Post on 04-Jun-2018
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Multiple Regression
(SW Chapter 5)
OLS estimate of the Test Score/STRrelation:
TestScore= 698.9 2.28STR, R2= .05, SER= 18.6
(10.4) (0.52)
Is this a credible estimate of the causal effect o test
scores of a cha!e i the studet"teacher ratio#
No$ there are omitted cofoudi! factors (famil%
icome& 'hether the studets are atie !lishs*ea+ers) that bias the - estimator$ STRcould be
/*ic+i! u* the effect of these cofoudi! factors.
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Omitted Variable ias
(SW Se!tion 5"#)
he bias i the - estimator that occurs as a result of
a omitted factor is called omitted variablebias. or
omitted ariable bias to occur, the omitted factor /Z
must be$1. a determiat of Y& and
2. correlated 'ith the re!ressorX.
Both conditions must hold for the omission of Z to result
in omitted variable bias.
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I the test score e3am*le$
1. !lish la!ua!e abilit% ('hether the studet has
!lish as a secod la!ua!e) *lausibl% affects
stadardied test scores$ Zis a determiat of Y.
2. Immi!rat commuities ted to be less affluet ad
thus hae smaller school bud!ets ad hi!her STR$
Zis correlated 'ithX.
ccordi!l%, 16 is biased
7hat is the directio of this bias# 7hat does commo sese su!!est#
If commo sese fails %ou, there is a formula
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formula for omitted ariable bias$ recall the e:uatio,
1
6
1=
1
2
1
( )
( )
n
i i
i
n
i
i
X X u
X X
=
=
=1
2
1
1
n
i
i
X
vn
n sn
=
'here vi= (Xi X)ui
(XiX)ui. ;der -east :uaresssum*tio = co(Xi,ui) = 0.
?ut 'hat ifE(XiX)ui> = co(Xi,ui) = Xu0#
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he
16 1=1
2
1
( )
( )
n
i i
i n
i
i
X X u
X X=
=
=
1
2
1
1
n
i
i
X
vn
ns
n
=
so
E( 16 ) 1=
1
2
1
( )
( )
n
i i
i
n
i
i
X X u
E
X X
=
=
2
Xu
X
=
u Xu
X X u
'here holds 'ith e:ualit% 'he nis lar!e& s*ecificall%,
16
p
1@u
Xu
X
, 'hereXu= corr(X,u)
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Omitted $ariable bias formula$ 16
p
1@u
Xu
X
.
If a omitted factorZis both$
(1) a determiat of Y(that is, it is cotaied i u)&
and
(2) correlated 'ithX,
theXu0 ad the - estimator 16 is biased.
he math ma+es *recise the idea that districts 'ith fe'
- studets (1) do better o stadardied tests ad (2)hae smaller classes (bi!!er bud!ets), so i!ori! the
- factor results i oerstati! the class sie effect.
Is this is actually going on in the C data#
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Aistricts 'ith fe'er !lish -earers hae hi!her testscores
Aistricts 'ith lo'er *ercetE!("ctE!) hae smallerclasses
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mo! districts 'ith com*arable"ctE!, the effect of classsie is small (recall oerall /test score !a* = B.4)
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%hree &a's to o$er!ome omitted $ariable bias
1. Cu a radomied cotrolled e3*erimet i 'hich
treatmet (STR) is radoml% assi!ed$ the"ctE!is
still a determiat of TestScore, but"ctE!is
ucorrelated 'ith STR. (#ut this is unrealistic in
practice$)
2. do*t the /cross tabulatio a**roach, 'ith fier!radatios of STRad"ctE!(#ut soon %e %ill run
out of data& and %hat about other determinants li'e
family income and parental education#)
. ;se a method i 'hich the omitted ariable ("ctE!) is
o lo!er omitted$ iclude"ctE!as a additioal
re!ressor i a multi*le re!ressio.
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%he opulation Multiple Regression Model
(SW Se!tion 5")
Dosider the case of t'o re!ressors$
Yi= 0@ 1X1i@ 2X2i@ ui, i= 1,,n
X1,X2are the t'o independent variables(regressors)
(Yi,X1i,X2i) deote the ithobseratio o Y,X1, adX2.
0= u+o' *o*ulatio iterce*t 1= effect o Yof a cha!e iX1, holdi!X2costat
2= effect o Yof a cha!e iX2, holdi!X1costat
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ui= /error term (omitted factors)
Interpretation of multiple regression coefficients
Yi= 0@ 1X1i@ 2X2i@ ui, i= 1,,n
Dosider cha!i!X1b% X1'hile holdi!X2costat$
Eo*ulatio re!ressio lie before the cha!e$
Y= 0@ 1X1@ 2X2
Eo*ulatio re!ressio lie, after the cha!e$
Y@
Y= 0@ 1(X1@
X1) @ 2X2
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Before$ Y= 0@ 1(X1@ X1) @ 2X2
After$ Y@ Y= 0@ 1(X1@ X1) @ 2X2
Difference$ Y= 1X1hat is,
1=1
Y
X
, holdingX!onstant
also,
2=2
Y
X
, holdingX#!onstant
ad
0= *redicted alue of Y'heX1=X2= 0.
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%he OLS *stimator in Multiple Regression
(SW Se!tion 5"+)
7ith t'o re!ressors, the - estimator soles$
0 1 2
2
, , 0 1 1 2 2
1
mi ( )>n
b b b i i i
i
Y b b X b X
=
+ +
he - estimator miimies the aera!e s:uareddifferece bet'ee the actual alues of Yiad the
*redictio (*redicted alue) based o the estimated lie.
his miimiatio *roblem is soled usi! calculus
he result is the OLS estimators of ,and #.5"1
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*-ample: the California test s!ore data
Ce!ressio of TestScorea!aist STR$
TestScore= 698.9 2.28STR
Fo' iclude *ercet !lish -earers i the district("ctE!)$
TestScore= 696.0 1.10STR 0.65"ctE!
7hat ha**es to the coefficiet o STR#
7h%# (Note$ corr(STR,"ctE!) = 0.19)5"14
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Multiple regression in S%.%.
reg testscr str pctel, robust;
Regression with robust standard errors Number of obs = 420 F( 2, 41! = 22"#$2 %rob & F = 0#0000 R'suared = 0#42)4 Root *+ = 14#4)4
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' - Robust
testscr - .oef# +td# rr# t %&-t- / .onf# 3nteral5'''''''''''''6'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' str - '1#1012) #4"2$42 '2#4 0#011 '1#21" '#204)1) pctel - '#)4)$ #0"10"1$ '20#4 0#000 '#10 '#$$$) 7cons - )$)#0"22 $#2$224 $#)0 0#000 ))$#$4 0"#1$''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
TestScore= 696.0 1.10STR 0.65"ctE!
7hat are the sam*li! distributio of 16 ad 2
#
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%he Least Suares .ssumptions for Multiple
Regression (SW Se!tion 5"0)
Yi= 0@ 1X1i@ 2X2i@ @ 'X'i@ ui, i= 1,,n
1. he coditioal distributio of u!ie theXGs has
mea ero, that is,E(uHX1=(1,,X'=(') = 0.2. (X1i,,X'i&Yi), i=1,,n, are i.i.d.
. X1,,X', ad uhae four momets$E( 4
1iX ) ,,
E(
4
'iX ) ,E(
4
iu ) .4. here is o *erfect multicolliearit%.
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.ssumption 1#: the !onditional mean of ugi$en the
in!ludedX2s is 3ero"
his has the same iter*retatio as i re!ressio'ith a si!le re!ressor.
If a omitted ariable (1) belo!s i the e:uatio (so
is i u) ad (2) is correlated 'ith a icludedX, the
this coditio fails
ailure of this coditio leads to omitted ariable
bias he solutio if possible is to iclude the omitted
ariable i the re!ressio.
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.ssumption 1: (X#i44Xki,Yi)4 i6#44n4 are i"i"d"
his is satisfied automaticall% if the data are collected
b% sim*le radom sam*li!.
.ssumption 1+: finite fourth moments
his is techical assum*tio is satisfied automaticall%
b% ariables 'ith a bouded domai (test scores,
"ctE!, etc.)
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.ssumption 10: %here is no perfe!t multi!ollinearit'
erfect multicollinearit!is 'he oe of the re!ressors is
a e3act liear fuctio of the other re!ressors.
"#ample$ u**ose %ou accidetall% iclude STRt'ice$
regress testscr str str, robust
Regression with robust standard errors Number of obs = 420 F( 1, 41$! = 1#2)
%rob & F = 0#0000
R'suared = 0#012
Root *+ = 1$#$1
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
- Robust
testscr - .oef# +td# rr# t %&-t- / .onf# 3nteral5
''''''''6''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
str - '2#2$0$ #14$2 '4#" 0#000 '"#"004 '1#2$)1
str- (dropped!
7cons - )$#"" 10#")4") )#44 0#000 )$#)02 1#"0
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
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erfect multicollinearit!is 'he oe of the re!ressors is
a e3act liear fuctio of the other re!ressors.
I the *reious re!ressio, 1is the effect o TestScoreof a uit cha!e i STR, holdi! STRcostat (###)
ecod e3am*le$ re!ress TestScoreo a costat,),ad#, 'here$)i= 1 if STRJ 20, = 0 other'isei= 1
if STRK20, = 0 other'ise, so#i= 1 )iad there is
*erfect multicolliearit%
7ould there be *erfect multicolliearit% if the iterce*t
(costat) 'ere someho' dro**ed (that is, omitted orsu**ressed) i the re!ressio#
Eerfect multicolliearit% usuall% reflects a mista+e ithe defiitios of the re!ressors, or a oddit% i the data
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%he Sampling 7istribution of the OLS *stimator
(SW Se!tion 5"5)
;der the four -east :uares ssum*tios,
he e3act (fiite sam*le) distributio of 16 has mea
1, ar( 16 ) is iersel% *ro*ortioal to n& so too for 2
.
ther tha its mea ad ariace, the e3act
distributio of 16 is er% com*licated
1
6 is cosistet$ 16
p
1(la' of lar!e umbers)
1 1
1
( )
ar( )
E
is a**ro3imatel% distributedN(0,1) (D-)
o too for 2 ,, '
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8'pothesis %ests and Confiden!e 9nter$als for a
Single Coeffi!ient in Multiple Regression
(SW Se!tion 5")
1 1
1
( )
ar( )
E
is a**ro3imatel% distributedN(0,1) (D-).
hus h%*otheses o 1ca be tested usi! the usual t"statistic, ad cofidece iterals are costructed as L
16 1.96SE( 1
6)M.
o too for 2,, '.
16 ad 2 are !eerall% ot ide*edetl% distributed
so either are their t"statistics (more o this later).
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"#ample: he Daliforia class sie data
(1)TestScore= 698.9 2.28STR
(10.4) (0.52)
(2) TestScore= 696.0 1.10STR 0.650"ctE!
(8.B) (0.4) (0.01)
he coefficiet o STRi (2) is the effect oTestScoresof a uit cha!e i STR, holdi! costat
the *erceta!e of !lish -earers i the district
Doefficiet o STRfalls b% oe"half 95N cofidece iteral for coefficiet o STRi (2)
is L1.10 1.960.4M = (1.95, 0.26)
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%ests of ;oint 8'potheses
(SW Se!tion 5"
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TestScorei= 0@ 1STRi@ 2E(pni@ "ctE!i@ ui
*0$ 1= 0 and2= 0
s.*1$ either10 or20 or both
$oint h!pothesiss*ecifies a alue for t'o or more
coefficiets, that is, it im*oses a restrictio o t'o or
more coefficiets.
/commo sese test is to reOect if either of the
idiidual t"statistics e3ceeds 1.96 i absolute alue. ?ut this /commo sese a**roach doesGt 'or+P
he resulti! test doesGt hae the ri!ht si!ificace
leelP5"25
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*ere+s %hy$ Dalculatio of the *robabilit% of icorrectl%
reOecti! the ull usi! the /commo sese test based o
the t'o idiidual t"statistics. o sim*lif% the
calculatio, su**ose that 16 ad 2
are ide*edetl%
distributed. -et t1ad t2be the t"statistics$
t1=1
1
0( )SE
ad t2= 2
2
0( )SE
he /commo sese test is$
reOect*0$ 1= 2= 0 if Ht1H K 1.96 adQor Ht2H K 1.96
7hat is the *robabilit% that this /commo sese test
reOects*0, 'he*0is actuall% true# (Itshouldbe 5N.)5"26
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Erobabilit% of icorrectl% reOecti! the ull
=0
Er* Ht1H K 1.96 adQor Ht2H K 1.96>
=0
Er* Ht1H K 1.96, Ht2H K 1.96>
@0
Er* Ht1H K 1.96, Ht2H J 1.96>
@0
Er* Ht1H J 1.96, Ht2H K 1.96> (disOoit eets)
=0
Er* Ht1H K 1.96> 0Er* Ht2H K 1.96>
@0
Er* Ht1H K 1.96> 0Er* Ht2H J 1.96>
@0
Er* Ht1H J 1.96> 0Er* Ht2H K 1.96>
(t1, t2are ide*edet b% assum*tio)
= .05.05 @ .05.95 @ .95.05
= .09B5 = 9.B5N 'hich is notthe desired 5NPP
he si%eof a test is the actual reOectio rate uder the ull
h%*othesis.5"2B
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%he &=statisti!
he,"statistic tests all *arts of a Ooit h%*othesis at oce.
;*leasat formula for the s*ecial case of the Ooit
h%*othesis 1= 1,0ad 2= 2,0i a re!ressio 'ith t'o
re!ressors$
,=1 2
1 2
2 2
1 2 , 1 2
2
,
21
2 1
t t
t t
t t t t
+
'here1 2,
t t estimates the correlatio bet'ee t1ad t2.
CeOect 'he,is /lar!e5"29
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he,"statistic testi! 1ad 2(s*ecial case)$
,=1 2
1 2
2 2
1 2 , 1 2
2
,
21
2 1
t t
t t
t t t t
+
he,"statistic is lar!e 'he t1adQor t2is lar!e
he,"statistic corrects (i Oust the ri!ht 'a%) for thecorrelatio bet'ee t1ad t2.
he formula for more tha t'o Gs is reall% ast%
uless %ou use matri3 al!ebra.
his !ies the,"statistic a ice lar!e"sam*lea**ro3imate distributio, 'hich is
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Large=sample distribution of the &=statisti!
Dosider s*ecial case that t1ad t2are ide*edet, so
1 2,
t t
p
0& i lar!e sam*les the formula becomes
,=1 2
1 2
2 2
1 2 , 1 2
2
,
621
62 1
t t
t t
t t t t
+
2 2
1 2
1( )
2t t+
;der the ull, t1ad t2hae stadard ormaldistributios that, i this s*ecial case, are ide*edet
he lar!e"sam*le distributio of the,"statistic is thedistributio of the aera!e of t'o ide*edetl%
distributed s:uared stadard ormal radom ariables.
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he chi's(uareddistributio 'ith -de!rees of freedom (2
- ) is defied to be the distributio of the sum of -
ide*edet s:uared stadard ormal radom ariables.
I lar!e sam*les,,is distributed as2
- Q-.
Sele!ted large=sample !riti!al $alues of 2- /(
- 5N critical alue
1 .84 (%hy#)
2 .00 (the case -=2 aboe) 2.60
4 2.B
5 2.215"2
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p.value using the ,.statistic$
p"alue = tail *robabilit% of the2
- Q-distributio
be%od the,"statistic actuall% com*uted.
9mplementation in S%.%.
;se the /test commad after the re!ressio
E(ample/ est the Ooit h%*othesis that the *o*ulatio
coefficiets o STRad e3*editures *er *u*il
(e(pn0stu) are both ero, a!aist the alteratie that at
least oe of the *o*ulatio coefficiets is oero.
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,.test e(ample& California class si1e data$
reg testscr str e8pn7stu pctel, r;
Regression with robust standard errors Number of obs = 420
F( ", 41)! = 14#20 %rob & F = 0#0000 R'suared = 0#4")) Root *+ = 14#""
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' - Robust testscr - .oef# +td# rr# t %&-t- / .onf# 3nteral5
'''''''''''''6'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' str - '#2$)"2 #4$202$ '0# 0#" '1#2"4001 #))120" e8pn7stu - #00"$) #001$0 2#4 0#01 #000)0 #00)1 pctel - '#))022 #0"1$44 '20#)4 0#000 '#1$00$ '#"44) 7cons - )4# 1#4$"4 42#02 0#000 )1#11 )#)41''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
NOTE
test str e8pn7stu; The test command follows the regression
( 1! str = 0#0 There are q=2 restrictions being tested( 2! e8pn7stu = 0#0
F( 2, 41)! = #4" The 5% critical value for q=2 is "#00 %rob & F = 0#004 Stata computes the pvalue for !ou
5"4
2 l d3 l d
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T%o 2related3 loose ends$
1. Romos+edasticit%"ol% ersios of the,"statistic
2. he /, distributio
%he homos>edasti!it'=onl' (?rule=of=thumb@) &=
statisti!
o com*ute the homos+edasticit%"ol% "statistic$ ;se the *reious formulas, but usi!
homos+edasticit%"ol% stadard errors& or
Cu t'o re!ressios, oe uder the ull h%*othesis(the /restricted re!ressio) ad oe uder the
alteratie h%*othesis (the /urestricted re!ressio).
he secod method !ies a sim*le formula5"5
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? h h h R2 i f h ffi i
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?% ho' much must theR2icrease for the coefficiets o
E(pnad"ctE!to be Oud!ed statisticall% si!ificat#
Simple formula for the homos'edasticity.only ,.statistic/
,=2 2
2
( ) Q
(1 ) Q( 1)
unrestricted restricted
unrestricted unrestricted
R R -
R n '
'here$2
restrictedR = theR2for the restricted re!ressio
2
unrestrictedR = theR2for the urestricted re!ressio
-= the umber of restrictios uder the ull
'unrestricted= the umber of re!ressors i the
urestricted re!ressio.
5"B
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E(ample$
Cestricted re!ressio$
TestScore= 644.B 0.6B1"ctE!,2
restrictedR = 0.4149
(1.0) (0.02)
;restricted re!ressio$
TestScore= 649.6 0.29STR@ .8BE(pn 0.656"ctE!
(15.5) (0.48) (1.59) (0.02)2
unrestrictedR = 0.466, 'unrestricted= , -= 2
so$
,=
2 2
2
( ) Q
(1 ) Q( 1)
unrestricted restricted
unrestricted unrestricted
R R -
R n '
=(.466 .4149) Q 2
(1 .466) Q(420 1)
= 8.01
5"8
Th h ' d i i l , i i
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The homos'edasticity.only ,.statistic
,=
2 2
2
( ) Q
(1 ) Q( 1)
unrestricted restricted
unrestricted unrestricted
R R -
R n '
he homos+edasticit%"ol%,"statistic reOects 'headdi! the t'o ariables icreased theR2b% /eou!h
that is, 'he addi! the t'o ariables im*roes thefit of the re!ressio b% /eou!h
If the errors are homos+edastic, the the
homos+edasticit%"ol%,"statistic has a lar!e"sam*le
distributio that is2
- Q-.
?ut if the errors are heteros+edastic, the lar!e"sam*le
distributio is a mess ad is ot2
- Q-5"9
%h & di t ib ti
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%he &distribution
If$
1. u1,,unare ormall% distributed& ad
2. Xiis distributed ide*edetl% of ui(so i
*articular uiis homos+edastic)
the the homos+edasticit%"ol%,"statistic has the
/,-&n.'61 distributio, 'here -= the umber of
restrictios ad '= the umber of re!ressors uder the
alteratie (the urestricted model).
5"40
h , di t ib ti
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he,-,n6'61distributio$
he,distributio is tabulated ma% *laces
7he n!ets lar!e the,-&n.'61distributio as%m*totes
to the2
- Q- distributio$
&(4 is another name for2
-/(
or -ot too bi! ad nS100, the,-,n6'61distributio
ad the2
- Q-distributio are essetiall% idetical.
Ta% re!ressio *ac+a!es com*utep"alues of,"statistics usi! the,distributio ('hich is U if the
sam*le sie is 100
Vou 'ill ecouter the /,"distributio i *ublishedem*irical 'or+.
5"41
)i i littl hi t f t ti ti
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)igression/ little history of statistics7
he theor% of the homos+edasticit%"ol%,"statisticad the,-,n6'61distributios rests o im*lausibl%
stro! assum*tios (are eari!s ormall%
distributed#)
hese statistics dates to the earl% 20thcetur%, 'he
/com*uter 'as a Oob descri*tio ad obseratiosumbered i the does.
he,"statistic ad,-,n6'61distributio 'ere maOor
brea+throu!hs$ a easil% com*uted formula& a si!leset of tables that could be *ublished oce, the
a**lied i ma% setti!s& ad a *recise,
mathematicall% ele!at Oustificatio.
5"42
littl hi t f t ti ti td
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little history of statistics& ctd7
he stro! assum*tios seemed a mior *rice for thisbrea+throu!h.
?ut 'ith moder com*uters ad lar!e sam*les 'e cause the heteros+edasticit%"robust,"statistic ad the
,-,distributio, 'hich ol% re:uire the four least
s:uares assum*tios.
his historical le!ac% *ersists i moder soft'are, i'hich homos+edasticit%"ol% stadard errors (ad,"
statistics) are the default, ad i 'hichp"alues arecom*uted usi! the,-,n6'61distributio.
5"4
Summar': the homos>edasti!it' onl' (?rule of
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Summar': the homos>edasti!it'=onl' (?rule of
thumb@) &=statisti! and the &distribution
hese are Oustified ol% uder er% stro! coditios stro!er tha are realistic i *ractice.
Vet, the% are 'idel% used.
Youshould use the heteros+edasticit%"robust,"
statistic, 'ith 2- Q-(that is,,-,) critical alues.
or n S 100, the,"distributio essetiall% is the 2- Q-
distributio.
or small n, the,distributio isGt ecessaril% a/better a**ro3imatio to the sam*li! distributio of
the,"statistic ol% if the stro! coditios are true.
5"44
Summar': testing Aoint h'potheses
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Summar': testing Aoint h'potheses
he /commo"sese a**roach of reOecti! if eitherof the t"statistics e3ceeds 1.96 reOects more tha 5N of
the time uder the ull (thesi1ee3ceeds the desired
si!ificace leel)
he heteros+edasticit%"robust,"statistic is built i to
(/test commad)& this tests all -restrictiosat oce.
or nlar!e,,is distributed as2
- Q-(=,-,)
he homos+edasticit%"ol%,"statistic is im*ortathistoricall% (ad thus i *ractice), ad is ituitiel%
a**eali!, but ialid 'he there is heteros+edasticit%
5"45
%esting Single Restri!tions on Multiple Coeffi!ients
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%esting Single Restri!tions on Multiple Coeffi!ients
(SW Se!tion 5"B)
Yi= 0@ 1X1i@ 2X2i@ ui, i= 1,,n
Dosider the ull ad alteratie h%*othesis,
*0$ 1= 2 s. *1$ 12
his ull im*oses asinglerestrictio (-= 1) o multiple
coefficiets it is ot a Ooit h%*othesis 'ith multi*le
restrictios (com*are 'ith 1= 0 ad 2= 0).
5"46
'o methods for testi! si!le restrictios o multi*le
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'o methods for testi! si!le restrictios o multi*le
coefficiets$
1. Cearra!e (/trasform) the re!ressio
Cearra!e the re!ressors so that the restrictio
becomes a restrictio o a si!le coefficiet i
a e:uialet re!ressio
2. Eerform the test directl%
ome soft'are, icludi! , lets %ou test
restrictios usi! multi*le coefficiets directl%
5"4B
8ethod 9/ Rearrange 24transform53 the regression
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8ethod 9/Rearrange 2 transform 3 the regression
Yi= 0@ 1X1i@ 2X2i@ ui
*0$ 1= 2 s. *1$ 12
dd ad subtract 2X1i$
Yi= 0@ (1 2)X1i@ 2(X1i@X2i) @ ui
or
Yi= 0@ 1X1i@ 2:i@ ui
'here1= 1 2
:i=X1i@X2i
5"48
2a3 ;riginal system$
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2a3 ;riginal system$
Yi= 0@ 1X1i@ 2X2i@ ui
*0$ 1= 2 s. *1$ 12
2b3 Rearranged 24transformed53 system$
Yi= 0@ 1X1i@ 2:i@ ui
'here 1= 1 2ad :i=X1i@X2i
so
*0$ 1= 0 s. *1$ 10
he testi! *roblem is o' a sim*le oe$
test 'hether 1= 0 i s*ecificatio (b).
5"49
8ethod
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8ethod
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he coverage rate of a cofidece set is the *robabilit%
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he coverage rateof a cofidece set is the *robabilit%
that the cofidece set cotais the true *arameter alues
/commo sese cofidece set is the uio of the
95N cofidece iterals for 1ad 2, that is, the
recta!le$
L 16 1.96SE( 1
6 ), 2 1.96 SE( 2
)M
7hat is the coera!e rate of this cofidece set#
Aes its coera!e rate e:ual the desired cofideceleel of 95N#
5"52
Doera!e rate of /commo sese cofidece set$
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Doera!e rate of commo sese cofidece set$
Er(1, 2) L 161.96SE( 1
6), 21.96 SE( 2
)M>
= Er 16 1.96SE( 1
6 ) 1 16@ 1.96SE( 1
6),
2
1.96SE(2
) 2 2 @ 1.96SE(
2 )>
= Er1.96 1 1
1
( )SE
1.96, 1.96
2 2
2
( )SE
1.96>
= ErHt1H 1.96 ad Ht2H 1.96>
= 1 ErHt1H K1.96 adQor Ht2H K1.96> 95N P
7h%#
This confidence set 4inverts5 a test for %hich the si1edoesn+t e-ual the significance level>
5"5
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-et,(1,0,2,0) be the (heteros+edasticit%"robust),"
statistic testi! the h%*othesis that 1= 1,0ad 2= 2,0$
95N cofidece set = L1,0, 2,0$ ,(1,0, 2,0) .00M
.00 is the 5N critical alue of the 2,distributio
his set has coera!e rate 95N because the test o'hich it is based (the test it /ierts) has sie of 5N.
5"55
The confidence set based on the ,.statistic is an ellipse
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The confidence set based on the , statistic is an ellipse
L1, 2$ ,=1 2
1 2
2 2
1 2 , 1 2
2
,
21
2 1
t t
t t
t t t t
+
J .00M
Fo'
,= 1 21 2
2 2
1 2 , 1 22
,
12
2(1 ) t t
t t
t t t t
+
1 2
1 2
2
,
2 2
2 2,0 1 1,0 1 1,0 2 2,0
,
2 1 1 2
12(1 )
2
( ) ( ) ( ) ( )
t t
t t
SE SE SE SE
=
+ +
his is a :uadratic form i 1,0ad 2,0 thus the
boudar% of the set,= .00 is a elli*se.
5"56
Confidence set based on inverting the ,.statistic
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Confidence set based on inverting the , statistic
5"5B
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%heR4S"R4 and 2R for Multiple Regression
(SW Se!tion 5"#,)
ctual = *redicted @ residual$ Yi= 6iY @ iu
s i re!ressio 'ith a si!le re!ressor, the SER(ad theR8SE) is a measure of the s*read of the YGs aroud the
re!ressio lie$
SER=2
1
1
1
n
i
i
un ' =
5"58
he R2 is the fractio of the ariace e3*laied$
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heR is the fractio of the ariace e3*laied$
R2=ESS
TSS= 1
SSR
TSS ,
'hereESS=2
1
( )n
i
i
Y Y=
, SSR=2
1
n
i
i
u= , ad TSS=
2
1
( )n
i
i
Y Y=
Oust as for re!ressio 'ith oe re!ressor.
heR2al'a%s icreases 'he %ou add aotherre!ressor a bit of a *roblem for a measure of /fit
he 2R corrects this *roblem b% /*ealii! %ou for
icludi! aother re!ressor$
5"59
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5"61
"#ample: . Closer Loo> at the %est S!ore 7ata
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p
(SW Se!tion 5"##4 5"#)
general approach to variable selection and model
specification$
*ecif% a /base or /bechmar+ model.
*ecif% a ra!e of *lausible alteratie models, 'hichiclude additioal cadidate ariables.
Aoes a cadidate ariable cha!e the coefficiet of
iterest (1)# Is a cadidate ariable statisticall% si!ificat#
;se Oud!met, ot a mechaical reci*e
5"62
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?ariables %e %ould li'e to see in the California data set$
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f
S!hool !hara!teristi!s:
studet"teacher ratio
teacher :ualit%
com*uters (o"teachi! resources) *er studet
measures of curriculum desi!
Student !hara!teristi!s:
!lish *roficiec% aailabilit% of e3tracurricular erichmet
home leari! eiromet
5"64
*aretGs educatio leel
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*
5"65
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?ariables actually in the California class si1e data set$
studet"teacher ratio (STR)
*ercet !lish learers i the district ("ctE!)
*ercet eli!ible for subsidiedQfree luch
*ercet o *ublic icome assistace
aera!e district icome
5"66
loo' at more of the California data
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f f
5"6B
)igression/ presentation of regression results in a table
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-isti! re!ressios i /e:uatio form ca becumbersome 'ith ma% re!ressors ad ma% re!ressios
ables of re!ressio results ca *reset the +e%iformatio com*actl%
Iformatio to iclude$
ariables i the re!ressio (de*edet ad
ide*edet)
estimated coefficiets
stadard errors results of,"tests of *ertiet Ooit h%*otheses
some measure of fit
5"68
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5"B0
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Summar': Multiple Regression
Tulti*le re!ressio allo's %ou to estimate the effecto Yof a cha!e iX1, holdi!X2 costat.
If %ou ca measure a ariable, %ou ca aoid omitted
ariable bias from that ariable b% icludi! it. here is o sim*le reci*e for decidi! 'hich ariablesbelo! i a re!ressio %ou must e3ercise Oud!met.
e a**roach is to s*ecif% a base model rel%i! o
a.priorireasoi! the e3*lore the sesitiit% of the
+e% estimate(s) i alteratie s*ecificatios.