2009 lpga performance statistics and prize winnings lpga

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Instrumental Variables: 2-Stage and 3-Stage Least Squares Regression of a Linear Systems of Equations 2009 LPGA Performance Statistics and Prize Winnings www.lpga.com S.J. Callan and J.M. Thomas (2007). “Modeling the Determinants of a Professional Golfer’s Tournament Earnings,” Journal of Sports Economics, Vol. 8, No. 4, pp. 394-411

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Instrumental Variables: 2-Stage and 3-Stage Least Squares Regression of a Linear Systems of Equations. 2009 LPGA Performance Statistics and Prize Winnings www.lpga.com - PowerPoint PPT Presentation

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Page 1: 2009 LPGA Performance Statistics and Prize Winnings lpga

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

Equations

2009 LPGA Performance Statistics and Prize Winnings

www.lpga.comS.J. Callan and J.M. Thomas (2007). “Modeling the Determinants of a Professional Golfer’s Tournament Earnings,” Journal of Sports Economics, Vol. 8, No. 4, pp. 394-411

Page 2: 2009 LPGA Performance Statistics and Prize Winnings lpga

Data Description

• Prize Winnings and Performance Statistics for n = 146 professional women (LPGA) golfers for 2009 season

• Exogenous Performance Variables: Average Driving Distance Percentage of Fairways reached on Drive Percentage of Greens Reached in Regulation Percentage of Sand Saves (in hole in 2 shots from close traps) Average Putts per hole on greens reached in regulation Numbers of Events, Events Completed, Rounds

• Endogenous Result (Dependent & Independent) Variables: Average Score per Round Average Rank (Percentile in Tournaments) Log(Prize Winnings)

Page 3: 2009 LPGA Performance Statistics and Prize Winnings lpga

Variables in Systems of Equations

• Endogenous Variables – Jointly dependent (response) variables that are system determined. They can also appear as predictor variables in other equations

• Exogenous Variables – Independent variables that do not depend on the endogenous variables

• Predetermined Variables – Exogenous and lagged Endogenous variables

• Instrumental Variables – Predetermined variables used to predict endogenous variables in first-stage regressions, with predicted values being used in place of the endogenous predictors in system of equations

Page 4: 2009 LPGA Performance Statistics and Prize Winnings lpga

System of Equations (Callan and Thomas, 2007)

1. Average Score (per 18 holes) is related to the golfers’ skills and experience (number of rounds played)

2. Average Rank (transformed to percentile) in tournaments is related to average score and the number of events she competed in

3. Season Earnings is related to average rank and the number of tournaments she completed

i 0 1

i 0 SCORE i 2

i 0 RANK i 3

SCORERank SCORE

ln Prize Rank

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

E i i

C i i

D F G S P RE

C

Page 5: 2009 LPGA Performance Statistics and Prize Winnings lpga

Potential Problems with Endogenous Predictors

• When endogenous variables are included as predictors, they can be correlated with error terms for that equation, particularly when there are omitted variables that may be related to the outcome. This causes Ordinary Least Squares Estimates to be biased and inconsistent. In equation 2, SCORE may be correlated with the error term

without a variable measuring average course difficulty (Callan and Thomas, p. 402).

In equation 3, Rank may be correlated with the error term without a variable measuring golfer’s human capital investment such as diet and concentration level (Callan and Thomas, p. 402).

Page 6: 2009 LPGA Performance Statistics and Prize Winnings lpga

Model Building Process1. Regress all endogenous variables (Score, Rank, and

ln(Prize)) on all exogenous variables2. Obtain the predicted values for each endogenous

variable, based on the Regressions from 1.3. In the system of equations, replace any “right hand

side” endogenous predictors with their fitted values from 2.

4. Note that software (e.g. SAS and STATA) will fit all the regressions in 1., even if that variable does not appear as a predictor (ln(Prize) in this example).

5. This method provides correct estimates, but not ANOVA table or correct standard errors

Page 7: 2009 LPGA Performance Statistics and Prize Winnings lpga

First Stage Regressions for Score and Rank SUMMARY OUTPUTDep. Var. = SCORERegression StatisticsMultiple R 0.969534R Square 0.939996Adjusted R Square0.936492Standard Error0.288069Observations 146

ANOVAdf SS MS F

Regression 8 178.0979 22.26223 268.2725Residual 137 11.36876 0.082984Total 145 189.4666

CoefficientsStandard Error t Stat P-valueIntercept 63.22496 1.969431 32.10316 1.3E-65Drive -0.00496 0.004252 -1.16683 0.245304Fairway -0.01622 0.005784 -2.80373 0.005786Green -0.11232 0.010222 -10.9876 1.57E-20sandsv -0.01448 0.003307 -4.3771 2.37E-05GIRPuttsHole10.8415 0.893289 12.13661 1.8E-23Rounds -0.02758 0.010894 -2.5313 0.012494Events 0.096164 0.027269 3.526557 0.000573Completed -0.02314 0.018163 -1.27401 0.204818

SUMMARY OUTPUTDep. Var. = RANKRegression StatisticsMultiple R 0.971418R Square 0.943652Adjusted R Square0.940362Standard Error3.699136Observations 146

ANOVAdf SS MS F

Regression 8 31394.75 3924.344 286.7916Residual 137 1874.655 13.68361Total 145 33269.41

CoefficientsStandard Error t Stat P-valueIntercept 127.4395 25.28977 5.039173 1.45E-06Drive 0.006845 0.054606 0.125355 0.900426Fairway 0.051758 0.074276 0.696836 0.487086Green 1.11684 0.131266 8.508233 2.74E-14sandsv 0.118137 0.042471 2.781584 0.006172GIRPuttsHole-88.1679 11.47086 -7.68625 2.63E-12Rounds 0.475649 0.139888 3.400216 0.000882Events -1.84119 0.35016 -5.25814 5.46E-07Completed0.894404 0.233229 3.834869 0.000191

The fitted (predicted) values for SCORE will be used in equation 2 in place of SCORE, and the fitted values for RANK in equation 3. Equation 1 has no right hand side endogenous variables

Page 8: 2009 LPGA Performance Statistics and Prize Winnings lpga

Equation 1) - SCORE is related to SKILLS and experienceSUMMARY OUTPUT Model 1

Regression StatisticsMultiple R 0.963573R Square 0.928473Adjusted R Square 0.925386Standard Error 0.312244Observations 146

ANOVAdf SS MS F

Regression 6 175.9147 29.31911 300.7211Residual 139 13.55195 0.097496Total 145 189.4666

CoefficientsStandard Error t Stat P-valueIntercept 61.2801 2.093701 29.26879 2.71E-61Drive -0.00526 0.004609 -1.14194 0.255442Fairway -0.01897 0.006242 -3.03902 0.002836Green -0.13583 0.009903 -13.7162 1.3E-27sandsv -0.01557 0.003563 -4.37003 2.41E-05GIRPuttsHole 13.15201 0.835199 15.74715 1.05E-32Rounds -0.00837 0.002169 -3.86102 0.000172

All variables except average driving distance are significant.All else equal: Average SCORE decreases as

Percent Fairways Hit Increases (a 10% increase in fairways hit corresponds to a 0.19 decrease in SCORE)

Average SCORE decreases by 1.36 with a 10% increase in Greens in regulation

Average SCORE decreases by 0.16 with a 10% increase in Sand Saves

Average SCORE increases by 1.32 with a 0.1 increase in putts per Green in Regulation hole

Average SCORE decreases by 0.08 for 10 Round Increase in Rounds played

Page 9: 2009 LPGA Performance Statistics and Prize Winnings lpga

Equation 2) - Rank is related to SCORE and Events

SUMMARY OUTPUT Model 2

Regression StatisticsMultiple R 0.963379R Square 0.928099Adjusted R Square 0.927093Standard Error 4.089985Observations 146

ANOVAdf SS MS F

Regression 2 30877.31 15438.65 922.9243Residual 143 2392.1 16.72798Total 145 33269.41

CoefficientsStandard Error t Stat P-valueIntercept 956.9965 29.14945 32.83069 1.82E-68Score-hat -12.5125 0.384185 -32.5689 5E-68Events 0.281134 0.103653 2.712259 0.007503

Rank (as Percentile, with 100 meaning golfer won every tournament she played in) is: Negative associated with

predicted SCORE (decreases by 12.5 with unit increase in average SCORE)

Positively associated with number of Events (increases by 0.28 with a unit increase in # of EVENTS played)

Note: The estimated coefficients are correct, but the standard errors, t-tests, and Analysis of Variance are incorrect (see slide 11)

Page 10: 2009 LPGA Performance Statistics and Prize Winnings lpga

Equation 3) – ln(Prize) is related to Rank and Completed Events

SUMMARY OUTPUT Model 3

Regression StatisticsMultiple R 0.932864R Square 0.870235Adjusted R Square 0.86842Standard Error 0.507583Observations 146

ANOVAdf SS MS F

Regression 2 247.075 123.5375 479.4961Residual 143 36.84256 0.25764Total 145 283.9176

CoefficientsStandard Error t Stat P-valueIntercept 7.881995 0.228644 34.47279 3.69E-71Rank-hat 0.055812 0.007667 7.279667 2.05E-11Completed 0.079741 0.017742 4.494427 1.43E-05

Prize Winnings (in log form): Increase with (Predicted)

Rank. A 10% increase in Rank (percentile) increases ln(Prize) by 0.56

Increase with Completed Events. For each tournament completed, ln(Prize) increases by 0.080.

Note: The estimated coefficients are correct, but the standard errors, t-tests, and Analysis of Variance are incorrect (see slide 11)

Page 11: 2009 LPGA Performance Statistics and Prize Winnings lpga

Matrix Approach: Models w/ Endogenous PredictorsMatrix of Instrumental Variables: Intercept and 8 Exogenous variables

Intercept, Drive, Fairway, Greens, SandSave, Putts, Rounds, Events, Completed

Matrix of Predictors for Model:Model 2: Intercept

Z

X, Score (Actual, not predicted), Events

Model 3: Intercept, Rank, Completed

Vector of Responses:Model 2: Rank Model 3: ln(Prize)

2-Stage Least Squares Estimator and Estimated Variance-Covariance

Y

^2 2

2

Matrix:

'

( )1

SSEV s s SSEn rank X

SSRSSR Rn SSR SSE

^ -1-1 -1 -1Z Z2SLS

-1Z

^ ^ ^-1Z2SLS 2SLS 2SLS

-1Z Z Z n

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

P = Z Z'Z Z'

β X'P X Y - Xβ Y -Xβ

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

Page 12: 2009 LPGA Performance Statistics and Prize Winnings lpga

Model 2 – Rank = f(Score, Events)Z Intercept Drive Fairway Green sandsv Putts Rounds Events Completed

1 251 73.80 64.70 36.50 1.79 61 20 131 256.7 73.30 69.60 30.20 1.78 65 20 141 250.1 65.90 64.10 32.70 1.78 56 18 12

... ... ... ... ... ... ... ... ...1 249.8 70.10 67.60 26.30 1.83 67 22 141 239.8 77.70 62.30 30.60 1.88 40 17 41 256.1 74.50 72.40 31.40 1.8 89 25 23

X Intercept AveStrokesEvents Y Rank(Pctile)1 72.492 20 55.177571 71.477 20 66.574071 72.25 18 55.50107

... ... ... ...1 72.657 22 52.171181 74.225 17 31.274891 71.157 25 75.77619

X'P_ZX X'P_ZY146 10610.59 2767 7734.555

10610.59 771305.2 200694.4 559770.42767 200694.4 54887 152254.3

INV(X'P_ZX) Beta_2SLS50.79456 -0.66845 -0.1165 956.9965-0.66845 0.008823 0.001436 -12.5125

-0.1165 0.001436 0.000642 0.281134

SSE s^21806.234 12.63101

V^(Beta_2SLS) SE(B_2SLS)641.5866 -8.4432 -1.47152 25.32956

-8.4432 0.111449 0.018132 0.333839-1.47152 0.018132 0.008113 0.09007

SSModel ybar SSReg R^2440626.2 52.9764 30877.31 0.944736

Page 13: 2009 LPGA Performance Statistics and Prize Winnings lpga

Model 3: ln(Prize) = f(Rank,Completed)

Z Intercept Drive Fairway Green sandsv Putts Rounds Events Completed1 251 73.80 64.70 36.50 1.79 61 20 131 256.7 73.30 69.60 30.20 1.78 65 20 141 250.1 65.90 64.10 32.70 1.78 56 18 12

... ... ... ... ... ... ... ... ...1 249.8 70.10 67.60 26.30 1.83 67 22 141 239.8 77.70 62.30 30.60 1.88 40 17 41 256.1 74.50 72.40 31.40 1.8 89 25 23

X Intercept Rank(Pctile)Completed Y lnPrize1 55.17757 13 12.221 66.57407 14 12.861 55.50107 12 11.74

... ... ... ...1 52.17118 14 12.661 31.27489 4 9.361 75.77619 23 13.33

X'P_ZX X'P_ZY146 7734.555 1736 1720.879

7734.555 441143.7 104550.7 93921.621736 104550.7 26504 21631.74

INV(X'P_ZX) Beta_2SLS0.202911 -0.00626 0.011416 7.881995-0.00626 0.000228 -0.00049 0.0558120.011416 -0.00049 0.001222 0.079741

SSE s^236.66316 0.256386

V^(Beta_2SLS) SE(B_2SLS)0.052024 -0.00161 0.002927 0.228087-0.00161 5.85E-05 -0.00013 0.0076480.002927 -0.00013 0.000313 0.017699

Page 14: 2009 LPGA Performance Statistics and Prize Winnings lpga

Robust Estimate of Variance of 2SLS Estimator

221 21

2222 22

2 2

22 2

1 1 1

1 1

0 00 0

0 0

Replacing with its est

i i

n n

V V

V V

^

Z Z Z Z Z Z2SLS

-1 -1Z Z

Σ

β X'P X X'P Y X'P X X'P ΣP X X'P X

X'P X X'Z Z'Z Z'ΣZ Z'Z Z'X X'P X

Z'ΣZ

221

22222 2 2

122

^ 1 1

imator:

0 00 0

0 0

n

i i ii

n

ee

e e Y

e

V

1 1

^ ^2 2

i i i 2SLS

n n

^ ^-1 -1Z Z2SLS

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

' 'z x

β X'P X X'Z Z'Z S Z'Z Z'X X'P X

Exact same method for equation 3

Page 15: 2009 LPGA Performance Statistics and Prize Winnings lpga

Results for Model 2: Rank = f(Score, Events)S-hat12.37147 3102.712 850.8146 795.9452 462.1251 22.63899 638.5061 210.1805 125.07673102.712 778916.5 213337.1 199840.8 115696.1 5676.833 160482.6 52740.96 31513.16850.8146 213337.1 58857.24 54843.46 31745.12 1556.302 44235.96 14519.52 8719.568795.9452 199840.8 54843.46 51415.23 29684.13 1455.89 41635.56 13637.83 8236.469462.1251 115696.1 31745.12 29684.13 18204.05 844.6313 24297.06 7958.194 4765.15322.63899 5676.833 1556.302 1455.89 844.6313 41.4413 1164.82 384.0072 227.7275638.5061 160482.6 44235.96 41635.56 24297.06 1164.82 36888.23 11763.14 7644.445210.1805 52740.96 14519.52 13637.83 7958.194 384.0072 11763.14 3807.094 2383.913125.0767 31513.16 8719.568 8236.469 4765.153 227.7275 7644.445 2383.913 1659.86

Homoskedastic Errors Heteroskedastic ErrorsBeta_2SLS SE(B_2SLS) t SE(B_2SLS) t

956.9965 25.3296 37.7818 23.4808 40.7566-12.5125 0.3338 -37.4806 0.3046 -41.0801

0.2811 0.0901 3.1213 0.1053 2.6707

V(B_2SLS) V(B_2SLS)641.5866 -8.4432 -1.4715 551.3468 -7.1334 -1.6873

-8.4432 0.1114 0.0181 -7.1334 0.0928 0.0202-1.4715 0.0181 0.0081 -1.6873 0.0202 0.0111

Page 16: 2009 LPGA Performance Statistics and Prize Winnings lpga

Results for Model 3: ln(Prize) = f(Rank,Completed)S-hat0.251118 62.19775 17.16093 16.13331 9.791545 0.461501568 13.36902 4.551543 2.36998162.19775 15434.73 4247.927 4004.667 2420.563 114.2658074 3342.212 1132.968 597.98317.16093 4247.927 1179.906 1103.003 668.8473 31.54885121 911.7178 310.2781 161.277616.13331 4004.667 1103.003 1042.178 627.3954 29.633223 874.5716 294.8918 158.29789.791545 2420.563 668.8473 627.3954 397.5982 17.9908815 521.5487 177.865 91.501510.461502 114.2658 31.54885 29.63322 17.99088 0.848513025 24.44991 8.342034 4.31534713.36902 3342.212 911.7178 874.5716 521.5487 24.44990949 812.2522 263.6976 157.16734.551543 1132.968 310.2781 294.8918 177.865 8.342033579 263.6976 87.66156 49.096442.369981 597.983 161.2776 158.2978 91.50151 4.315346873 157.1673 49.09644 34.27717

Homoskedastic Errors Heteroskedastic ErrorsBeta_2SLS SE(B_2SLS) t SE(B_2SLS) t

7.8820 0.2281 34.5570 0.2544 30.98700.0558 0.0076 7.2975 0.0086 6.51690.0797 0.0177 4.5054 0.0205 3.8845

V(B_2SLS) V(B_2SLS)0.05202 -0.00161 0.00293 0.06470 -0.00196 0.00359

-0.00161 0.00006 -0.00013 -0.00196 0.00007 -0.000160.00293 -0.00013 0.00031 0.00359 -0.00016 0.00042

Page 17: 2009 LPGA Performance Statistics and Prize Winnings lpga

3-Stage Least Squares

• Extension of 2-Stage Least Squares that allows for a covariance structure among the system of equations

• Errors from 2SLS are obtained, and used to estimate the within individual (golfer) variance-covariance structure among the equations

• The response vector is stacked with the n responses from model 1, being stacked over the n responses from model 2, which are stacked over the n responses from model 3.

• The X matrices are “blocked” out diagonally, with 0 matrices off the blocked diagonal

Page 18: 2009 LPGA Performance Statistics and Prize Winnings lpga

Model Description - I

i 0 1 1

i 0 SCORE i 2 2

i 0 RANK i 3 3

11 21

12 221 2

1,146 2,146

Model 1: SCOREModel 2: Rank SCORE

Model 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 YE Y

C Y

Y YY Y

Y Y

Y Y

31

323

3,146

1 1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2 23

146 146 146 146 146 146 146 146 146 146

1 1 11 1 1

1 1 1

YY

Y

D F G S P R SC E RA CD F G S P R SC E RA C

D F G S P R SC E RA C

1

2

3

1 2

YY Y Y

Y

X X X

^

146 1462

11 1 12 1 2 13 22 23 331 1

11 12 13

21 22 23

1,2,3 are residuals from 2-Stage Least Squares Regressions1 1= = and so on for , , ,

146 7 146 (7 3) / 2

kiki ki

i i ii i

e Y Y k

S e S e e S S S S

S S SS S S

1

2

3

X 0 0X 0 X 0

0 0 X

S 11 1

31 32 33S S S

ZW S Z Z'Z Z' S P

Page 19: 2009 LPGA Performance Statistics and Prize Winnings lpga

Model Description - II

1

^ 11

11 12 13 11 12 13

1 21 22 23 21 22 23

31 32 33 31 32 33

11

where:

V

S S S S S SS S S S S SS S S S S S

S

^ -1 -1 -1-1 -13SLS

^ -1-13SLS

Z Z Z

Z Z Z

Z Z Z

1 Z

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

β X'WX X'S Z Z'Z Z'X

P P PS W P P P

P P P

X 'PX'WX

12 13

21 22 23

31 32 33

11 12 13

21 22 23

31 32 33

S SS S SS S S

S S SS S SS 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 XX 'P X X 'P X X 'P XX 'P X X 'P X X 'P X

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

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

Page 20: 2009 LPGA Performance Statistics and Prize Winnings lpga

Estimation ResultsX'WX

1554.094 389488.97 108250.1 101857 58810.75 2837.925 90861.28 -25.0786 -1822.6 -475.292 -152.235 -8064.85 -1810.13389489 97734135 27100830 25559022 14728936 711073.2 22874738 -6285.24 -456663 -119415 -38153.3 -2030746 -457357

108250.1 27100830 7588745 7104946 4096401 197673.2 6343826 -1746.85 -126916 -33121.7 -10603.9 -564282 -126816101857 25559022 7104946 6705381 3853879 185907.4 6033962 -1643.68 -119354 -31362.4 -9977.64 -536619 -121572

58810.75 14728936 4096401 3853879 2319387 107296.4 3488898 -949.037 -68922.7 -18155.7 -5760.94 -308831 -69983.92837.925 711073.22 197673.2 185907.4 107296.4 5184.691 165259.8 -45.796 -3329.07 -866.152 -277.995 -14662.7 -3281.3290861.28 22874738 6343826 6033962 3488898 165259.8 5838242 -1466.24 -106149 -29567.5 -8900.52 -506396 -122817-25.0786 -6285.237 -1746.85 -1643.68 -949.037 -45.796 -1466.24 13.9931 1016.952 265.1981 39.31672 2082.859 467.492

-1822.6 -456663.2 -126916 -119354 -68922.7 -3329.07 -106149 1016.952 73924.34 19235.18 2857.353 150742 33733.38-475.292 -119415.2 -33121.7 -31362.4 -18155.7 -866.152 -29567.5 265.1981 19235.18 5260.544 745.1327 41000.96 9711.5-152.235 -38153.28 -10603.9 -9977.64 -5760.94 -277.995 -8900.52 39.31672 2857.353 745.1327 684.3542 36254.62 8137.252-8064.85 -2030746 -564282 -536619 -308831 -14662.7 -506396 2082.859 150742 41000.96 36254.62 2067798 490066.5-1810.13 -457357.3 -126816 -121572 -69983.9 -3281.32 -122817 467.492 33733.38 9711.5 8137.252 490066.5 124233.7

X'WY109821.227513834

764674471895544151912

200612.56386177-617.871-45213.8-10931.8

-914.47-24634

-1065.64

V(Beta_3SLS)4.234876 -0.00573 -0.00541 0.004066 -0.00191 -1.39939 -0.00103 1.641475 -0.02114 -0.00549 -0.0096 0.000355 -0.00077-0.00573 2.047E-05 1.76E-05 -2.5E-05 2.81E-06 0.000515 -3.5E-07 0.000709 -9.3E-06 -1.7E-06 1.38E-06 -4.4E-08 7.95E-08-0.00541 1.756E-05 3.76E-05 -3.2E-05 1.76E-06 0.000212 3.51E-07 0.002615 -3.4E-05 -7.2E-06 -3.5E-07 1.73E-09 2.15E-080.004066 -2.52E-05 -3.2E-05 9.56E-05 2.35E-06 -0.00073 -9.6E-06 0.014586 -0.00019 -4E-05 -0.00012 4.31E-06 -8.9E-06-0.00191 2.809E-06 1.76E-06 2.35E-06 1.22E-05 0.00031 -1.7E-06 0.001482 -2E-05 -3E-06 -1.4E-05 5.59E-07 -1.3E-06-1.39939 0.0005153 0.000212 -0.00073 0.00031 0.680887 0.000829 -1.53791 0.020033 0.004326 0.00906 -0.00032 0.000649-0.00103 -3.54E-07 3.51E-07 -9.6E-06 -1.7E-06 0.000829 4.68E-06 -0.00349 4.24E-05 2.17E-05 2.43E-05 -1.2E-06 3.26E-061.641475 0.000709 0.002615 0.014586 0.001482 -1.53791 -0.00349 631.5903 -8.32114 -1.41211 2.067496 -0.06317 0.107152-0.02114 -9.32E-06 -3.4E-05 -0.00019 -2E-05 0.020033 4.24E-05 -8.32114 0.109957 0.017411 -0.02734 0.000817 -0.00134-0.00549 -1.67E-06 -7.2E-06 -4E-05 -3E-06 0.004326 2.17E-05 -1.41211 0.017411 0.007743 -0.00451 0.0002 -0.00051

-0.0096 1.378E-06 -3.5E-07 -0.00012 -1.4E-05 0.00906 2.43E-05 2.067496 -0.02734 -0.00451 0.051242 -0.00157 0.0028270.000355 -4.38E-08 1.73E-09 4.31E-06 5.59E-07 -0.00032 -1.2E-06 -0.06317 0.000817 0.0002 -0.00157 5.67E-05 -0.00012-0.00077 7.95E-08 2.15E-08 -8.9E-06 -1.3E-06 0.000649 3.26E-06 0.107152 -0.00134 -0.00051 0.002827 -0.00012 0.0003

Beta_3SLS StdErr60.66021 2.057881-0.00305 0.004524-0.01449 0.006129-0.1377 0.009775-0.01484 0.00349913.09106 0.825158-0.00905 0.002163954.3673 25.13146-12.4821 0.3315980.303414 0.0879927.96384 0.2263670.050763 0.0075290.095351 0.017317

EQ1

EQ2

EQ3

Page 21: 2009 LPGA Performance Statistics and Prize Winnings lpga

SAS Programdata lpga2009;infile 'lpga2009.dat';input golfer drive fairway green putts sandsv prize lnprize events girputts complete aveposrank rounds strokes;lnprize1=log(prize);run;

proc syslin 2sls out=regout;instruments drive fairway green girputts sandsv rounds events complete;strokes: model strokes = drive fairway green girputts sandsv rounds; output residual=e1;rank: model aveposrank = strokes events; output residual=e2;prize: model lnprize1 = aveposrank complete; output residual=e3;run;

proc syslin 3sls data=lpga2009 itprint out=regout3;instruments drive fairway green girputts sandsv rounds events complete;strokes: model strokes = drive fairway green girputts sandsv rounds / xpx;output residual=e1;rank: model aveposrank = strokes events / xpx;output residual=e2;prize: model lnprize1 = aveposrank complete / xpx;output residual=e3;run;

Page 22: 2009 LPGA Performance Statistics and Prize Winnings lpga

STATA Programinsheet using lpga_2009_meq.csv generate lnprize=ln(prize) reg3 (avestrokes=drive fairway green sandsvpct girputtshole rounds) ///(averagepospct=avestrokes events) (lnprize=averagepospct completed), ///2sls reg3 (avestrokes=drive fairway green sandsvpct girputtshole rounds) ///(averagepospct=avestrokes events) (lnprize=averagepospct completed), ///3sls