
Union FE=8%, Union LS= 17%
Educ is now included. Check whether the gap in union is significant between RE and FE model. Note that
EDUC is significant in RE model. Since educ is no longer part of Ui, Ui is now smaller.
F test that all u_i=0: F(544, 3811) = 8.12 Prob > F = 0.0000
rho .56467849 (fraction of variance due to u_i)
sigma_e .35125535
sigma_u .4000539
_cons 1.06488 .0266607 39.94 0.000 1.012609 1.11715
union .0820871 .0192907 4.26 0.000 .044266 .1199083
married .0453033 .0183097 2.47 0.013 .0094056 .081201
expersq -.0043009 .0006053 -7.11 0.000 -.0054876 -.0031142
exper .1168467 .0084197 13.88 0.000 .1003392 .1333542
educ (omitted)
lwage Coef. Std. Err. t P>|t| [95 Conf. Interval]
corr(u_i, Xb) = -0.1139 Prob > F = 0.0000
F(4,3811) = 206.38
overall = 0.0638 max = 8
between = 0.0005 avg = 8.0
R-sq: within = 0.1780 Obs per group: min = 8
Group variable: crossid Number of groups = 545
Fixed-effects (within) regression Number of obs = 4360
note: educ omitted because of collinearity
. xtreg lwage educ exper expersq married union, fe
rho .46395167 (fraction of variance due to u_i)
sigma_e .35125535
sigma_u .32678141
_cons -.1186803 .1071673 -1.11 0.268 -.3287243 .0913637
union .1041501 .0178144 5.85 0.000 .0692346 .1390657
married .0668301 .0167367 3.99 0.000 .0340268 .0996335
expersq -.0040453 .000592 -6.83 0.000 -.0052056 -.002885
exper .1114758 .0082611 13.49 0.000 .0952843 .1276672
educ .101201 .0087763 11.53 0.000 .0839998 .1184022
lwage Coef. Std. Err. z P>|z| [95 Conf. Interval]
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
Random effects u_i ~ Gaussian Wald chi2(5) = 932.13
overall = 0.1729 max = 8
between = 0.1690 avg = 8.0
R-sq: within = 0.1774 Obs per group: min = 8
Group variable: crossid Number of groups = 545
Random-effects GLS regression Number of obs = 4360
. xtreg lwage educ exper expersq married union, re
8/10/2019 Panel Data Notes
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Educ is now removed in both model.
Reject null hypothesis, therefore there is significant difference between FE and RE estimate. Therefore
we select the FE estimate.
What if we have reverse causality?? An option is to use lagged variables. Here we use union lag 1.
[L.UNION]. note that union lag 1 is not significant
Another option is to treat it as a endogenous variable and use panel data-Instrumental Variable
estimation. Commonly we use lagged variables as instruments. We can easily argue that lagged variables
are exogenous, but we still have to meet the relevance condition.
Prob>chi2 = 0.0000
= 33.71
chi2(4) = (b-B)'[(V_b-V_B)^(-1)](b-B)
Test: Ho: difference in coefficients not systematic
B = inconsistent under Ha, efficient under Ho; obtained from xtreg
b = consistent under Ho and Ha; obtained from xtreg
union .0820871 .1041501 -.022063 .0074013
married .0453033 .0668301 -.0215268 .0074247
expersq -.0043009 -.0040453 -.0002556 .0001261
exper .1168467 .1114758 .0053709 .0016265
FE RE Difference S.E.
(b) (B) (b-B) sqrt(diag(V_b-V_B))
Coefficients
. hausman FE RE
.
F test that all u_i=0: F(544, 3266) = 8.96 Prob > F = 0.0000
rho .61946546 (fraction of variance due to u_i)
sigma_e .32665813
sigma_u .41677819
_cons 1.126349 .0360629 31.23 0.000 1.055641 1.197057
L1. .0227485 .0198253 1.15 0.251 -.0161228 .0616198
union
married .0547899 .0190767 2.87 0.004 .0173864 .0921933
expersq -.0032345 .0007133 -4.53 0.000 -.0046331 -.0018359
exper .1010353 .0105081 9.61 0.000 .0804321 .1216384
educ (omitted)
lwage Coef. Std. Err. t P>|t| [95 Conf. Interval]
corr(u_i, Xb) = -0.1428 Prob > F = 0.0000
F(4,3266) = 130.65
overall = 0.0295 max = 7
between = 0.0016 avg = 7.0
R-sq: within = 0.1379 Obs per group: min = 7
Group variable: crossid Number of groups = 545
Fixed-effects (within) regression Number of obs = 3815
note: educ omitted because of collinearity
. xtreg lwage educ exper expersq married l.union, fe
8/10/2019 Panel Data Notes
7/9
In the model above, we instrumented the endogenous variable union, with union lag 1 as instrument.
The model below adds married lag 1 as an instrument. Although married lag 1 is relevant, notice thatunion standard error increased. Notice too that union is too large, it is most likely very biased.
Instruments: educ exper expersq married L.union
Instrumented: union
F test that all u_i=0: F(544,3266) = 8.61 Prob > F = 0.0000
rho .60828168 (fraction of variance due to u_i)
sigma_e .33147153
sigma_u .41305859
_cons 1.055652 .0763501 13.83 0.000 .9060084 1.205296
married .0484844 .0200249 2.42 0.015 .0092363 .0877324
expersq -.0033162 .0007247 -4.58 0.000 -.0047367 -.0018958
exper .1031398 .0107769 9.57 0.000 .0820174 .1242622
educ (omitted)
union .2839369 .2510969 1.13 0.258 -.208204 .7760777
lwage Coef. Std. Err. z P>|z| [95 Conf. Interval]
corr(u_i, Xb) = -0.1561 Prob > chi2 = 0.0000
Wald chi2(4) = 99169.08
overall = 0.0447 max = 7
between = 0.0100 avg = 7.0
R-sq: within = 0.1123 Obs per group: min = 7
Group variable: crossid Number of groups = 545
Fixed-effects (within) IV regression Number of obs = 3815
F test that all u_i=0: F(544, 3266) = 3.71 Prob > F = 0.0000
rho .55719951 (fraction of variance due to u_i)
sigma_e .28060659
sigma_u .31477424
_cons .248989 .0309789 8.04 0.000 .1882491 .309729
L1. .0801182 .0170304 4.70 0.000 .0467269 .1135094
union
married .0222075 .0163873 1.36 0.175 -.0099229 .0543379
expersq .0002877 .0006127 0.47 0.639 -.0009137 .0014891
exper -.0074118 .0090267 -0.82 0.412 -.0251104 .0102867
educ (omitted)
union Coef. Std. Err. t P>|t| [95 Conf. Interval]
corr(u_i, Xb) = 0.7123 Prob > F = 0.0000
F(4,3266) = 6.28
overall = 0.3687 max = 7
between = 0.8580 avg = 7.0
R-sq: within = 0.0076 Obs per group: min = 7
Group variable: crossid Number of groups = 545
Fixed-effects (within) regression Number of obs = 3815
First-stage within regression
. xtivreg lwage educ exper expersq married (union = l.union), fe first
8/10/2019 Panel Data Notes
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BE Option
What if the dates of your values do not coincide? You interpolate the data for time gaps. E.g. cross
country data, you project/interpolate. But there will be measurement error. In this case, use the BE
(between) estimator.
Using BE (Between Estimator). This is an option used when there is a lot of interpolations.
Instruments: educ exper expersq married L.union L.married
Instrumented: union
F test that all u_i=0: F(544,3266) = 8.57 Prob > F = 0.0000
rho .60766203 (fraction of variance due to u_i)
sigma_e .33216294
sigma_u .41338247
_cons 1.052101 .0683068 15.40 0.000 .9182222 1.18598
married .0482124 .0198923 2.42 0.015 .0092242 .0872007
expersq -.0033186 .0007259 -4.57 0.000 -.0047413 -.0018959
exper .1032237 .0107687 9.59 0.000 .0821175 .1243298
educ (omitted)
union .2972112 .2161324 1.38 0.169 -.1264006 .720823
lwage Coef. Std. Err. z P>|z| [95 Conf. Interval]
corr(u_i, Xb) = -0.1605 Prob > chi2 = 0.0000
Wald chi2(4) = 98757.27
overall = 0.0447 max = 7
between = 0.0108 avg = 7.0
R-sq: within = 0.1086 Obs per group: min = 7
Group variable: crossid Number of groups = 545
Fixed-effects (within) IV regression Number of obs = 3815
F test that all u_i=0: F(544, 3265) = 3.73 Prob > F = 0.0000
rho .55799696 (fraction of variance due to u_i)
sigma_e .28031145
sigma_u .31495182
_cons .2575573 .0310964 8.28 0.000 .1965869 .3185278
L1. .0501667 .0178699 2.81 0.005 .0151295 .085204
married
L1. .078896 .017018 4.64 0.000 .0455289 .1122631
union
married .0018189 .0179088 0.10 0.919 -.0332947 .0369324
expersq .0003844 .0006131 0.63 0.531 -.0008176 .0015865
exper -.0109412 .0091044 -1.20 0.230 -.0287921 .0069097
educ (omitted)
union Coef. Std. Err. t P>|t| [95 Conf. Interval]
corr(u_i, Xb) = 0.5909 Prob > F = 0.0000
F(5,3265) = 6.61
overall = 0.2841 max = 7
between = 0.6227 avg = 7.0
R-sq: within = 0.0100 Obs per group: min = 7
Group variable: crossid Number of groups = 545
Fixed-effects (within) regression Number of obs = 3815
First-stage within regression
. xtivreg lwage educ exper expersq married (union = l.union l.married), fe first
8/10/2019 Panel Data Notes
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HAUSMAN TAYLOR ESTIMATOR
What it essentially does is that it separates the time invariant to time varying variables, but you do not
want to go into full RE. Hausman Taylor is a go-between model. We cannot use RE because union iscorrelated with u. we cannot use FE because educ is time-invariant (that is, if you really want to show
educ in the first place). So we use the Hausman Taylor as a go-between.
_cons .5248524 .2190544 2.40 0.017 .0945474 .9551574
union .252298 .0462559 5.45 0.000 .1614341 .3431619
married .1674925 .0405987 4.13 0.000 .0877415 .2472436
expersq .0055953 .0032176 1.74 0.083 -.0007253 .0119159
exper -.0608253 .0503829 -1.21 0.228 -.1597962 .0381456
educ .0937637 .0108543 8.64 0.000 .0724418 .1150855
lwage Coef. Std. Err. t P>|t| [95 Conf. Interval]
sd(u_i + avg(e_i.))= .3495835 Prob > F = 0.0000
F(5,539) = 28.13
overall = 0.1259 max = 8
between = 0.2069 avg = 8.0
R-sq: within = 0.0391 Obs per group: min = 8
Group variable: crossid Number of groups = 545
Between regression (regression on group means) Number of obs = 4360
. xtreg lwage educ exper expersq married union, be
.
Note: TV refers to time varying; TI refers to time invariant.
rho .47760032 (fraction of variance due to u_i)
sigma_e .35107115
sigma_u .33568041
_cons -.1122956 .1092657 -1.03 0.304 -.3264525 .1018613
educ .1011122 .0089567 11.29 0.000 .0835575 .118667
TIexogenous
union .0786858 .019242 4.09 0.000 .0409722 .1163994
TVendogenous
married .0666511 .0167574 3.98 0.000 .0338072 .099495
expersq -.0040726 .000591 -6.89 0.000 -.005231 -.0029142
exper .1118329 .0082459 13.56 0.000 .0956713 .1279946
TVexogenous
lwage Coef. Std. Err. z P>|z| [95 Conf. Interval]
Prob > chi2 = 0.0000
Random effects u_i ~ i.i.d. Wald chi2(5) = 915.18
max = 8
avg = 8
Obs per group: min = 8
Group variable: crossid Number of groups = 545
Hausman-Taylor estimation Number of obs = 4360
. xthtaylor lwage exper expersq married union educ, endog(union) constant(educ)