Introduc tion to E c onome tric s Cha pte r 6 E ze quie l Urie l - - PowerPoint PPT Presentation
Introduc tion to E c onome tric s Cha pte r 6 E ze quie l Urie l Jim ne z Unive rsity of Va le nc ia Va le nc ia , Se pte mbe r 2013 6 Re la xing the a ssumptions in the line a r c la ssic a l mode l 6.1 Re la xing the a ssumptions
Unive rsity of Va le nc ia Va le nc ia , Se pte mbe r 2013
6.1 Re la xing the a ssumptions in the line a r c la ssic a l mode l: a n
6.2 Misspe c ific a tion 6.3 Multic olline a rity 6.4 Norma lity te st 6.5 He te roske da stic ity 6.6 Autoc orre la tion E xe rc ise s Appe ndix
T ABL E 6.1. Summa ry of bia s in whe n x2 is omitte d in e stima ting e qua tion.
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6 Relaxing the assumptions in the linear classical model
2
Corr(x 2,x 3)>0 Corr(x 2,x 3)<0 3>0 Positive bias Negative bias 3<0 Negative bias Positive bias
E XAMPL E 6.1 Misspe c ific a tion in a mode l for de te rmina tion of wa g e s (file wa g e 06sp)
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6 Relaxing the assumptions in the linear classical model
1 2 3
Initial model wage educ tenure u b b b = + + +
(1.55) (0.146) (0.071) 2
4.679 0.681 0.293 0.249 150
i i i init
wage educ tenure R n = + + = =
2 3 1 2 3 1 1 2
Augmented model 0.289
augm
wage educ tenure wage wage u R b b b a a = + + + + + =
2 2 2
( ) / 4.18 (1 ) / ( )
augm init augm
R R r F R n h
E XAMPL E 6.2 Ana lyzing multic olline a rity in the c a se of la bor a bse nte e ism (file a bse nt)
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6 Relaxing the assumptions in the linear classical model
T ABL E 6.2. T
.
E XAMPL E 6.3 Ana lyzing the multic olline a rity of fa c tors de te rmining time de vote d to house work (file timuse 03)
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6 Relaxing the assumptions in the linear classical model
1 2 3 4 5 max min
542.14 8782 7.06 06 houswork educ hhinc age paidwork u E
T ABL E 6.3. E ig e nva lue s a nd va ria nc e de c omposition proportions.
Va ria nc e de c omposition proportions
Associated Eigenvalue Variable 1 2 3 4 5 C 0.999995 4.72E-06 8.36E-09 1.23E-13 1.90E-15 EDUC 0.295742 0.704216 4.22E-05 2.32E-09 3.72E-11 HHINC 0.064857 0.385022 0.209016 0.100193 0.240913 AGE 0.651909 0.084285 0.263805 5.85E-07 1.86E-08 PAIDWORK 0.015405 0.031823 0.007178 0.945516 7.80E-05
Eigenvalues 7.03E-06 0.000498 0.025701 1.861396 542.1400
E XAMPL E 6.4 Is the hypothe sis of norma lity a c c e pta ble in the mode l to a na lyze the e ffic ie nc y of the Ma drid Stoc k E xc ha ng e ? (file bolma de f)
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6 Relaxing the assumptions in the linear classical model
T ABL E 6.4. Norma lity te st in the mode l on the Ma drid Stoc k E xc ha ng e .
n=247
skewness coefficient kurtosis coefficient Bera and Jarque statistic
4.4268 21.0232
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6 Relaxing the assumptions in the linear classical model
F IGURE 6.1. Sc a tte r dia g ra m c orre sponding to a mode l with homoske da stic disturba nc e s. F IGURE 6.2. Sc a tte r dia g ra m c orre sponding to a mode l with he te roske da stic disturba nc e s.
y x
y x
E XAMPL E 6.5 Applic a tion of the Bre usc h- Pa g a n- Godfre y te st
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6 Relaxing the assumptions in the linear classical model
T ABL E 6.5. Hoste l a nd inc da ta .
i hostel inc 1 17 500 2 24 700 3 7 250 4 17 430 5 31 810 6 3 200 7 8 300 8 42 760 9 30 650 10 9 320
S te p 1. Applying OL Sto the mo de l,
1 2
hostel inc u b b + +
(3.48) (0.0065)
7.427 0.0533
i i
hostel inc = - +
using data fro m table 6.5, the fo llo wing e stimate d mo de l is o btaine d: T he re siduals c o rre spo nding to this fitte d mo de l appe ar in table 6.6.
E XAMPL E 6.5 Applic a tion of the Bre usc h- Pa g a n- Godfre y te st. (Cont.)
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6 Relaxing the assumptions in the linear classical model
T ABL E 6.6. Re sidua ls of the re g re ssion of hoste l on inc .
S te p 2. T he auxiliary re gre ssio n
2 1 2 2
ˆ ˆ 23.93 0.0799
i i i 2 i
u inc u inc R 5 45
( )
2
10 0.56 5.05
ar
BPG nR = = =
i 1 2 3 4 5 6 7 8 9 10
1.1 1.505
8.913 2.777
S te p 3. T he BPG statistic s is: S te p 4. Give n that =3.84, the null hypo the sis o f ho mo ske dastic ity is re je c te d fo r a signific anc e le ve l o f 5%, but no t fo r the signific anc e le ve l o f 1%.
2(0.05) 1
E XAMPL E 6.6 Applic a tion of the White te st
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6 Relaxing the assumptions in the linear classical model
S te p 1. T his ste p is the same as in the Bre usc h-Pagan-Go dfre y te st.
1 2 2 3 2 2 1 2 3 2 2
1 1 ˆ ˆ 14.29 0.10 0.00018 0.
i i i i i i i i i 2 i i i
i inc inc u inc inc u inc inc R 56
( )
2
10 0.56 5.60 W nR = = =
S te p 2. T he re gre sso rs o f the auxiliary re gre ssio n will be S te p 4. Give n that =4.61, the null hypo the sis o f ho mo ske dastic ity is re je c te d fo r a 10% signific anc e le ve l be c ause W=nR2>4.61, but no t fo r signific anc e le ve ls o f 5% and 1%.
2(0.10) 2
S te p 3. T he W statistic :
E XAMPL E 6.7 He te roske da stic ity te sts in mode ls e xpla ining the ma rke t va lue of the Spa nish ba nks (file bolma d95)
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6 Relaxing the assumptions in the linear classical model
H e te ro ske dastic ity in the line ar mo de l
1 2 (30.85) (0.127)
29.42 1.219 20 marktval bookval u marktval bookval n b b = + + + =
50 100 150 200 250 300 350 400 100 200 300 400 500 600 700
Residuals in absolute value bookval
GRAPHIC 6.1. Sc a tte r plot be twe e n the re sidua ls in a bsolute va lue a nd the va ria ble bookval in the line a r mode l.
As =6.64<10.44, the null hypo the sis o f ho mo ske dastic ity is re je c te d fo r a signific anc e le ve l o f 1%, and the re fo re fo r=0.05 and fo r =0.10.l As =9.21<12.03, the null hypo the sis o f ho mo ske dastic ity is re je c te d fo r a signific anc e le ve l o f 1%.
2
ar
2(0.01) 1
2
20 0.6017 12.03
ar
W nR
2(0.01) 2
E XAMPL E 6.7 He te roske da stic ity te sts in mode ls e xpla ining the ma rke t va lue of the Spa nish ba nks (Cont.)
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6 Relaxing the assumptions in the linear classical model
H e te ro ske dastic ity in the lo g-lo g mo de l
(0.265) (0.062)
ln( ) 0.676 0.9384ln( ) marktval bookval +
GRAPHIC 6.2. Sc a tte r plot be twe e n the re sidua ls in a bsolute va lue a nd the va ria ble bookval in the log - log mode l.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
Residuals in absolute value ln(bookval)
T ABL E 6.7. T e sts of he te roske da stic ity on the log -log mode l to e xpla in the ma rke t va lue of Spa nish ba nks.
Test Statistic Table values Breusch-Pagan BP = =1.05 =4.61 White W= =2.64 =4.61
2 ra
nR
2(0.10) 2
2 ra
nR
2(0.10) 2
E XAMPL E 6.8 Is the re he te roske da stic ity in de ma nd of hoste l se rvic e s? (file hoste l)
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6 Relaxing the assumptions in the linear classical model
GRAPHIC 6.3. Sc a tte r plot be twe e n the re sidua ls in a bsolute va lue a nd the va ria ble ln(inc ) in the hoste l mode l. T ABL E 6.8. T e sts of he te roske da stic ity in the mode l of de ma nd for hoste l se rvic e s.
( )
1 2 3 4 5 (2.26) (0.324) (0.258) (0.088) (0.333) 2
ln ln( ) ln( ) 16.37 2.732ln( ) 1.398 2.972 0.444 0.921 40
i i i i i
hostel inc secstud terstud hhsize u hostel inc secstud terstud hhsize R n b b b b b + + + + +
+ +
=
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8
Residuals in absolute value ln(inc)
Test Statistic Table values Breusch-Pagan- Godfrey BPG= =7.83 =5.99 White W= =12.24 =9.21
2(0.05) 2
2(0.01) 2
2 ra
nR
2 ra
nR
E XAMPL E 6.9 He te roske da stic ity c onsiste nt sta nda rd e rrors in the mode ls e xpla ining the ma rke t va lue of Spa nish ba nks (Continua tion of e xa mple 6.7) (file bolma d95)
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6 Relaxing the assumptions in the linear classical model
(30.85) (0.127) (0.265) (0.062)
29.42 1.219 ln( ) 0.676 0.9384ln( ) marktval bookval marktval bookval + +
(18.67) (0.249) (0.3218) (0.0698)
29.42 1.219 ln( ) 0.676 0.9384ln( ) marktval bookval marktval bookval + +
No n c o nsiste nt White pro c e dure
E XAMPL E 6.10 Applic a tion of we ig hte d le a st squa re s in the de ma nd of hote l se rvic e s (Continua tion of e xa mple 6.8) (file hoste l)
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6 Relaxing the assumptions in the linear classical model
2 (0.143) (2.73) 2 ( 1.34) (2.82) 2 (5.39) ( 2.87) ( 2.46)
ˆ 0.0239 0.0003 0.1638 ˆ 0.4198 0.0235 0.1733 1 ˆ 0.8857 532.1 0.1780 ˆ 2.7033 0.438
i i i i
u inc R u inc R u R inc u
2 (2.88) 9ln(
) 0.1788 inc R
(2.15) (0.309) (0.247) (0.085) (0.326) 2
ln( ) 16.21 2.709ln( ) 1.401 2.982 0.445 0.914 40
i i i i i
hostel inc secstud terstud hhsize R n - + + +
=
WL S e stimatio n
F IGURE 6.3. Plot of non- a utoc orre la te d disturba nc e s.
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6 Relaxing the assumptions in the linear classical model
1 2 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
u
time
F IGURE 6.4. Plot of positive a utoc or r e la te d distur ba nc e s.
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6 Relaxing the assumptions in the linear classical model
1 2 3 4 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
u
time
1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
u
time
F IGURE 6.5. Plot of ne g a tive a utoc or r e la te d distur ba nc e s.
F IGURE 6.6. Autoc orre la te d disturba nc e s due to a spe c ific a tion bia s.
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6 Relaxing the assumptions in the linear classical model
y x
E XAMPL E 6.11 Autoc orre la tion in the mode l to de te rmine the e ffic ie nc y of the Ma drid Stoc k E xc ha ng e (file bolma de f)
[20]
6 Relaxing the assumptions in the linear classical model Sinc e DW=2.04>dU, we do no t re je c t the null hypo the sis that the disturbanc e s are no t auto c o rre late d fo r a signific anc e le ve l o f =0.01, i.e .
dL=1.664; dU=1.684
1 2 3 4
GRAPHIC 6.4. Sta nda rdize d re sidua ls in the e stima tion of the mode l to de te rmine the e ffic ie nc y of the Ma drid Stoc k E xc ha ng e .
E XAMPL E 6.12 Autoc orre la tion in the mode l for the de ma nd for fish (file fishde m)
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6 Relaxing the assumptions in the linear classical model Sinc e dL<1.202<dU, the re is no t e no ugh e vide nc e to ac c e pt the null hypo the sis, o r to re je c t it. F
GRAPHIC 6.5. Sta nda rdize d re sidua ls in the mode l on the de ma nd for fish.
dL=0.969; dU=1.415
1 2 3 2 4 6 8 10 12 14 16 18 20 22 24 26 28
E XAMPL E 6.13 Autoc orre la tion in the c a se of L ydia E . Pinkha m (file pinkha m)
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6 Relaxing the assumptions in the linear classical model Give n this value o f h, the null hypo the sis o f no auto c o rre latio n is re je c te d fo r =0.01 o r, e ve n, fo r =0.001, ac c o rding to the table o f the no rmal distributio n.
GRAPHIC 6.6. Sta nda rdize d re sidua ls in the e stima tion of the mode l of the L ydia E . Pinkha m c a se .
( ) ( )
2
1.2012 53 ˆ 1 1 ˆ ˆ 2 2 1 53 0.0814 1 var 1 var
j j
n d n h n n r b b é ù é ù ê ú ê ú
ú ê ú
û ë û
0,0 1,0 2,0 3,0 4,0 5,0 8 13 18 23 28 33 38 43 48 53 58
E XAMPL E 6.14 Autoc orre la tion in a mode l to e xpla in the e xpe nditure s of re side nts a broa d (file qna ta c sp)
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6 Relaxing the assumptions in the linear classical model
F
null hypo the sis o f no auto c o rre latio n is re je c te d fo r =0.01, sinc e =15.09.
GRAPHIC 6.7. Sta nda rdize d re sidua ls in the e stima tion of the mode l e xpla ining the e xpe nditure s of re side nts a broa d.
(3.43) (0.276) 2
t t
0.0 0.5 1.0 1.5 2.0 2.5 5 10 15 20 25 30 35 40 45
2 ar
2( ) 5
E XAMPL E 6.15 HAC sta nda rd e rrors in the c a se of L ydia E . Pinkha m (Continua tion of e xa mple 6.13) (file pinkha m)
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6 Relaxing the assumptions in the linear classical model
T ABL E 6.9.T he t sta tistic s, c onve ntiona l a nd HAC, in the c a se of L ydia E . Pinkha m.