Introduction to Econometrics- Stock & Watson -Ch 5 Slides.doc

8/14/2019 Introduction to Econometrics- Stock & Watson -Ch 5 Slides.doc

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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"


25/71

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


35/71

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


37/71

?% 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)



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


39/71

The homos'edasticity.only ,.statistic

,=

2 2

2

( ) Q

(1 ) Q( 1)



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


40/71

%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


41/71

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


42/71

)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


43/71

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


44/71

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%

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%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.

Documents

Introduction to Econometrics- Stock & Watson -Ch 5 Slides.doc