First Difference and Fixed Effects
Moulton-UNC Chapel Hill
This log file provides an example of Fixed Effects and First Differences. We are investigating
the determinants of the probability that someone will leave a bequest. Please email me
(moulton@email.unc.edu) if you see any errors or have questions.
This is important since the Fixed Effects model that is estimated by including all the
individual fixed effects will not run if the matrix size is not large enough to accommodate all
the fixed effects.
. set matsize 11000
First open the Health and Retirement Study data, but just the variables that are necessary.
. use hhidpn r*cenreg r*beqany h*atota ragender raedyrs r*shlt h*hhres r*sayret h*hhres using
rndhrs_p.dta, clear
I convert all the i.w, i.u, etc missing value flags to instead just be “.”. I only do this for
waves 3 to 8 (using a loop) since I will drop all other waves later.
. replace ragender = . if ragender > 2
(7 real changes made, 7 to missing)
. replace raedyrs = . if raedyrs > 17
(131 real changes made, 131 to missing)
. forvalues w = 3/8 {
2. *make retirement 1 if fully retired and 0 otherwise
. replace r`w'sayret = (r`w'sayret == 1) if r`w'sayret <= 3
3. replace r`w'sayret = . if r`w'sayret > 1
4. *remove any oddly coded missing values
. replace r`w'beqany = . if r`w'beqany > 100
5. replace h`w'atota = log(h`w'atota)
6. su h`w'hhres
7. replace h`w'hhres = . if h`w'hhres > r(max)
8. replace r`w'shlt = . if r`w'shlt > 5
9. replace r`w'cenreg = . if r`w'cenreg > 5
10. }
Note that the output from this loop has been redacted from the log file.
Next use a loop to create a balanced panel, where individuals with non-missing values for all
relevant variables in waves 3 to 8 are retained.
. forvalues w = 3/8 {
2. foreach x in r`w'beqany h`w'hhres h`w'atota r`w'sayret r`w'shlt r`w'cenreg {
3. keep if `x' != .
4. }
5. }
(25,349 observations deleted)
(0 observations deleted)
(818 observations deleted)
(2,651 observations deleted)
(2 observations deleted)
(0 observations deleted)
(3,466 observations deleted)
(0 observations deleted)
(170 observations deleted)
(5 observations deleted)
(2 observations deleted)
(0 observations deleted)
(1,619 observations deleted)
(0 observations deleted)
(61 observations deleted)
(8 observations deleted)
(0 observations deleted)
(1 observation deleted)
(935 observations deleted)
(0 observations deleted)
(45 observations deleted)
(19 observations deleted)
(0 observations deleted)
(1 observation deleted)
(614 observations deleted)
(0 observations deleted)
(21 observations deleted)
(1 observation deleted)
(1 observation deleted)
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
(0 observations deleted)
(331 observations deleted)
(0 observations deleted)
(12 observations deleted)
(3 observations deleted)
(1 observation deleted)
(0 observations deleted)
Then save the data that we used in class.
. save FE-FD-updated.dta, replace
file FE-FD-updated.dta saved
Then use that data.
. use "/Users/jgmoulton/Downloads/FE-FD-updated.dta", clear
We will estimate the First Difference model by hand by differencing several of the variables
between waves 3 and 4.
. gen beqdiff = r4beqany - r3beqany
. gen hhresdiff = h4hhres - h3hhres
. gen wealthdiff = h4atota - h3atota
. gen retireddiff = r4sayret - r3sayret
Then reshape the wide data to be long.
. reshape long h@hhres r@shlt r@cenreg r@beqany h@atota r@sayret, i(hhidpn) j(wave)
(note: j = 1 2 3 4 5 6 7 8 9 10 11 12)
(note: r1beqany not found)
Data
wide
->
long
----------------------------------------------------------------------------Number of obs.
1359
->
16308
Number of variables
78
->
14
j variable (12 values)
->
wave
xij variables:
h1hhres h2hhres ... h12hhres
->
hhhres
r1shlt r2shlt ... r12shlt
->
rshlt
r1cenreg r2cenreg ... r12cenreg
->
rcenreg
r1beqany r2beqany ... r12beqany
->
rbeqany
h1atota h2atota ... h12atota
->
hatota
r1sayret r2sayret ... r12sayret
->
rsayret
----------------------------------------------------------------------------Xtset the data so that we can use the xtreg command to estimate a fixed effects model later.
. xtset hhidpn wave
panel variable: hhidpn (strongly balanced)
time variable: wave, 1 to 12
delta: 1 unit
Keep only waves 3 to 8.
. keep if wave >= 3 & wave <= 8
(8,154 observations deleted)
Before running regressions, we first check that we have within variation for the variables
included in our regression. It looks like we have a good amount for most of the variables.
Although variables like education do not have any within variation in this dataset.
. xtsum
Variable
|
Mean
Std. Dev.
Min
Max |
Observations
-----------------+--------------------------------------------+---------------hhidpn
overall | 7.76e+07
6.61e+07
1.00e+07
2.09e+08 |
N =
8154
between |
6.62e+07
1.00e+07
2.09e+08 |
n =
1359
within |
0
7.76e+07
7.76e+07 |
T =
6
|
|
wave
overall |
5.5
1.70793
3
8 |
N =
8154
between |
0
5.5
5.5 |
n =
1359
within |
1.70793
3
8 |
T =
6
|
|
rcenreg overall | 2.660657
.9895839
1
5 |
N =
8154
between |
.9710592
1
5 |
n =
1359
First Difference and Fixed Effects
within
ragender overall
between
within
raedyrs
overall
between
within
rshlt
overall
between
within
hatota
overall
between
within
rsayret
overall
between
within
rbeqany
overall
between
within
hhhres
overall
between
within
beqdiff
overall
between
within
hhresd~f overall
between
within
wealth~f overall
between
within
retire~f overall
between
within
|
|
| 1.499632
|
|
|
| 13.35173
|
|
|
| 2.429605
|
|
|
| 12.73514
|
|
|
| .4540103
|
|
|
| 92.97768
|
|
|
| 2.126686
|
|
|
| 4.025018
|
|
|
| -.1103753
|
|
|
| .0836954
|
|
|
| -.0036792
|
|
Moulton-UNC Chapel Hill
.1920898
-.5060093
.5000305
.5001839
0
1
1
1.499632
2.846221
2.847094
0
0
0
13.35173
1.052416
.869697
.5930178
1
1
-.2370616
1.520786
1.412563
.5645154
.6931472
5.408678
3.577643
.497911
.3717857
.3313239
0
0
-.379323
24.62895
21.54188
11.95061
0
0
9.644346
.8770996
.7058009
.5210127
1
1
-1.206647
20.12526
20.13144
0
-100
-100
4.025018
.7670273
.7672627
0
-10
-10
-.1103753
.677269
.6774768
0
-6.339477
-6.339477
.0836954
.4331829
.4333158
0
-1
-1
-.0036792
4.660657 |
|
2 |
2 |
1.499632 |
|
17 |
17 |
13.35173 |
|
5 |
5 |
5.596272 |
|
17.78957 |
17.07953 |
16.13795 |
|
1 |
1 |
1.287344 |
|
100 |
100 |
176.311 |
|
12 |
6.5 |
10.46002 |
|
100 |
100 |
4.025018 |
|
6 |
6 |
-.1103753 |
|
3.128419 |
3.128419 |
.0836954 |
|
1 |
1 |
-.0036792 |
T =
6
N =
n =
T =
8154
1359
6
N =
n =
T =
8154
1359
6
N =
n =
T =
8154
1359
6
N =
n =
T =
8154
1359
6
N =
n =
T =
8154
1359
6
N =
n =
T =
8154
1359
6
N =
n =
T =
8154
1359
6
N =
n =
T =
8154
1359
6
N =
n =
T =
8154
1359
6
N =
n =
T =
8154
1359
6
N =
n =
T =
8154
1359
6
One way to create indicator variables, is to use the tab and gen option.
. tab wave, gen(TTT)
wave |
Freq.
Percent
Cum.
------------+----------------------------------3 |
1,359
16.67
16.67
4 |
1,359
16.67
33.33
5 |
1,359
16.67
50.00
6 |
1,359
16.67
66.67
7 |
1,359
16.67
83.33
8 |
1,359
16.67
100.00
------------+----------------------------------Total |
8,154
100.00
Estimate the two-period first difference model using the manually differenced variables. Note
that I am only include wave 4 since we only created the difference between waves 4 and 3. We are
also omitting the constant since we differenced it out.
. reg beqdiff hhresdiff wealthdiff retireddiff TTT* if wave == 4, nocons
note: TTT1 omitted because of collinearity
note: TTT3 omitted because of collinearity
note: TTT4 omitted because of collinearity
note: TTT5 omitted because of collinearity
note: TTT6 omitted because of collinearity
Source |
SS
df
MS
Number of obs
=
1,359
First Difference and Fixed Effects
-------------+---------------------------------Model | 23207.0244
4 5801.75611
Residual | 549172.976
1,355 405.293709
-------------+---------------------------------Total |
572380
1,359 421.177336
Moulton-UNC Chapel Hill
F(4, 1355)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
14.31
0.0000
0.0405
0.0377
20.132
-----------------------------------------------------------------------------beqdiff |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhresdiff |
.2287623
.7136789
0.32
0.749
-1.171273
1.628798
wealthdiff |
1.066715
.8078159
1.32
0.187
-.5179905
2.651421
retireddiff | -1.396982
1.262236
-1.11
0.269
-3.873131
1.079166
TTT1 |
0 (omitted)
TTT2 |
3.955849
.5553654
7.12
0.000
2.86638
5.045319
TTT3 |
0 (omitted)
TTT4 |
0 (omitted)
TTT5 |
0 (omitted)
TTT6 |
0 (omitted)
-----------------------------------------------------------------------------The coefficients are the same if we use the D. command that differences the data across one time
period (you need to xtset the data first).
. reg D.(rbeqany hhhres hatota rsayret TTT*) if wave == 4, nocons
note: D.TTT2 omitted because of collinearity
note: D.TTT3 omitted because of collinearity
note: D.TTT4 omitted because of collinearity
note: D.TTT5 omitted because of collinearity
note: D.TTT6 omitted because of collinearity
Source |
SS
df
MS
-------------+---------------------------------Model | 23207.0244
4 5801.75611
Residual | 549172.976
1,355 405.293709
-------------+---------------------------------Total |
572380
1,359 421.177336
Number of obs
F(4, 1355)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
1,359
14.31
0.0000
0.0405
0.0377
20.132
-----------------------------------------------------------------------------D.rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
D1. |
.2287623
.7136789
0.32
0.749
-1.171273
1.628798
|
hatota |
D1. |
1.066715
.8078159
1.32
0.187
-.5179906
2.651421
|
rsayret |
D1. | -1.396982
1.262236
-1.11
0.269
-3.873131
1.079166
|
TTT1 |
D1. | -3.955849
.5553654
-7.12
0.000
-5.045319
-2.86638
|
TTT2 |
D1. |
0 (omitted)
|
TTT3 |
D1. |
0 (omitted)
|
TTT4 |
D1. |
0 (omitted)
|
TTT5 |
D1. |
0 (omitted)
|
TTT6 |
D1. |
0 (omitted)
-----------------------------------------------------------------------------The coefficients are also the same if we estimate a Fixed Effects model using only waves 3 and
4. First Difference and Fixed Effects provide the same estimates when using two time periods.
Note that with xtreg you do not specify noconstant since we are estimating all the different
intercepts.
First Difference and Fixed Effects
. xtreg rbeqany hhhres hatota
note: TTT2 omitted because of
note: TTT3 omitted because of
note: TTT4 omitted because of
note: TTT5 omitted because of
note: TTT6 omitted because of
Moulton-UNC Chapel Hill
rsayret TTT* if wave == 4 | wave == 3, fe
collinearity
collinearity
collinearity
collinearity
collinearity
Fixed-effects (within) regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within = 0.0405
between = 0.3288
overall = 0.1508
corr(u_i, Xb)
= 0.3236
=
=
2,718
1,359
min =
avg =
max =
2
2.0
2
=
=
14.31
0.0000
F(4,1355)
Prob > F
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
.2287623
.7136789
0.32
0.749
-1.171273
1.628798
hatota |
1.066715
.8078159
1.32
0.187
-.5179906
2.651421
rsayret | -1.396982
1.262236
-1.11
0.269
-3.873131
1.079166
TTT1 | -3.955849
.5553654
-7.12
0.000
-5.045319
-2.86638
TTT2 |
0 (omitted)
TTT3 |
0 (omitted)
TTT4 |
0 (omitted)
TTT5 |
0 (omitted)
TTT6 |
0 (omitted)
_cons |
80.53884
10.37666
7.76
0.000
60.18278
100.8949
-------------+---------------------------------------------------------------sigma_u |
22.26177
sigma_e | 14.235408
rho | .70977198
(fraction of variance due to u_i)
-----------------------------------------------------------------------------F test that all u_i=0: F(1358, 1355) = 3.49
Prob > F = 0.0000
Now estimate the First Difference model using all the time periods.
. reg D.(rbeqany hhhres hatota rsayret TTT*), nocons
note: D.TTT6 omitted because of collinearity
Source |
SS
df
MS
-------------+---------------------------------Model |
30619.435
8 3827.42938
Residual | 1918271.56
6,787 282.639099
-------------+---------------------------------Total |
1948891
6,795 286.812509
Number of obs
F(8, 6787)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
6,795
13.54
0.0000
0.0157
0.0146
16.812
-----------------------------------------------------------------------------D.rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
D1. | -.0198774
.3279133
-0.06
0.952
-.6626902
.6229354
|
hatota |
D1. |
1.456645
.2958633
4.92
0.000
.8766604
2.03663
|
rsayret |
D1. | -.9030089
.4722086
-1.91
0.056
-1.828686
.0226679
|
TTT1 |
D1. | -2.469052
1.043154
-2.37
0.018
-4.513962
-.4241425
|
TTT2 |
D1. |
1.428535
.931412
1.53
0.125
-.3973244
3.254395
|
TTT3 |
D1. |
1.065853
.8029547
1.33
0.184
-.5081904
2.639896
|
TTT4 |
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
D1. |
1.12836
.6513059
1.73
0.083
-.1484037
2.405124
|
TTT5 |
D1. |
.6005762
.4574299
1.31
0.189
-.2961298
1.497282
|
TTT6 |
D1. |
0 (omitted)
-----------------------------------------------------------------------------Note that the coefficients on the other variables are the same if you difference the time
indicators or include them as is.
. reg D.(rbeqany hhhres hatota rsayret) TTT*, nocons
note: TTT1 omitted because of collinearity
Source |
SS
df
MS
-------------+---------------------------------Model |
30619.435
8 3827.42938
Residual | 1918271.56
6,787 282.639099
-------------+---------------------------------Total |
1948891
6,795 286.812509
Number of obs
F(8, 6787)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
6,795
13.54
0.0000
0.0157
0.0146
16.812
-----------------------------------------------------------------------------D.rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
D1. | -.0198774
.3279133
-0.06
0.952
-.6626902
.6229354
|
hatota |
D1. |
1.456645
.2958633
4.92
0.000
.8766604
2.03663
|
rsayret |
D1. | -.9030089
.4722086
-1.91
0.056
-1.828686
.0226679
|
TTT1 |
0 (omitted)
TTT2 |
3.897588
.4581274
8.51
0.000
2.999514
4.795661
TTT3 | -.3626826
.4582761
-0.79
0.429
-1.261048
.5356823
TTT4 |
.0625076
.4592435
0.14
0.892
-.8377537
.962769
TTT5 |
-.527784
.4596993
-1.15
0.251
-1.428939
.3733708
TTT6 | -.6005762
.4574299
-1.31
0.189
-1.497282
.2961298
-----------------------------------------------------------------------------The coefficients for the Fixed Effects model using all the time periods are similar to the First
Difference model, but not exactly the same.
. xtreg rbeqany hhhres hatota rsayret TTT*, fe
note: TTT6 omitted because of collinearity
Fixed-effects (within) regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within = 0.0220
between = 0.4381
overall = 0.2768
corr(u_i, Xb)
= 0.4883
F(8,6787)
Prob > F
=
=
8,154
1,359
min =
avg =
max =
6
6.0
6
=
=
19.05
0.0000
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres | -.1103689
.2828504
-0.39
0.696
-.6648443
.4441066
hatota |
1.937789
.2655226
7.30
0.000
1.417281
2.458296
rsayret | -1.031235
.4576656
-2.25
0.024
-1.928403
-.1340665
TTT1 |
-2.24378
.5352282
-4.19
0.000
-3.292995
-1.194565
TTT2 |
1.603078
.5263248
3.05
0.002
.5713163
2.63484
TTT3 |
1.191522
.514469
2.32
0.021
.1830017
2.200043
TTT4 |
1.21655
.5039036
2.41
0.016
.2287408
2.204359
TTT5 |
.6473157
.4979766
1.30
0.194
-.3288745
1.623506
TTT6 |
0 (omitted)
_cons |
68.60014
3.534376
19.41
0.000
61.67166
75.52863
-------------+----------------------------------------------------------------
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
sigma_u | 19.819517
sigma_e |
12.95355
rho | .70069183
(fraction of variance due to u_i)
-----------------------------------------------------------------------------F test that all u_i=0: F(1358, 6787) = 9.81
Prob > F = 0.0000
But both Fixed Effects and First Difference are quite different from the Pooled OLS model.
. reg rbeqany hhhres hatota rsayret TTT*
note: TTT6 omitted because of collinearity
Source |
SS
df
MS
-------------+---------------------------------Model | 1571509.72
8 196438.715
Residual | 3373978.22
8,145 414.239192
-------------+---------------------------------Total | 4945487.94
8,153 606.585053
Number of obs
F(8, 8145)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
8,154
474.22
0.0000
0.3178
0.3171
20.353
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
.5956975
.2603045
2.29
0.022
.0854342
1.105961
hatota |
9.131058
.1491015
61.24
0.000
8.838781
9.423335
rsayret | -.7810734
.4652072
-1.68
0.093
-1.692998
.1308515
TTT1 |
1.001606
.7975332
1.26
0.209
-.5617631
2.564974
TTT2 |
4.325273
.7945231
5.44
0.000
2.767805
5.882741
TTT3 |
3.201472
.7891959
4.06
0.000
1.654447
4.748498
TTT4 |
2.614182
.7839107
3.33
0.001
1.077517
4.150847
TTT5 |
1.371202
.7811275
1.76
0.079
-.1600075
2.902411
TTT6 |
0 (omitted)
_cons | -26.30545
2.112393
-12.45
0.000
-30.44628
-22.16462
-----------------------------------------------------------------------------We can observe the different intecepts for each individual if we run the Fixed Effects model by
including all the indicators for the identifier (i.hhidpn in this case). You will need to use
the command “set matsize 11000” if you have not done so already since there are a lot of
individuals in the dataset and the default matsize is not large enough to estimate the
regression.
. reg rbeqany hhhres hatota rsayret TTT* i.hhidpn
note: TTT6 omitted because of collinearity
Source |
SS
df
MS
-------------+---------------------------------Model | 3806666.92
1,366 2786.72542
Residual | 1138821.02
6,787 167.794463
-------------+---------------------------------Total | 4945487.94
8,153 606.585053
Number of obs
F(1366, 6787)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
8,154
16.61
0.0000
0.7697
0.7234
12.954
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres | -.1103689
.2828504
-0.39
0.696
-.6648443
.4441066
hatota |
1.937789
.2655226
7.30
0.000
1.417281
2.458296
rsayret | -1.031235
.4576656
-2.25
0.024
-1.928403
-.1340665
TTT1 |
-2.24378
.5352282
-4.19
0.000
-3.292995
-1.194565
TTT2 |
1.603078
.5263248
3.05
0.002
.5713163
2.63484
TTT3 |
1.191522
.514469
2.32
0.021
.1830017
2.200043
TTT4 |
1.21655
.5039036
2.41
0.016
.2287408
2.204359
TTT5 |
.6473157
.4979766
1.30
0.194
-.3288745
1.623506
TTT6 |
0 (omitted)
|
hhidpn |
10038040 | -4.91e-12
7.478736
-0.00
1.000
-14.66067
14.66067
10059020 | -3.406031
7.493284
-0.45
0.649
-18.09522
11.28316
10097040 | -18.37875
7.482767
-2.46
0.014
-33.04732
-3.710178
10394010 |
1.938642
7.503271
0.26
0.796
-12.77012
16.64741
30 pages later…
208504010
208675010
|
|
-86.01498
.3376332
7.709259
7.5091
-11.16
0.04
0.000
0.964
-101.1275
-14.38256
-70.90242
15.05782
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
208725020
208773020
208827020
208896010
|
-34.3129
7.543641
-4.55
0.000
-49.1008
-19.525
| -4.518654
7.505793
-0.60
0.547
-19.23236
10.19505
|
-79.631
7.728287
-10.30
0.000
-94.78086
-64.48113
| -24.63045
7.566187
-3.26
0.001
-39.46255
-9.798353
|
_cons |
72.39976
6.562587
11.03
0.000
59.53504
85.26449
-----------------------------------------------------------------------------Stata has stored all of those intercepts temporarily.
. return list
scalars:
r(level) =
95
macros:
r(label10) : "(base)"
r(label9) : "(omitted)"
matrices:
r(table) :
9 x 1369
And we can test if they are different. The easiest way to do that is to conduct an F-test with a
null that the intercepts are all equal to 0, meaning that each individual’s intercept is not
different from the omitted individual’s intercept. Since the p-value is smaller than 0.05 in
this case, we would reject the null that they are equal to 0 and accept the alternative that
they are different. This suggests that Fixed Effects may be preferred to Pooled OLS.
. testparm i.*hhidpn
( 1)
( 2)
( 3)
10038040.hhidpn = 0
10059020.hhidpn = 0
10097040.hhidpn = 0
Several pages later…
(1357)
(1358)
208827020.hhidpn = 0
208896010.hhidpn = 0
F(1358, 6787) =
Prob > F =
9.81
0.0000
But it is a lot easier to conduct this test, it shows up at the bottom of the xtreg Fixed
Effects output.
. xtreg rbeqany hhhres hatota rsayret TTT*, fe
note: TTT6 omitted because of collinearity
Fixed-effects (within) regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within = 0.0220
between = 0.4381
overall = 0.2768
corr(u_i, Xb)
= 0.4883
F(8,6787)
Prob > F
=
=
8,154
1,359
min =
avg =
max =
6
6.0
6
=
=
19.05
0.0000
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres | -.1103689
.2828504
-0.39
0.696
-.6648443
.4441066
hatota |
1.937789
.2655226
7.30
0.000
1.417281
2.458296
rsayret | -1.031235
.4576656
-2.25
0.024
-1.928403
-.1340665
TTT1 |
-2.24378
.5352282
-4.19
0.000
-3.292995
-1.194565
TTT2 |
1.603078
.5263248
3.05
0.002
.5713163
2.63484
TTT3 |
1.191522
.514469
2.32
0.021
.1830017
2.200043
TTT4 |
1.21655
.5039036
2.41
0.016
.2287408
2.204359
TTT5 |
.6473157
.4979766
1.30
0.194
-.3288745
1.623506
TTT6 |
0 (omitted)
_cons |
68.60014
3.534376
19.41
0.000
61.67166
75.52863
-------------+---------------------------------------------------------------sigma_u | 19.819517
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
sigma_e |
12.95355
rho | .70069183
(fraction of variance due to u_i)
-----------------------------------------------------------------------------F test that all u_i=0: F(1358, 6787) = 9.81
Prob > F = 0.0000
We estimated the First Difference by manually differencing the data, but we can also manually
mean difference the data to estimate the Fixed Effects model. We start with generating the mean
value of a variable for each individual.
. egen MEANbeqany = mean(rbeqany), by(hhidpn)
Then creating another variable that is the difference between the value of the variable and the
individual’s mean.
. gen MDbeqany = rbeqany - MEANbeqany
Check to make sure that it worked.
. br hhidpn wave rbeqany MEANbeqany MDbeqany
Then use a loop to do it for every variable.
. foreach x in hhhres hatota rsayret {
2. egen MEAN`x' = mean(`x'), by(hhidpn)
3. gen MD`x' = `x' - MEAN`x'
4. }
Then regress the individual mean differenced variables. Note that we are omitting the constant
since we differenced it out (just like in First Differences).
. reg MDbeqany MDhhhres MDhatota MDrsayret, nocons
Source |
SS
df
MS
-------------+---------------------------------Model | 12270.6371
3 4090.21236
Residual | 1152117.71
8,151 141.346793
-------------+---------------------------------Total | 1164388.34
8,154
142.79965
Number of obs
F(3, 8151)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
8,154
28.94
0.0000
0.0105
0.0102
11.889
-----------------------------------------------------------------------------MDbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------MDhhhres | -.3409778
.2541839
-1.34
0.180
-.839243
.1572874
MDhatota |
2.108028
.2351886
8.96
0.000
1.646998
2.569057
MDrsayret | -.9354061
.3998503
-2.34
0.019
-1.719215
-.1515975
-----------------------------------------------------------------------------And rerun our xtreg, fe to make sure that the results are the same. The coefficients are the
same, however the standard errors are too small in the manually estimated version since the
degrees of freedom are not adjusted for the individual indicator variables.
. xtreg rbeqany hhhres hatota rsayret, fe
Fixed-effects (within) regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within = 0.0105
between = 0.4364
overall = 0.3065
corr(u_i, Xb)
= 0.5278
F(3,6792)
Prob > F
=
=
8,154
1,359
min =
avg =
max =
6
6.0
6
=
=
24.11
0.0000
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres | -.3409778
.2784547
-1.22
0.221
-.8868363
.2048807
hatota |
2.108028
.2576457
8.18
0.000
1.602961
2.613094
rsayret | -.9354061
.4380302
-2.14
0.033
-1.794083
-.0767297
_cons |
67.2815
3.377482
19.92
0.000
60.66057
73.90242
-------------+---------------------------------------------------------------sigma_u | 19.684013
sigma_e | 13.024156
rho | .69550892
(fraction of variance due to u_i)
-----------------------------------------------------------------------------F test that all u_i=0: F(1358, 6792) = 9.72
Prob > F = 0.0000
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
We can predict the different intercepts for each individual
. predict MU, u
And regress them on the time-invariant variables in the model to get an idea of how much of the
variation in the intercepts (or time invariant unobservables) we can explain with our observable
time invariant variables. In this case, about 10% (see R squared).
. gen female = (ragender == 2)
. reg MU raedyrs female
Source |
SS
df
MS
-------------+---------------------------------Model | 313772.768
2 156886.384
Residual | 2843254.39
8,151 348.822769
-------------+---------------------------------Total | 3157027.16
8,153 387.222759
Number of obs
F(2, 8151)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
8,154
449.76
0.0000
0.0994
0.0992
18.677
-----------------------------------------------------------------------------MU |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------raedyrs |
1.802196
.0730644
24.67
0.000
1.658971
1.945421
female | -5.998004
.4158896
-14.42
0.000
-6.813253
-5.182754
_cons | -21.06563
1.038999
-20.27
0.000
-23.10234
-19.02893
-----------------------------------------------------------------------------With Fixed Effects models we cannot include time invariant variables, however we can interact
them with time-varying variables. This allows us to estimate if the effect varies across groups.
In this case we are using the ## option to include the female indicator interacted with the
wealth variable.
. xtreg rbeqany hhhres i.female##c.hatota rsayret, fe
note: 1.female omitted because of collinearity
Fixed-effects (within) regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within = 0.0110
between = 0.0318
overall = 0.0265
corr(u_i, Xb)
= -0.1099
=
=
8,154
1,359
min =
avg =
max =
6
6.0
6
=
=
18.84
0.0000
F(4,6791)
Prob > F
--------------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
----------------+---------------------------------------------------------------hhhres | -.3438924
.2784191
-1.24
0.217
-.8896812
.2018964
1.female |
0 (omitted)
hatota |
1.570832
.4037838
3.89
0.000
.7792891
2.362375
|
female#c.hatota |
1 |
.8996671
.5207344
1.73
0.084
-.1211355
1.92047
|
rsayret | -.9205838
.4380502
-2.10
0.036
-1.7793
-.0618681
_cons |
68.48225
3.447765
19.86
0.000
61.72355
75.24095
----------------+---------------------------------------------------------------sigma_u | 21.331345
sigma_e | 13.022253
rho | .72850204
(fraction of variance due to u_i)
--------------------------------------------------------------------------------F test that all u_i=0: F(1358, 6791) = 9.62
Prob > F = 0.0000
Note that if you run the model separately for men…
. xtreg rbeqany hhhres hatota rsayret if female == 0, fe
Fixed-effects (within) regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within
= 0.0080
=
=
4,080
680
min =
6
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
between = 0.2380
overall = 0.1471
corr(u_i, Xb)
= 0.3311
avg =
max =
6.0
6
=
=
9.18
0.0000
F(3,3397)
Prob > F
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres | -.3191085
.3364081
-0.95
0.343
-.9786912
.3404742
hatota |
1.59187
.3324218
4.79
0.000
.940103
2.243637
rsayret | -1.187565
.5279356
-2.25
0.025
-2.222668
-.1524612
_cons |
77.55832
4.406477
17.60
0.000
68.91871
86.19793
-------------+---------------------------------------------------------------sigma_u | 12.630731
sigma_e | 10.663882
rho | .58383597
(fraction of variance due to u_i)
-----------------------------------------------------------------------------F test that all u_i=0: F(679, 3397) = 7.22
Prob > F = 0.0000
and women, you get different coefficients because we only interacted one of the variables in the
combined model, but when we run them separately it is like we interacted every variable.
. xtreg rbeqany hhhres hatota rsayret if female == 1, fe
Fixed-effects (within) regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within = 0.0125
between = 0.5441
overall = 0.3939
corr(u_i, Xb)
= 0.6129
=
=
4,074
679
min =
avg =
max =
6
6.0
6
=
=
14.33
0.0000
F(3,3392)
Prob > F
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres | -.3678956
.4367785
-0.84
0.400
-1.224271
.4884802
hatota |
2.457914
.38449
6.39
0.000
1.704059
3.21177
rsayret | -.6905625
.6887097
-1.00
0.316
-2.040891
.6597656
_cons |
59.31754
4.979433
11.91
0.000
49.55455
69.08053
-------------+---------------------------------------------------------------sigma_u |
24.08111
sigma_e | 15.020166
rho | .71992064
(fraction of variance due to u_i)
-----------------------------------------------------------------------------F test that all u_i=0: F(678, 3392) = 9.59
Prob > F = 0.0000
For instance, we get the same results as the stratified models (where we ran them separately) if
we interact every variable in the model.
. xtreg rbeqany i.female##c.hhhres i.female##c.hatota i.female##i.rsayret, fe
note: 1.female omitted because of collinearity
Fixed-effects (within) regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within = 0.0110
between = 0.0344
overall = 0.0286
corr(u_i, Xb)
= -0.0969
F(6,6789)
Prob > F
=
=
8,154
1,359
min =
avg =
max =
6
6.0
6
=
=
12.61
0.0000
-------------------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
---------------------+---------------------------------------------------------------1.female |
0 (omitted)
hhhres | -.3191085
.4108568
-0.78
0.437
-1.124517
.4862996
|
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
female#c.hhhres |
1 | -.0487871
.5587819
-0.09
0.930
-1.144175
1.046601
|
hatota |
1.59187
.4059883
3.92
0.000
.7960055
2.387734
|
female#c.hatota |
1 |
.8660443
.5253322
1.65
0.099
-.1637716
1.89586
|
rsayret |
1.completely reti.. | -1.187565
.6447703
-1.84
0.066
-2.451517
.0763871
|
female#rsayret |
1 #|
1.completely reti.. |
.4970022
.8788319
0.57
0.572
-1.225784
2.219788
|
_cons |
68.44464
3.450335
19.84
0.000
61.6809
75.20838
---------------------+---------------------------------------------------------------sigma_u |
21.27283
sigma_e | 13.023851
rho | .72736543
(fraction of variance due to u_i)
-------------------------------------------------------------------------------------F test that all u_i=0: F(1358, 6789) = 9.42
Prob > F = 0.0000
Note that the coefficient for the individual being retired is -1.18 for males and:
. di -1.187565 + .4970022
-.6905628
for females. Which is what we see in the stratified results above.
We can also include self-reported health using i. (correctly this time, unlike the example from
a prior lecture).
. reg rbeqany hhhres hatota rsayret i.rshlt
Source |
SS
df
MS
-------------+---------------------------------Model | 1591584.52
7 227369.218
Residual | 3353903.41
8,146 411.723964
-------------+---------------------------------Total | 4945487.94
8,153 606.585053
Number of obs
F(7, 8146)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
8,154
552.24
0.0000
0.3218
0.3212
20.291
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
.6540828
.257867
2.54
0.011
.1485976
1.159568
hatota |
8.790527
.1526396
57.59
0.000
8.491315
9.08974
rsayret | -.7791816
.4576924
-1.70
0.089
-1.676375
.1180122
|
rshlt |
2.very good | -.3837923
.6246604
-0.61
0.539
-1.608286
.8407015
3.good | -.1905096
.6543037
-0.29
0.771
-1.473112
1.092093
4.fair | -3.576621
.8139137
-4.39
0.000
-5.1721
-1.981143
5.poor | -11.38068
1.390215
-8.19
0.000
-14.10586
-8.655506
|
_cons | -19.00389
2.156946
-8.81
0.000
-23.23206
-14.77573
-----------------------------------------------------------------------------We can also change the reference category using b3.
The reference category changed from the first category (excellent health) to 3 (good health) and
now all the coefficients are in relation to 3.
. reg rbeqany hhhres hatota rsayret b3.rshlt
Source |
SS
df
MS
-------------+---------------------------------Model | 1591584.52
7 227369.218
Residual | 3353903.41
8,146 411.723964
-------------+---------------------------------Total | 4945487.94
8,153 606.585053
Number of obs
F(7, 8146)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
8,154
552.24
0.0000
0.3218
0.3212
20.291
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
.6540828
.257867
2.54
0.011
.1485976
1.159568
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
hatota |
8.790527
.1526396
57.59
0.000
8.491315
9.08974
rsayret | -.7791816
.4576924
-1.70
0.089
-1.676375
.1180122
|
rshlt |
1.excellent |
.1905096
.6543037
0.29
0.771
-1.092093
1.473112
2.very good | -.1932827
.5709417
-0.34
0.735
-1.312474
.9259088
4.fair | -3.386112
.7572668
-4.47
0.000
-4.870548
-1.901676
5.poor | -11.19017
1.351554
-8.28
0.000
-13.83956
-8.54078
|
_cons |
-19.1944
2.069981
-9.27
0.000
-23.25209
-15.13671
-----------------------------------------------------------------------------This is important since when I change the reference category to 5 (poor health), the
coefficients appear to change quite a bit and become more statistically significant. This is
because they are all in relation to 5 (poor health).
. reg rbeqany hhhres hatota rsayret b5.rshlt
Source |
SS
df
MS
-------------+---------------------------------Model | 1591584.52
7 227369.218
Residual | 3353903.41
8,146 411.723964
-------------+---------------------------------Total | 4945487.94
8,153 606.585053
Number of obs
F(7, 8146)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
8,154
552.24
0.0000
0.3218
0.3212
20.291
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
.6540828
.257867
2.54
0.011
.1485976
1.159568
hatota |
8.790527
.1526396
57.59
0.000
8.491315
9.08974
rsayret | -.7791816
.4576924
-1.70
0.089
-1.676375
.1180122
|
rshlt |
1.excellent |
11.38068
1.390215
8.19
0.000
8.655506
14.10586
2.very good |
10.99689
1.348262
8.16
0.000
8.353952
13.63983
3.good |
11.19017
1.351554
8.28
0.000
8.54078
13.83956
4.fair |
7.80406
1.421072
5.49
0.000
5.018396
10.58972
|
_cons | -30.38457
2.275688
-13.35
0.000
-34.8455
-25.92365
-----------------------------------------------------------------------------If you use the xi: command to estimate a model with indicators or fixed effects, Stata will
create variables for each category (minus the reference category).
. xi: reg rbeqany hhhres hatota rsayret i.rshlt
i.rshlt
_Irshlt_1-5
(naturally coded; _Irshlt_1 omitted)
Source |
SS
df
MS
-------------+---------------------------------Model | 1591584.52
7 227369.218
Residual | 3353903.41
8,146 411.723964
-------------+---------------------------------Total | 4945487.94
8,153 606.585053
Number of obs
F(7, 8146)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
8,154
552.24
0.0000
0.3218
0.3212
20.291
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
.6540828
.257867
2.54
0.011
.1485976
1.159568
hatota |
8.790527
.1526396
57.59
0.000
8.491315
9.08974
rsayret | -.7791816
.4576924
-1.70
0.089
-1.676375
.1180122
_Irshlt_2 | -.3837923
.6246604
-0.61
0.539
-1.608286
.8407015
_Irshlt_3 | -.1905096
.6543037
-0.29
0.771
-1.473112
1.092093
_Irshlt_4 | -3.576621
.8139137
-4.39
0.000
-5.1721
-1.981143
_Irshlt_5 | -11.38068
1.390215
-8.19
0.000
-14.10586
-8.655506
_cons | -19.00389
2.156946
-8.81
0.000
-23.23206
-14.77573
-----------------------------------------------------------------------------As we can see in the summary output below.
. su
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+--------------------------------------------------------hhidpn |
8,154
7.76e+07
6.61e+07
1.00e+07
2.09e+08
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
wave |
8,154
5.5
1.70793
3
8
rcenreg |
8,154
2.660657
.9895839
1
5
ragender |
8,154
1.499632
.5000305
1
2
raedyrs |
8,154
13.35173
2.846221
0
17
-------------+--------------------------------------------------------rshlt |
8,154
2.429605
1.052416
1
5
hatota |
8,154
12.73514
1.520786
.6931472
17.78957
rsayret |
8,154
.4540103
.497911
0
1
rbeqany |
8,154
92.97768
24.62895
0
100
hhhres |
8,154
2.126686
.8770996
1
12
-------------+--------------------------------------------------------beqdiff |
8,154
4.025018
20.12526
-100
100
hhresdiff |
8,154
-.1103753
.7670273
-10
6
wealthdiff |
8,154
.0836954
.677269 -6.339477
3.128419
retireddiff |
8,154
-.0036792
.4331829
-1
1
TTT1 |
8,154
.1666667
.3727009
0
1
-------------+--------------------------------------------------------TTT2 |
8,154
.1666667
.3727009
0
1
TTT3 |
8,154
.1666667
.3727009
0
1
TTT4 |
8,154
.1666667
.3727009
0
1
TTT5 |
8,154
.1666667
.3727009
0
1
TTT6 |
8,154
.1666667
.3727009
0
1
-------------+--------------------------------------------------------MEANbeqany |
8,154
92.97768
21.53527
0
100
MDbeqany |
8,154
1.98e-08
11.95061 -83.33334
83.33334
MEANhhhres |
8,154
2.126686
.7055845
1
6.5
MDhhhres |
8,154
-1.04e-09
.5210127 -3.333333
8.333333
MEANhatota |
8,154
12.73514
1.41213
5.408678
17.07953
-------------+--------------------------------------------------------MDhatota |
8,154
-1.27e-09
.5645154 -9.157493
3.402811
MEANrsayret |
8,154
.4540103
.3716716
0
1
MDrsayret |
8,154
-3.01e-09
.3313239 -.8333333
.8333333
MU |
8,154
2.63e-08
19.67798 -91.83736
13.45221
female |
8,154
.4996321
.5000305
0
1
-------------+--------------------------------------------------------_Irshlt_2 |
8,154
.345352
.4755121
0
1
_Irshlt_3 |
8,154
.2857493
.4517983
0
1
_Irshlt_4 |
8,154
.1293844
.3356454
0
1
_Irshlt_5 |
8,154
.0311504
.1737346
0
1
Lastly, we will conduct the Hausman test between Fixed Effects and Pooled OLS. To do this, we
need to remove any time indicators from both models and any time-invariant controls from Pooled
OLS.
Fixed Effects
. xtreg rbeqany hhhres hatota rsayret i.rshlt, fe
Fixed-effects (within) regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within = 0.0113
between = 0.4260
overall = 0.2963
corr(u_i, Xb)
= 0.5169
F(7,6788)
Prob > F
=
=
8,154
1,359
min =
avg =
max =
6
6.0
6
=
=
11.09
0.0000
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
-.33284
.2785446
-1.19
0.232
-.8788747
.2131947
hatota |
2.100239
.2578482
8.15
0.000
1.594776
2.605702
rsayret | -.9805525
.4392543
-2.23
0.026
-1.841629
-.1194764
|
rshlt |
2.very good |
-.733578
.5591741
-1.31
0.190
-1.829735
.3625785
3.good |
.2267821
.6529866
0.35
0.728
-1.053276
1.506841
4.fair |
.3095366
.8211299
0.38
0.706
-1.300135
1.919209
5.poor |
.4336902
1.332549
0.33
0.745
-2.178524
3.045904
|
_cons |
67.51886
3.398302
19.87
0.000
60.85712
74.1806
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
-------------+---------------------------------------------------------------sigma_u | 19.748141
sigma_e | 13.022953
rho | .69692381
(fraction of variance due to u_i)
-----------------------------------------------------------------------------F test that all u_i=0: F(1358, 6788) = 9.56
Prob > F = 0.0000
We need to store the estimates.
. estimates store my_fe
Pooled OLS
. reg rbeqany hhhres hatota rsayret i.rshlt
Source |
SS
df
MS
-------------+---------------------------------Model | 1591584.52
7 227369.218
Residual | 3353903.41
8,146 411.723964
-------------+---------------------------------Total | 4945487.94
8,153 606.585053
Number of obs
F(7, 8146)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
8,154
552.24
0.0000
0.3218
0.3212
20.291
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
.6540828
.257867
2.54
0.011
.1485976
1.159568
hatota |
8.790527
.1526396
57.59
0.000
8.491315
9.08974
rsayret | -.7791816
.4576924
-1.70
0.089
-1.676375
.1180122
|
rshlt |
2.very good | -.3837923
.6246604
-0.61
0.539
-1.608286
.8407015
3.good | -.1905096
.6543037
-0.29
0.771
-1.473112
1.092093
4.fair | -3.576621
.8139137
-4.39
0.000
-5.1721
-1.981143
5.poor | -11.38068
1.390215
-8.19
0.000
-14.10586
-8.655506
|
_cons | -19.00389
2.156946
-8.81
0.000
-23.23206
-14.77573
-----------------------------------------------------------------------------. estimates store my_ols
To calculate the Hausman test you need a consistent estimator (Fixed Effects) and another
estimator that might be biased if the time-invariant unobservables are correlated with X, but is
more efficient (Pooled OLS). The test tells you if there are significant enough differences in
the coefficients that you should use the Fixed Effects model rather than the more efficient but
maybe biased OLS model. Type hausman followed by the fixed effects estimates, then the OLS
estimates, followed by the sigmamore option. Cameron & Trivedi state that it is better to use
the sigmamore option in their book on page 267. Sigmamore specifies that both covariance
matrices are based on the same estimated disturbance variance from the efficient estimator
(Pooled OLS). In this case, we reject the null hypothesis that the coefficients are the same and
we should the use the consistent estimator (Fixed Effects).
. hausman my_fe my_ols, sigmamore
---- Coefficients ---|
(b)
(B)
(b-B)
sqrt(diag(V_b-V_B))
|
my_fe
my_ols
Difference
S.E.
-------------+---------------------------------------------------------------hhhres |
-.33284
.6540828
-.9869228
.3490836
hatota |
2.100239
8.790527
-6.690288
.3716256
rsayret |
-.9805525
-.7791816
-.2013709
.5088421
rshlt |
2 |
-.733578
-.3837923
-.3497857
.6073454
3 |
.2267821
-.1905096
.4172917
.779114
4 |
.3095366
-3.576621
3.886158
.9871182
5 |
.4336902
-11.38068
11.81437
1.542096
-----------------------------------------------------------------------------b = consistent under Ho and Ha; obtained from xtreg
B = inconsistent under Ha, efficient under Ho; obtained from regress
Test:
Ho:
difference in coefficients not systematic
chi2(7) = (b-B)'[(V_b-V_B)^(-1)](b-B)
=
397.37
Prob>chi2 =
0.0000
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
Calculating a Random Effects is as easy as changing , fe to , re. Random Effects allows the
inclusion of time invariant variables, such as female and education.
. xtreg rbeqany hhhres hatota rsayret female raedyrs, re
Random-effects GLS regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within = 0.0102
between = 0.4109
overall = 0.3052
corr(u_i, X)
=
=
8,154
1,359
min =
avg =
max =
6
6.0
6
=
=
853.22
0.0000
Wald chi2(5)
Prob > chi2
= 0 (assumed)
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
.0712011
.2586166
0.28
0.783
-.4356781
.5780802
hatota |
4.954339
.2075139
23.87
0.000
4.547619
5.361059
rsayret | -1.397019
.4181785
-3.34
0.001
-2.216634
-.5774039
female | -5.245325
.8875721
-5.91
0.000
-6.984934
-3.505716
raedyrs |
1.277051
.1601563
7.97
0.000
.9631506
1.590952
_cons |
15.93623
3.176974
5.02
0.000
9.709478
22.16299
-------------+---------------------------------------------------------------sigma_u | 14.983751
sigma_e | 13.024156
rho | .56962494
(fraction of variance due to u_i)
-----------------------------------------------------------------------------I am using a slimmed down version of the regression equation and will conduct the test triangle.
First, we can see the lambda (or theta in Stata) of the Random Effects model using the theta
option. Theta is included in the output, but you can also…
. xtreg rbeqany hhhres hatota rsayret, re theta
Random-effects GLS regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within = 0.0102
between = 0.4448
overall = 0.3137
corr(u_i, X)
theta
=
=
8,154
1,359
min =
avg =
max =
6
6.0
6
=
=
723.69
0.0000
Wald chi2(3)
Prob > chi2
= 0 (assumed)
= .66861607
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
.0908525
.2599972
0.35
0.727
-.4187327
.6004377
hatota |
5.429998
.2023974
26.83
0.000
5.033306
5.82669
rsayret | -1.349901
.4200215
-3.21
0.001
-2.173128
-.5266739
_cons |
24.24557
2.705589
8.96
0.000
18.94272
29.54843
-------------+---------------------------------------------------------------sigma_u |
15.13849
sigma_e | 13.024156
rho | .57465506
(fraction of variance due to u_i)
-----------------------------------------------------------------------------Use the formula for lambda from class using the sigma_u (standard deviation of the time
invariant, individual specific portion of the error – MU) and sigma_e (standard deviation of the
time-variant – NU).
. di 1 - (e(sigma_e)^2/(e(sigma_e)^2+6*e(sigma_u)^2))^.5
.66861607
Start with the OLS model and save the output.
. reg rbeqany hhhres hatota rsayret
Source |
SS
df
MS
Number of obs
=
8,154
First Difference and Fixed Effects
-------------+---------------------------------Model | 1554814.13
3 518271.378
Residual |
3390673.8
8,150 416.033596
-------------+---------------------------------Total | 4945487.94
8,153 606.585053
Moulton-UNC Chapel Hill
F(3, 8150)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
1245.74
0.0000
0.3144
0.3141
20.397
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
.6481612
.2591728
2.50
0.012
.1401164
1.156206
hatota |
9.083049
.1486387
61.11
0.000
8.791679
9.374419
rsayret | -1.173936
.4563964
-2.57
0.010
-2.068589
-.2792826
_cons | -23.54163
2.014055
-11.69
0.000
-27.4897
-19.59357
-----------------------------------------------------------------------------. est store my_OLS
Estimate the fixed effects model and check the outcome of Specification Test 1. The null is that
the intercepts are = 0. We would reject the null at greater than 99% confidence. This leads us
to think that fixed effects might be preferred to OLS.
. xtreg rbeqany hhhres hatota rsayret, fe
Fixed-effects (within) regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within = 0.0105
between = 0.4364
overall = 0.3065
corr(u_i, Xb)
= 0.5278
=
=
8,154
1,359
min =
avg =
max =
6
6.0
6
=
=
24.11
0.0000
F(3,6792)
Prob > F
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres | -.3409778
.2784547
-1.22
0.221
-.8868363
.2048807
hatota |
2.108028
.2576457
8.18
0.000
1.602961
2.613094
rsayret | -.9354061
.4380302
-2.14
0.033
-1.794083
-.0767297
_cons |
67.2815
3.377482
19.92
0.000
60.66057
73.90242
-------------+---------------------------------------------------------------sigma_u | 19.684013
sigma_e | 13.024156
rho | .69550892
(fraction of variance due to u_i)
-----------------------------------------------------------------------------F test that all u_i=0: F(1358, 6792) = 9.72
Prob > F = 0.0000
. est store my_FE
Then run the random effects model and store the output.
. xtreg rbeqany hhhres hatota rsayret, re
Random-effects GLS regression
Group variable: hhidpn
Number of obs
Number of groups
R-sq:
Obs per group:
within = 0.0102
between = 0.4448
overall = 0.3137
corr(u_i, X)
= 0 (assumed)
Wald chi2(3)
Prob > chi2
=
=
8,154
1,359
min =
avg =
max =
6
6.0
6
=
=
723.69
0.0000
-----------------------------------------------------------------------------rbeqany |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hhhres |
.0908525
.2599972
0.35
0.727
-.4187327
.6004377
hatota |
5.429998
.2023974
26.83
0.000
5.033306
5.82669
rsayret | -1.349901
.4200215
-3.21
0.001
-2.173128
-.5266739
_cons |
24.24557
2.705589
8.96
0.000
18.94272
29.54843
-------------+----------------------------------------------------------------
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
sigma_u |
15.13849
sigma_e | 13.024156
rho | .57465506
(fraction of variance due to u_i)
-----------------------------------------------------------------------------. est store my_RE
This command calculates the Breusch-Pagan test, with a null that the variance of the timeinvariant, individual specific errors are = 0. We would reject the null at greater than 99%
confidence and indicates that Random Effects might be preferred to OLS.
. xttest0
Breusch and Pagan Lagrangian multiplier test for random effects
rbeqany[hhidpn,t] = Xb + u[hhidpn] + e[hhidpn,t]
Estimated results:
|
Var
sd = sqrt(Var)
---------+----------------------------rbeqany |
606.5851
24.62895
e |
169.6286
13.02416
u |
229.1739
15.13849
Test:
Var(u) = 0
chibar2(01) =
Prob > chibar2 =
6120.85
0.0000
Here are the estimates for each of the different models.
. est table my_OLS my_RE my_FE
----------------------------------------------------Variable |
my_OLS
my_RE
my_FE
-------------+--------------------------------------hhhres | .64816117
.09085249
-.34097782
hatota | 9.0830489
5.4299981
2.1080277
rsayret | -1.1739359
-1.349901
-.9354061
_cons | -23.541634
24.245572
67.281498
----------------------------------------------------We are calculating the Hausman test by comparing a consistent estimator (Fixed Effects) under
both the null (that mu and X are uncorrelated) and that alternative hypothesis (mu and X are
correlated) to an estimator (OLS) that is more efficient under the null but might be biased
under the alternative. We reject the null that the slopes are not different at least at 99%
confidence. This indicates that Fixed Effects might be preferred to OLS.
. hausman my_FE my_OLS, sigmamore
---- Coefficients ---|
(b)
(B)
(b-B)
sqrt(diag(V_b-V_B))
|
my_FE
my_OLS
Difference
S.E.
-------------+---------------------------------------------------------------hhhres |
-.3409778
.6481612
-.989139
.3507106
hatota |
2.108028
9.083049
-6.975021
.375119
rsayret |
-.9354061
-1.173936
.2385298
.5121392
-----------------------------------------------------------------------------b = consistent under Ho and Ha; obtained from xtreg
B = inconsistent under Ha, efficient under Ho; obtained from regress
Test:
Ho:
difference in coefficients not systematic
chi2(3) = (b-B)'[(V_b-V_B)^(-1)](b-B)
=
349.79
Prob>chi2 =
0.0000
Just as above, we are calculating the Hausman test by comparing a consistent estimator (Fixed
Effects) under both the null (that mu and X are uncorrelated) and that alternative hypothesis
(mu and X are correlated) to an estimator (Random Effects) that is more efficient under the null
but might be biased under the alternative. We reject the null that the slopes are not different
at least at 99% confidence. This indicates that Fixed Effects might be preferred to Random.
. hausman my_FE my_RE, sigmamore
---- Coefficients ----
First Difference and Fixed Effects
Moulton-UNC Chapel Hill
|
(b)
(B)
(b-B)
sqrt(diag(V_b-V_B))
|
my_FE
my_RE
Difference
S.E.
-------------+---------------------------------------------------------------hhhres |
-.3409778
.0908525
-.4318303
.1173452
hatota |
2.108028
5.429998
-3.32197
.1694012
rsayret |
-.9354061
-1.349901
.4144949
.1579034
-----------------------------------------------------------------------------b = consistent under Ho and Ha; obtained from xtreg
B = inconsistent under Ha, efficient under Ho; obtained from xtreg
Test:
Ho:
difference in coefficients not systematic
chi2(3) = (b-B)'[(V_b-V_B)^(-1)](b-B)
=
386.62
Prob>chi2 =
0.0000
The tests all lead to using the Fixed Effects model.