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Instrumental Variables: 2-Stage and 3-Stage Least Squares Regression of a

Linear Systems of Equations

2009 LP! Performan"e Statisti"s andPri#e $innings

%%%&l'ga&"om

S&(& )allan and (&*& +,omas 200./& *odeling t,e 1eterminants of a Professionalolfers +ournament Earnings4 (ournal of S'orts E"onomi"s Vol& 5 6o& 7 ''& 397-788

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1ata 1es"ri'tion

Pri#e $innings and Performan"e Statisti"s for n 87

'rofessional %omen LP!/ golfers for 2009 season E;ogenous Performan"e Variables:

!

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System of Equations )allan and +,omas 200./

8& !

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Potential Problems %it, Endogenous Predi"tors

$,en endogenous may be "orrelated %it, t,e errorterm %it,out a

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*odel Cuilding Pro"ess

8& Regress all endogenous

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irst Stage Regressions for S"ore and Ran>

+,e @tted 'redi"ted/

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Equation 8/ - S)ARE is related to SFILLS and e;'erien"e

!ll

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Equation 2/ - Ran> is related to S)ARE and E as Per"entile %it,800 meaning golfer %one

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Equation 3/ ? lnPri#e/ is related to Ran> and)om'leted E& !80H in"rease inRan> 'er"entile/in"reases lnPri#e/by 0&D

In"rease %it,)om'leted E

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*atri; !''roa",: *odels %J Endogenous Predi"tors

Matrix of Instrumental aria!les" Inter#e$t an% & Exo'enous (aria!les

Inter#e$t) *ri(e) +air,a-) .reens) San%Sa(e) Putts) Roun%s) E(ents) Com$lete%

Matrix of Pre%i#tors for Mo%el"

Mo%el 2" Inter#e$t

Z

X

) S#ore /A#tual) not $re%i#te%) E(ents

Mo%el 3" Inter#e$t) Rank) Com$lete%

e#tor of Res$onses"Mo%el 2" Rank Mo%el 3" ln/Prize

2Sta'e east Suares Estimator an% Estimate% arian#eCo(arian#e

Y

( )( ) ( ) ( )

( )

{ } ( )

( )

42 2

2

Matrix"

5

/

1

SSEV s s SSE

n rank X

SSRSSR R

n SSR SSE

= = =

= = +

^ -1-1 -1 -1

Z Z2SLS

-1

Z

^ ^ ^-1

Z2SLS 2SLS 2SLS

-1

Z Z Z n

= X'Z Z'Z Z'X X'Z Z'Z Z'Y = X'P X X'P Y

P = Z Z'Z Z'

X'P X Y - X Y - X

Y' P X X'P X X'P J Y

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*odel 2 ? Ran> fS"ore E

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*odel 3: lnPri#e/ fRan>)om'leted/

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Robust Estimate of Varian"e of 2SLS Estimator

E;a"t same met,od for equation 3

{ }

{ } [ ] [ ]{ } [ ] [ ]

[ ] ( ) ( ) [ ]

221 21

2

222 222 2

22 2

1 1 1

1 1

0 0

0 0

0 0

Re$la#in' ,it6 its est

i i

n n

V V

V V

= = =

= =

=

^

Z Z Z Z Z Z2SLS

-1 -1

Z Z

X'P X X'P Y X'P X X'P P X X'P X

X'P X X'Z Z'Z Z'Z Z'Z Z'X X'P X

Z'Z

L

L

* * * A *

L

{ } [ ] ( ) ( ) [ ]

2

21

2

222

2 2 2

12

2

41 1

imator"

0 0

0 0

0 0

n

i i i

i

n

e

ee e Y

e

V

=

= =

=

1 1

^ ^

2 2i i i 2SLS

n n

^ ^-1 -1

Z Z2SLS

' 'z x

' 'z x' 'S = Z' Z z z x Z = X =

' 'z x

X'P X X'Z Z'Z S Z'Z Z'X X'P X

L

L

* * A * * *L

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Results for *odel 2: Ran> fS"ore E

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Results for *odel 3: lnPri#e/ fRan>)om'leted/

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3-Stage Least Squares

E;tension of 2-Stage Least Squares t,atallo%s for a "o

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*odel 1es"ri'tion - I

( )

i 0 1 1

i 0 SCORE i 2 2

i 0 RANK i 3 3

11 21

12 22

1 2

1)178 2)178

Mo%el 1" SCORE

Mo%el 2" Rank SCORE

Mo%el 3" ln Prize Rank

D i F i G i S i P i R i i i

E i i i

C i i i

D F G S P R Y

E Y

C Y

Y Y

Y Y

Y Y

= + + + + + + + == + + + =

= + + + = = =

Y Y* *

31

32

3

3)178

1 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2 2

3

178 178 178 178 178 178 178 178 178 178

1 1 1

1 1 1

1 1 1

Y

Y

Y

D F G S P R SC E RA C

D F G S P R SC E RA C

D F G S P R SC E RA C

= =

= = =

1

2

3

1 2

Y

Y Y Y

Y

X X X

*

* * * * * * * * * * * * *

4

178 1782

11 1 12 1 2 13 22 23 33

1 1

11 12 13

21 22 23

1) 2)3 are resi%uals from 2Sta'e east Suares Re'ressions

1 19 9 an% so on for ) ) )

178 : 178 /: 3 ; 2

kiki ki

i i i

i i

e Y Y k

S e S e e S S S S

S S S

S S S

= =

=

= =

+

=

1

2

3

X 0 0

X 0 X 0

0 0 X

S ( )11 1

31 32 33S S S

= =

ZW S Z Z'Z Z' S P

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*odel 1es"ri'tion - II

( ) ( )( ) ( )

{ } ( ) ( )( )

1

4 11

11 12 13 11 12 13

1 21 22 23 21 22 23

31 32 33 31 32 33

11

,6ere"

V

S S S S S S

S S S S S S

S S S S S S

S

= =

= =

= =

=

^-1 -1 -1-1 -1

3SLS

^-1-1

3SLS

Z Z Z

Z Z Z

Z Z Z

1 Z

X'WX X'WY X'S Z Z'Z Z'X X'S Z Z'Z Z'Y

X'WX X'S Z Z'Z Z'X

P P P

S W P P P

P P P

X 'P

X'WX

12 13

21 22 23

31 32 33

11 12 13

21 22 23

31 32 33

S S

S S S

S S S

S S S

S S S

S S S

+ += + + + +

1 1 Z 2 1 Z 3

2 Z 1 2 Z 2 2 Z 3

3 Z 1 3 Z 2 3 Z 3

1 Z 1 1 Z 2 1 Z 3

2 Z 1 2 Z 2 2 Z 3

3 Z 1 3 Z 2 3 Z 3

X X 'P X X 'P X

X 'P X X 'P X X 'P X

X 'P X X 'P X X 'P X

X 'P Y X 'P Y X 'P Y

X'WY X 'P Y X 'P Y X 'P Y

X 'P Y X 'P Y X 'P Y

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Estimation Results

EM8

EM2

EM3

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S!S Programdata l'ga2009Nin@le Ol'ga2009&datONin'ut golfer dries dri: model a: model a "om'lete J ;';Nout'ut residuale3N

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S+!+! Program

ins,eet using l'ga2009meq&"s