Econometrics
Homework 4 Solutions
Computer Question
(Optional, no need to hand in)
(a)
ci may capture some state-speci…c factor that contributes to higher or low rate of accident or fatality. For example, geographical feature, culture in driving, etc.
(b)
Pooled OLS with clustered standard errors.
. reg fatalityrate sb_useage speed65 speed70 ba08 drinkage21 lnincome age yr1984-yr1997, vce(robust)
Linear regression
Number of obs
F( 21,
534)
Prob > F
R-squared
Root MSE
fatalityrate
Coef.
sb_useage
speed65
speed70
ba08
drinkage21
lnincome
age
yr1984
yr1985
yr1986
yr1987
yr1988
yr1989
yr1990
yr1991
yr1992
yr1993
yr1994
yr1995
yr1996
yr1997
_cons
.0045798
.0002029
.0019266
-.001981
-.0005615
-.0184
-.0000179
.0032855
.0037588
.004711
.0044055
.0048785
.003858
.0048058
.0037004
.0029305
.003368
.0037277
.0039587
.0044151
.0050835
.1957515
Robust
Std. Err.
.0013516
.0005631
.0005544
.0003625
.0010652
.0013082
.0001694
.0015651
.0016269
.0016913
.0018591
.0019334
.0018718
.0019123
.0019337
.0019717
.0019972
.0020124
.0021007
.0020994
.0021389
.0122827
t
P>|t|
3.39
0.36
3.47
-5.47
-0.53
-14.07
-0.11
2.10
2.31
2.79
2.37
2.52
2.06
2.51
1.91
1.49
1.69
1.85
1.88
2.10
2.38
15.94
0.001
0.719
0.001
0.000
0.598
0.000
0.916
0.036
0.021
0.006
0.018
0.012
0.040
0.012
0.056
0.138
0.092
0.065
0.060
0.036
0.018
0.000
=
=
=
=
=
556
34.73
0.0000
0.5679
.00337
[95% Conf. Interval]
.0019247
-.0009033
.0008374
-.0026931
-.002654
-.0209698
-.0003506
.000211
.000563
.0013885
.0007534
.0010806
.0001809
.0010492
-.0000981
-.0009427
-.0005554
-.0002255
-.0001679
.0002909
.0008819
.1716231
.0072349
.0013092
.0030158
-.001269
.0015309
-.0158302
.0003148
.00636
.0069547
.0080334
.0080575
.0086764
.007535
.0085623
.007499
.0068037
.0072914
.0076808
.0080854
.0085393
.0092851
.2198799
The seat belt usage here has a positive e¤ect on fatality, which is not as expected. We expect
seat belt can save the driver and passengers even when there are accidents. Higher speed limit
(the base group has a 55mph limit.) leads to a higher fatality, which make sense, as lower speed
can reduce the impact of accidents. A lower blood alchohol limit (missing group is a higher blood
alcohol level is 0.1.) is related to a lower fatality, which also makes sense because alcohol reduces
the ability of judgement for drivers, which increases number of accidents. A higher legal drinking
age is associated to a lower fatality rate, which also makes sense, though the coe¢ cient is not
signi…cantly di¤erent from zero. States with higher income is associated with a lower fatality rate,
which makes sense, as richer states may have better roads, or cars better maintained or with better
safety measures, or with people driving more carefully. The age e¤ect is not signi…cant and close
to zero (it the average age of a state, which has little variation across states). The year dummies
1
generally involves small coe¢ cients, and there is no clear trend over the decade.
(c)
Random E¤ect Estimator:
. *Random effect model
. xtreg fatalityrate sb_useage speed65 speed70 ba08 drinkage21 lnincome age yr1984-yr1997, re vce(robust)
Random-effects GLS regression
Group variable: fips
Number of obs
Number of groups
=
=
556
51
R-sq:
Obs per group: min =
avg =
max =
8
10.9
15
within = 0.7376
between = 0.2761
overall = 0.4521
corr(u_i, X)
Wald chi2(21)
Prob > chi2
= 0 (assumed)
=
=
1148.23
0.0000
(Std. Err. adjusted for 51 clusters in fips)
Robust
Std. Err.
fatalityrate
Coef.
z
sb_useage
speed65
speed70
ba08
drinkage21
lnincome
age
yr1984
yr1985
yr1986
yr1987
yr1988
yr1989
yr1990
yr1991
yr1992
yr1993
yr1994
yr1995
yr1996
yr1997
_cons
-.0030776
-.0009825
.0007266
-.0011603
-.0007095
-.0093569
.0002356
.0012335
.0015175
.0028411
.0036311
.0037392
.0025717
.0024855
.0016075
.0006467
.0009072
.0010045
.0013854
.0013555
.0016532
.1051933
.0015007
.0006319
.0004319
.0003748
.000573
.0041158
.0005193
.0011133
.0012717
.001364
.001633
.0018478
.0020652
.0021993
.0023077
.0024266
.0025183
.0027394
.0029253
.003066
.0032724
.0374545
sigma_u
sigma_e
rho
.00310665
.00161752
.78672599
(fraction of variance due to u_i)
-2.05
-1.55
1.68
-3.10
-1.24
-2.27
0.45
1.11
1.19
2.08
2.22
2.02
1.25
1.13
0.70
0.27
0.36
0.37
0.47
0.44
0.51
2.81
P>|z|
0.040
0.120
0.092
0.002
0.216
0.023
0.650
0.268
0.233
0.037
0.026
0.043
0.213
0.258
0.486
0.790
0.719
0.714
0.636
0.658
0.613
0.005
[95% Conf. Interval]
-.0060189
-.002221
-.0001199
-.0018948
-.0018327
-.0174237
-.0007822
-.0009486
-.000975
.0001677
.0004305
.0001176
-.0014759
-.0018249
-.0029155
-.0041094
-.0040286
-.0043647
-.0043481
-.0046537
-.0047607
.0317839
-.0001363
.000256
.0015731
-.0004257
.0004136
-.0012901
.0012534
.0034155
.0040099
.0055145
.0068316
.0073609
.0066194
.006796
.0061305
.0054028
.0058431
.0063737
.0071189
.0073647
.008067
.1786028
The most notable change is the coe¢ cient on seat belt usage, from positive to negative, and
marginally signi…cant. Coe¢ cient on speed65 has also become negative, but still insigni…cant. The
e¤ect of age has changed sign, but it is still very small.
(d)
Fixed E¤ect Estimator:
2
. xtreg fatalityrate sb_useage speed65 speed70 ba08 drinkage21 lnincome age yr1984-yr1997, fe vce(robust)
Fixed-effects (within) regression
Group variable: fips
Number of obs
Number of groups
=
=
556
51
R-sq:
Obs per group: min =
avg =
max =
8
10.9
15
within = 0.7506
between = 0.1139
overall = 0.0338
corr(u_i, Xb)
F(21,50)
Prob > F
= -0.5086
=
=
52.30
0.0000
(Std. Err. adjusted for 51 clusters in fips)
Robust
Std. Err.
fatalityrate
Coef.
t
P>|t|
sb_useage
speed65
speed70
ba08
drinkage21
lnincome
age
yr1984
yr1985
yr1986
yr1987
yr1988
yr1989
yr1990
yr1991
yr1992
yr1993
yr1994
yr1995
yr1996
yr1997
_cons
-.0037186
-.0007833
.0008042
-.0008225
-.0011337
.0062643
.001318
-.0004319
-.0010707
-.0005777
-.0008722
-.001885
-.0041766
-.005266
-.0066622
-.008518
-.0089399
-.0096297
-.0101123
-.0110766
-.0116075
-.0779904
.0014515
.0005801
.0004572
.0004433
.0006221
.0066992
.0006937
.001378
.0017641
.0020078
.0024939
.002877
.0032564
.0035402
.0037593
.0039855
.004199
.0045934
.0048961
.0052089
.0055341
.0663611
sigma_u
sigma_e
rho
.00575371
.00161752
.92675648
(fraction of variance due to u_i)
-2.56
-1.35
1.76
-1.86
-1.82
0.94
1.90
-0.31
-0.61
-0.29
-0.35
-0.66
-1.28
-1.49
-1.77
-2.14
-2.13
-2.10
-2.07
-2.13
-2.10
-1.18
0.013
0.183
0.085
0.069
0.074
0.354
0.063
0.755
0.547
0.775
0.728
0.515
0.206
0.143
0.082
0.037
0.038
0.041
0.044
0.038
0.041
0.245
[95% Conf. Interval]
-.0066339
-.0019484
-.0001142
-.0017128
-.0023831
-.0071913
-.0000753
-.0031998
-.004614
-.0046106
-.0058813
-.0076636
-.0107172
-.0123767
-.0142131
-.0165232
-.0173738
-.0188559
-.0199464
-.0215389
-.0227231
-.2112805
-.0008032
.0003818
.0017225
.0000678
.0001158
.01972
.0027114
.002336
.0024726
.0034551
.0041368
.0038936
.0023641
.0018448
.0008886
-.0005128
-.000506
-.0004035
-.0002782
-.0006142
-.0004919
.0552998
In terms of order of magnitudes, the random e¤ect and …xed e¤ect models are similar and again
seatbelt usage has a negative e¤ect on fatality. However, the e¤ect of income now becomes positive
and insigni…cant, while age is positive and signi…cant.
(e)
Here we test all the time varying variables:
. *first generate the means
. egen sb_usem=mean(sb_useage) , by(fips)
. egen speed65m=mean(speed65), by(fips)
. egen speed70m=mean(speed70), by(fips)
. egen ba08m=mean(ba08), by(fips)
. egen drinkm=mean(drinkage21), by(fips)
. egen lnincm=mean(lnincome), by(fips)
. egen agem=mean(age), by(fips)
3
. xtreg fatalityrate sb_useage speed65 speed70 ba08 drinkage21 lnincome age yr1984-yr1997 sb_usem-agem, re vce(robust)
Random-effects GLS regression
Group variable: fips
Number of obs
Number of groups
=
=
556
51
R-sq:
Obs per group: min =
avg =
max =
8
10.9
15
within = 0.7506
between = 0.5441
overall = 0.6206
corr(u_i, X)
Wald chi2(28)
Prob > chi2
= 0 (assumed)
=
=
1293.30
0.0000
(Std. Err. adjusted for 51 clusters in fips)
Robust
Std. Err.
fatalityrate
Coef.
z
sb_useage
speed65
speed70
ba08
drinkage21
lnincome
age
yr1984
yr1985
yr1986
yr1987
yr1988
yr1989
yr1990
yr1991
yr1992
yr1993
yr1994
yr1995
yr1996
yr1997
sb_usem
speed65m
speed70m
ba08m
drinkm
lnincm
agem
_cons
-.0036794
-.0007732
.0008026
-.0008184
-.0011352
.0062112
.001349
-.0004117
-.0010593
-.0005755
-.000891
-.0019024
-.0042006
-.005289
-.0066889
-.008548
-.0089729
-.0096658
-.0101516
-.0111185
-.0116522
.0136737
.0027627
.0056357
-.0029648
.0006738
-.0239093
-.0014076
.1948447
.0014634
.0005808
.000459
.0004446
.0006323
.0063844
.00068
.0013856
.0017642
.0019916
.0024551
.0028145
.0031721
.0034377
.0036448
.0038626
.0040654
.0044412
.004732
.0050303
.0053424
.0037679
.0021271
.005749
.0018393
.0041371
.0061603
.0006089
.0346721
sigma_u
sigma_e
rho
.00307437
.00161752
.78319974
(fraction of variance due to u_i)
-2.51
-1.33
1.75
-1.84
-1.80
0.97
1.98
-0.30
-0.60
-0.29
-0.36
-0.68
-1.32
-1.54
-1.84
-2.21
-2.21
-2.18
-2.15
-2.21
-2.18
3.63
1.30
0.98
-1.61
0.16
-3.88
-2.31
5.62
P>|z|
[95% Conf. Interval]
0.012
0.183
0.080
0.066
0.073
0.331
0.047
0.766
0.548
0.773
0.717
0.499
0.185
0.124
0.066
0.027
0.027
0.030
0.032
0.027
0.029
0.000
0.194
0.327
0.107
0.871
0.000
0.021
0.000
-.0065476
-.0019117
-.000097
-.0016897
-.0023745
-.006302
.0000162
-.0031274
-.0045171
-.004479
-.0057029
-.0074186
-.0104178
-.0120268
-.0138325
-.0161184
-.016941
-.0183704
-.0194262
-.0209777
-.0221232
.0062888
-.0014062
-.005632
-.0065697
-.0074347
-.0359833
-.002601
.1268887
-.0008111
.0003652
.0017022
.0000529
.000104
.0187243
.0026817
.002304
.0023984
.003328
.0039209
.0036139
.0020165
.0014488
.0004547
-.0009775
-.0010048
-.0009613
-.0008771
-.0012592
-.0011812
.0210586
.0069316
.0169034
.0006401
.0087823
-.0118353
-.0002142
.2628007
. test sb_usem speed65m speed70m ba08m drinkm lnincm agem
(
(
(
(
(
(
(
1)
2)
3)
4)
5)
6)
7)
sb_usem = 0
speed65m = 0
speed70m = 0
ba08m = 0
drinkm = 0
lnincm = 0
agem = 0
chi2( 7) =
Prob > chi2 =
37.07
0.0000
So, we reject the null that ci and regressors are uncorrelated, and we should use …xed e¤ect.
(f)
Here I use the …xed e¤ect speci…cation.
4
( 1)
( 2)
( 3)
( 4)
( 5)
( 6)
( 7)
( 8)
( 9)
(10)
(11)
(12)
(13)
(14)
yr1984
yr1985
yr1986
yr1987
yr1988
yr1989
yr1990
yr1991
yr1992
yr1993
yr1994
yr1995
yr1996
yr1997
=
=
=
=
=
=
=
=
=
=
=
=
=
=
0
0
0
0
0
0
0
0
0
0
0
0
0
0
F( 14,
50) =
Prob > F =
9.82
0.0000
So we reject the null that there is no time e¤ect.
(g)
. reg d.fatalityrate d.sb_useage d.speed65 d.speed70 d.ba08 d.drinkage21 d.lnincome d.age yr1984-yr1997, vce(robust) noc
Linear regression
Number of obs
F( 21,
476)
Prob > F
R-squared
Root MSE
Robust
Std. Err.
=
=
=
=
=
497
6.30
0.0000
0.2343
.00174
D.
fatalityrate
Coef.
sb_useage
D1.
-.0026035
.0012698
-2.05
0.041
-.0050986
-.0001084
speed65
D1.
.0004715
.0005128
0.92
0.358
-.0005361
.0014792
speed70
D1.
.0000269
.0003734
0.07
0.943
-.0007068
.0007606
ba08
D1.
-.0002792
.0003967
-0.70
0.482
-.0010587
.0005003
drinkage21
D1.
-.0010048
.0005709
-1.76
0.079
-.0021266
.000117
lnincome
D1.
.0112131
.0061541
1.82
0.069
-.0008795
.0233056
age
D1.
.0018264
.0012464
1.47
0.143
-.0006227
.0042755
yr1984
yr1985
yr1986
yr1987
yr1988
yr1989
yr1990
yr1991
yr1992
yr1993
yr1994
yr1995
yr1996
yr1997
-.0014063
-.0019675
.0001575
-.0016881
-.0017077
-.002627
-.0016663
-.001701
-.0022125
-.0007109
-.0010295
-.0005935
-.0012865
-.0008694
.0011957
.0006068
.000565
.0006983
.0005591
.0005816
.0004864
.0003556
.0004291
.0003267
.0003975
.0003941
.0004066
.0003739
-1.18
-3.24
0.28
-2.42
-3.05
-4.52
-3.43
-4.78
-5.16
-2.18
-2.59
-1.51
-3.16
-2.33
0.240
0.001
0.781
0.016
0.002
0.000
0.001
0.000
0.000
0.030
0.010
0.133
0.002
0.020
-.0037559
-.0031598
-.0009528
-.0030603
-.0028062
-.0037699
-.002622
-.0023998
-.0030556
-.0013528
-.0018106
-.0013679
-.0020856
-.0016041
.0009432
-.0007752
.0012677
-.0003159
-.0006092
-.0014842
-.0007105
-.0010022
-.0013694
-.0000689
-.0002485
.0001808
-.0004875
-.0001347
t
P>|t|
5
[95% Conf. Interval]
. *use difference in year dummies too
. reg d.fatalityrate d.sb_useage d.speed65 d.speed70 d.ba08 d.drinkage21 d.lnincome d.age d.yr1984 d.yr1985 d.yr1986 d.y
> r1987 d.yr1988 d.yr1989 d.yr1990 d.yr1991 d.yr1992 d.yr1993 d.yr1994 d.yr1995 d.yr1996 d.yr1997, vce(robust) noc
Linear regression
Number of obs
F( 21,
476)
Prob > F
R-squared
Root MSE
Robust
Std. Err.
=
=
=
=
=
497
6.30
0.0000
0.2343
.00174
D.
fatalityrate
Coef.
sb_useage
D1.
-.0026035
.0012698
-2.05
0.041
-.0050986
-.0001084
speed65
D1.
.0004715
.0005128
0.92
0.358
-.0005361
.0014792
speed70
D1.
.0000269
.0003734
0.07
0.943
-.0007068
.0007606
ba08
D1.
-.0002792
.0003967
-0.70
0.482
-.0010587
.0005003
drinkage21
D1.
-.0010048
.0005709
-1.76
0.079
-.0021266
.000117
lnincome
D1.
.0112131
.0061541
1.82
0.069
-.0008795
.0233056
age
D1.
.0018264
.0012464
1.47
0.143
-.0006227
.0042755
yr1984
D1.
-.0014063
.0011957
-1.18
0.240
-.0037559
.0009432
yr1985
D1.
-.0033738
.0014915
-2.26
0.024
-.0063046
-.0004431
yr1986
D1.
-.0032164
.0018058
-1.78
0.076
-.0067647
.0003319
yr1987
D1.
-.0049045
.0021944
-2.23
0.026
-.0092165
-.0005925
yr1988
D1.
-.0066122
.0025908
-2.55
0.011
-.011703
-.0015213
yr1989
D1.
-.0092392
.0030107
-3.07
0.002
-.015155
-.0033234
yr1990
D1.
-.0109055
.0033457
-3.26
0.001
-.0174798
-.0043312
yr1991
D1.
-.0126065
.0035386
-3.56
0.000
-.0195597
-.0056533
yr1992
D1.
-.014819
.0038478
-3.85
0.000
-.0223798
-.0072582
yr1993
D1.
-.0155298
.0040818
-3.80
0.000
-.0235503
-.0075093
yr1994
D1.
-.0165594
.0043401
-3.82
0.000
-.0250875
-.0080313
yr1995
D1.
-.0171529
.0046398
-3.70
0.000
-.0262699
-.0080359
yr1996
D1.
-.0184395
.0049572
-3.72
0.000
-.0281802
-.0086987
yr1997
D1.
-.0193089
.0052399
-3.68
0.000
-.0296051
-.0090127
t
P>|t|
[95% Conf. Interval]
The result is similar to Fixed e¤ect estimator.
(h)
Using FE or FD estimator, it is found that seat belt, lower speed limit, lower alcohol
allowance, higher minimum drinking age can reduce fatality rate.
(i)
Using …xed e¤ect estimates, this means we increase sb_usage from 0.52 to 0.90, the
fatality rate then decreases by (0:90 0:52)( :0037186) = 1:413 1 10 3 : If there are 50000
million miles travelled per year, then the number of death reduced is 1:413 1 10 3 50000 ' 71:
This question is taken from Stock and Watson textbook. It comes from the paper Cohen and
Einav (2003) "The E¤ect of Mandatory Seat Belt Laws on Driving Behavior and Tra¢ c Fatality"
The Review of Economics and Statistics, 85(4): 828-843.
6