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
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)
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
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
SCORE i   0   D Di   F Fi   G Gi   S Si   P Pi   R Ri  1i
Rank i   0  SCORESCORE i   E Ei   2i
ln  Prizei    0   RANK Rank i   C Ci   3i
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).
Model Building Process
1. Regress all endogenous variables (Score, Rank, and
ln(Prize)) on all exogenous variables
2. 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
First Stage Regressions for Score and Rank
SUMMARY OUTPUT
Dep. Var. = SCORE
Regression Statistics
Multiple R 0.969534
R Square 0.939996
Adjusted R Square
0.936492
Standard Error
0.288069
Observations
146
SUMMARY OUTPUT
Dep. Var. = RANK
Regression Statistics
Multiple R 0.971418
R Square 0.943652
Adjusted R Square
0.940362
Standard Error
3.699136
Observations
146
ANOVA
ANOVA
df
Regression
Residual
Total
SS
MS
F
8 178.0979 22.26223 268.2725
137 11.36876 0.082984
145 189.4666
Coefficients
Standard Error t Stat
Intercept 63.22496 1.969431 32.10316
Drive
-0.00496 0.004252 -1.16683
Fairway
-0.01622 0.005784 -2.80373
Green
-0.11232 0.010222 -10.9876
sandsv
-0.01448 0.003307
-4.3771
GIRPuttsHole10.8415 0.893289 12.13661
Rounds
-0.02758 0.010894
-2.5313
Events
0.096164 0.027269 3.526557
Completed -0.02314 0.018163 -1.27401
P-value
1.3E-65
0.245304
0.005786
1.57E-20
2.37E-05
1.8E-23
0.012494
0.000573
0.204818
df
Regression
Residual
Total
SS
MS
F
8 31394.75 3924.344 286.7916
137 1874.655 13.68361
145 33269.41
Coefficients
Standard Error t Stat
Intercept 127.4395 25.28977 5.039173
Drive
0.006845 0.054606 0.125355
Fairway
0.051758 0.074276 0.696836
Green
1.11684 0.131266 8.508233
sandsv
0.118137 0.042471 2.781584
GIRPuttsHole
-88.1679 11.47086 -7.68625
Rounds
0.475649 0.139888 3.400216
Events
-1.84119 0.35016 -5.25814
Completed0.894404 0.233229 3.834869
P-value
1.45E-06
0.900426
0.487086
2.74E-14
0.006172
2.63E-12
0.000882
5.46E-07
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
Equation 1) - SCORE is related to SKILLS and experience
SUMMARY OUTPUT
Model 1
Regression Statistics
Multiple R
0.963573
R Square
0.928473
Adjusted R Square
0.925386
Standard Error
0.312244
Observations
146
ANOVA
df
Regression
Residual
Total
Intercept
Drive
Fairway
Green
sandsv
GIRPuttsHole
Rounds
SS
MS
F
6 175.9147 29.31911 300.7211
139 13.55195 0.097496
145 189.4666
Coefficients
Standard Error t Stat
61.2801 2.093701 29.26879
-0.00526 0.004609 -1.14194
-0.01897 0.006242 -3.03902
-0.13583 0.009903 -13.7162
-0.01557 0.003563 -4.37003
13.15201 0.835199 15.74715
-0.00837 0.002169 -3.86102
P-value
2.71E-61
0.255442
0.002836
1.3E-27
2.41E-05
1.05E-32
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
Equation 2) - Rank is related to SCORE and Events
SUMMARY OUTPUT
Model 2
Regression Statistics
Multiple R
0.963379
R Square
0.928099
Adjusted R Square
0.927093
Standard Error
4.089985
Observations
146
ANOVA
df
Regression
Residual
Total
Intercept
Score-hat
Events
SS
MS
F
2 30877.31 15438.65 922.9243
143
2392.1 16.72798
145 33269.41
Coefficients
Standard Error t Stat
P-value
956.9965 29.14945 32.83069 1.82E-68
-12.5125 0.384185 -32.5689
5E-68
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)
Equation 3) – ln(Prize) is related to Rank and Completed Events
SUMMARY OUTPUT
Model 3
Regression Statistics
Multiple R
0.932864
R Square
0.870235
Adjusted R Square
0.86842
Standard Error
0.507583
Observations
146
ANOVA
df
Regression
Residual
Total
Intercept
Rank-hat
Completed
SS
MS
F
2 247.075 123.5375 479.4961
143 36.84256 0.25764
145 283.9176
Coefficients
Standard Error t Stat
7.881995 0.228644 34.47279
0.055812 0.007667 7.279667
0.079741 0.017742 4.494427
P-value
3.69E-71
2.05E-11
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, ttests, and Analysis of
Variance are incorrect
(see slide 11)
Matrix Approach: Models w/ Endogenous Predictors
Z  Matrix of Instrumental Variables: Intercept and 8 Exogenous variables
Intercept, Drive, Fairway, Greens, SandSave, Putts, Rounds, Events, Completed
X  Matrix of Predictors for Model:
Model 2: Intercept, Score (Actual, not predicted), Events
Model 3: Intercept, Rank, Completed
Y  Vector of Responses:
Model 2: Rank
Model 3: ln(Prize)
2-Stage Least Squares Estimator and Estimated Variance-Covariance Matrix:

^
β 2SLS = X'Z  Z'Z  Z'X
-1

-1
X'Z  Z'Z  Z'Y =  X'PZ X  X'PZ Y
-1
-1
PZ = Z  Z'Z  Z'
-1
^
 
^
V β 2SLS  s  X'PZ X 
2
-1
^


SSE   Y - X β 2SLS 


SSR
R2 
SSR  SSE
SSE
s 
n  rank ( X )
2
1 
-1

SSR  Y'  PZ X  X'PZ X  X'PZ  J n  Y
n 

'
^


 Y - X β 2SLS 


Model 2 – Rank = f(Score, Events)
Z
X
Intercept Drive Fairway Green sandsv Putts Rounds Events Completed
1
251 73.80 64.70 36.50
1.79
61
20
13
1 256.7 73.30 69.60 30.20
1.78
65
20
14
1 250.1 65.90 64.10 32.70
1.78
56
18
12
...
...
...
...
...
...
...
...
...
1 249.8 70.10 67.60 26.30
1.83
67
22
14
1 239.8 77.70 62.30 30.60
1.88
40
17
4
1 256.1 74.50 72.40 31.40
1.8
89
25
23
X'P_ZX
146 10610.59
2767
10610.59 771305.2 200694.4
2767 200694.4
54887
X'P_ZY
7734.555
559770.4
152254.3
INV(X'P_ZX)
50.79456 -0.66845 -0.1165
-0.66845 0.008823 0.001436
-0.1165 0.001436 0.000642
Beta_2SLS
956.9965
-12.5125
0.281134
Intercept AveStrokesEvents
1 72.492
20
1 71.477
20
1 72.25
18
...
...
...
1 72.657
22
1 74.225
17
1 71.157
25
SSE
s^2
1806.234 12.63101
Y
Rank(Pctile)
55.17757
66.57407
55.50107
...
52.17118
31.27489
75.77619
V^(Beta_2SLS)
641.5866 -8.4432 -1.47152
-8.4432 0.111449 0.018132
-1.47152 0.018132 0.008113
SSModel ybar
SSReg
R^2
440626.2 52.9764 30877.31 0.944736
SE(B_2SLS)
25.32956
0.333839
0.09007
Model 3: ln(Prize) = f(Rank,Completed)
Z
X
Intercept Drive Fairway Green sandsv Putts Rounds Events Completed X'P_ZX
146 7734.555
1 251 73.80 64.70 36.50 1.79
61
20
13
1 256.7 73.30 69.60 30.20 1.78
65
20
14 7734.555 441143.7
1736 104550.7
1 250.1 65.90 64.10 32.70 1.78
56
18
12
...
...
...
...
...
...
...
...
... INV(X'P_ZX)
1 249.8 70.10 67.60 26.30 1.83
67
22
14 0.202911 -0.00626
1 239.8 77.70 62.30 30.60 1.88
40
17
4
-0.00626 0.000228
1 256.1 74.50 72.40 31.40
1.8
89
25
23 0.011416 -0.00049
Intercept Rank(Pctile)Completed
1 55.17757
13
1 66.57407
14
1 55.50107
12
...
...
...
1 52.17118
14
1 31.27489
4
1 75.77619
23
Y
lnPrize
12.22
12.86
11.74
...
12.66
9.36
13.33
1736
104550.7
26504
X'P_ZY
1720.879
93921.62
21631.74
0.011416
-0.00049
0.001222
Beta_2SLS
7.881995
0.055812
0.079741
V^(Beta_2SLS)
0.052024 -0.00161 0.002927
-0.00161 5.85E-05 -0.00013
0.002927 -0.00013 0.000313
SE(B_2SLS)
0.228087
0.007648
0.017699
SSE
s^2
36.66316 0.256386
Robust Estimate of Variance of 2SLS Estimator
V  2i    22i
 
2
   21    21
0
   
2
  22    0  22
 Σ V 




 


  2 n    0
0


^
 V β 2SLS  V
0 

0 


 22n 
 X'PZ X  X'PZY   X'PZ X
1
1
X'PZ ΣPZ X  X'PZ X 
1
1
1
-1
-1
  X'PZ X   X'Z  Z'Z  Z'ΣZ  Z'Z  Z'X   X'PZ X 


Replacing Z'ΣZ with its estimator:
2
e21
0

2
^
0 e22

S = Z'


0
 0
^
 
^
0

0
Z 


e22n 
V β 2SLS   X'PZ X 
1
n
 e z z'
i 1
2
2i i i
^
'
e2i  Y2i  x i β 2SLS
^
1
-1
-1


X'Z
Z'Z
S
Z'Z
Z'X
X'P
X






Z


Exact same method for equation 3
 z' 
 x' 
1
 
 1
 z' 
 x' 
2
Z =   X =  2
 
 
 
 
'
 z n 
 x'n 
Results for Model 2: Rank = f(Score, Events)
S-hat
12.37147
3102.712
850.8146
795.9452
462.1251
22.63899
638.5061
210.1805
125.0767
3102.712
778916.5
213337.1
199840.8
115696.1
5676.833
160482.6
52740.96
31513.16
850.8146
213337.1
58857.24
54843.46
31745.12
1556.302
44235.96
14519.52
8719.568
795.9452
199840.8
54843.46
51415.23
29684.13
1455.89
41635.56
13637.83
8236.469
462.1251
115696.1
31745.12
29684.13
18204.05
844.6313
24297.06
7958.194
4765.153
22.63899
5676.833
1556.302
1455.89
844.6313
41.4413
1164.82
384.0072
227.7275
638.5061
160482.6
44235.96
41635.56
24297.06
1164.82
36888.23
11763.14
7644.445
210.1805
52740.96
14519.52
13637.83
7958.194
384.0072
11763.14
3807.094
2383.913
125.0767
31513.16
8719.568
8236.469
4765.153
227.7275
7644.445
2383.913
1659.86
Homoskedastic Errors Heteroskedastic Errors
Beta_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)
641.5866
-8.4432
-1.4715
-8.4432
0.1114
0.0181
V(B_2SLS)
-1.4715 551.3468
0.0181
-7.1334
0.0081
-1.6873
-7.1334
0.0928
0.0202
-1.6873
0.0202
0.0111
Results for Model 3: ln(Prize) = f(Rank,Completed)
S-hat
0.251118
62.19775
17.16093
16.13331
9.791545
0.461502
13.36902
4.551543
2.369981
62.19775
15434.73
4247.927
4004.667
2420.563
114.2658
3342.212
1132.968
597.983
17.16093
4247.927
1179.906
1103.003
668.8473
31.54885
911.7178
310.2781
161.2776
16.13331
4004.667
1103.003
1042.178
627.3954
29.63322
874.5716
294.8918
158.2978
9.791545
2420.563
668.8473
627.3954
397.5982
17.99088
521.5487
177.865
91.50151
0.461501568
114.2658074
31.54885121
29.633223
17.9908815
0.848513025
24.44990949
8.342033579
4.315346873
13.36902
3342.212
911.7178
874.5716
521.5487
24.44991
812.2522
263.6976
157.1673
4.551543
1132.968
310.2781
294.8918
177.865
8.342034
263.6976
87.66156
49.09644
2.369981
597.983
161.2776
158.2978
91.50151
4.315347
157.1673
49.09644
34.27717
Homoskedastic Errors Heteroskedastic Errors
Beta_2SLS SE(B_2SLS) t
SE(B_2SLS) t
7.8820
0.2281
34.5570
0.2544
30.9870
0.0558
0.0076
7.2975
0.0086
6.5169
0.0797
0.0177
4.5054
0.0205
3.8845
V(B_2SLS)
0.05202
-0.00161
0.00293
-0.00161
0.00006
-0.00013
V(B_2SLS)
0.00293
0.06470
-0.00013
-0.00196
0.00031
0.00359
-0.00196
0.00007
-0.00016
0.00359
-0.00016
0.00042
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
Model Description - I
Model 1: SCORE i   0   D Di   F Fi   G Gi   S Si   P Pi   R Ri  1i  Y1i
Model 2: Rank i   0  SCORESCORE i   E Ei   2i  Y2i
Model 3: ln  Prizei    0   RANK Rank i   C Ci   3i  Y3i
 Y11 
 Y21 
 Y31 
 Y1 
Y 
Y 
Y 
12
22
32
 Y 
 Y 
 Y  Y 
Y1  
2
3
 2






 Y3 






Y
Y
Y
1,146
2,146
3,146






F1
G1
S1
P1
R1 
1 D1
1 SC1
1 D
1 SC
F2
G2
S2
P2
R2 
2
2

X1 
X2  






1 D146 F146 G146 S146 P146 R146 
1 SC146
0
 X1 0
X   0 X 2 0 
 0
0 X 3 
^
eki  Yki  Y ki
S11 =
k  1, 2,3 are residuals from 2-Stage Least Squares Regressions
1 146 2
 e1i
146  7 i 1
 S11
S   S 21
 S31
S12
S 22
S32
E1 
1 RA1
1 RA
E2 
2
X3  




E146 
1 RA146
S13 
S 23 
S33 
S12 =
146
1
 e1i e2i
146  (7  3) / 2 i 1
and so on for S13 , S 22 , S 23 , S33
W  S 1  Z  Z'Z  Z'  S 1  PZ
1
C1 
C2 


C146 
Model Description - II
^

β 3SLS   X'WX  X'WY  X'S  Z  Z'Z  Z'X
^
-1
 
^

-1
-1
V β 3SLS   X'WX   X'S  Z  Z'Z  Z'X
1
-1
-1


1
X'S -1  Z  Z'Z  Z'Y
1
where:
 S 11

S 1   S 21
 S 31

S 12
S 22
S 32
S 13 

S 23 
S 33 
 S 11X1'PZ X1

X'WX   S 21X 2'PZ X1
 S 31X 3'PZ X1

 S 11PZ

W   S 21PZ
 S 31PZ

S 12 X1'PZ X 2
S 22 X 2'PZ X 2
S 32 X 3'PZ X 2
S 12 PZ
S 22 PZ
S 32 PZ
S 13 PZ 

S 23 PZ 
S 33 PZ 
S 13 X1'PZ X 3 

S 23 X 2'PZ X 3 
S 33 X 3'PZ X 3 
 S 11X1'PZ Y1  S 12 X1'PZ Y2  S 13 X1'PZ Y3 


X'WY   S 21X 2'PZ Y1  S 22 X 2'PZ Y2  S 23 X 2'PZ Y3 
 S 31X 3'PZ Y1  S 32 X 3'PZ Y2  S 33 X 3'PZ Y3 


-1
Estimation Results
X'WX
1554.094
389489
108250.1
101857
58810.75
2837.925
90861.28
-25.0786
-1822.6
-475.292
-152.235
-8064.85
-1810.13
389488.97
97734135
27100830
25559022
14728936
711073.22
22874738
-6285.237
-456663.2
-119415.2
-38153.28
-2030746
-457357.3
V(Beta_3SLS)
4.234876 -0.00573
-0.00573 2.047E-05
-0.00541 1.756E-05
0.004066 -2.52E-05
-0.00191 2.809E-06
-1.39939 0.0005153
-0.00103 -3.54E-07
1.641475 0.000709
-0.02114 -9.32E-06
-0.00549 -1.67E-06
-0.0096 1.378E-06
0.000355 -4.38E-08
-0.00077 7.95E-08
108250.1
27100830
7588745
7104946
4096401
197673.2
6343826
-1746.85
-126916
-33121.7
-10603.9
-564282
-126816
-0.00541
1.76E-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-08
101857
25559022
7104946
6705381
3853879
185907.4
6033962
-1643.68
-119354
-31362.4
-9977.64
-536619
-121572
0.004066
-2.5E-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
58810.75
14728936
4096401
3853879
2319387
107296.4
3488898
-949.037
-68922.7
-18155.7
-5760.94
-308831
-69983.9
-0.00191
2.81E-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
2837.925
711073.2
197673.2
185907.4
107296.4
5184.691
165259.8
-45.796
-3329.07
-866.152
-277.995
-14662.7
-3281.32
-1.39939
0.000515
0.000212
-0.00073
0.00031
0.680887
0.000829
-1.53791
0.020033
0.004326
0.00906
-0.00032
0.000649
90861.28
22874738
6343826
6033962
3488898
165259.8
5838242
-1466.24
-106149
-29567.5
-8900.52
-506396
-122817
-0.00103
-3.5E-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-06
1.641475
0.000709
0.002615
0.014586
0.001482
-1.53791
-0.00349
631.5903
-8.32114
-1.41211
2.067496
-0.06317
0.107152
-25.0786
-6285.24
-1746.85
-1643.68
-949.037
-45.796
-1466.24
13.9931
1016.952
265.1981
39.31672
2082.859
467.492
-0.02114
-9.3E-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
-1822.6
-456663
-126916
-119354
-68922.7
-3329.07
-106149
1016.952
73924.34
19235.18
2857.353
150742
33733.38
-0.00549
-1.7E-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
-475.292
-119415
-33121.7
-31362.4
-18155.7
-866.152
-29567.5
265.1981
19235.18
5260.544
745.1327
41000.96
9711.5
-0.0096
1.38E-06
-3.5E-07
-0.00012
-1.4E-05
0.00906
2.43E-05
2.067496
-0.02734
-0.00451
0.051242
-0.00157
0.002827
-152.235
-38153.3
-10603.9
-9977.64
-5760.94
-277.995
-8900.52
39.31672
2857.353
745.1327
684.3542
36254.62
8137.252
0.000355
-4.4E-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
-8064.85
-2030746
-564282
-536619
-308831
-14662.7
-506396
2082.859
150742
41000.96
36254.62
2067798
490066.5
-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
-1810.13
-457357
-126816
-121572
-69983.9
-3281.32
-122817
467.492
33733.38
9711.5
8137.252
490066.5
124233.7
EQ1
EQ2
EQ3
X'WY
109821.2
27513834
7646744
7189554
4151912
200612.5
6386177
-617.871
-45213.8
-10931.8
-914.47
-24634
-1065.64
Beta_3SLS StdErr
60.66021 2.057881
-0.00305 0.004524
-0.01449 0.006129
-0.1377 0.009775
-0.01484 0.003499
13.09106 0.825158
-0.00905 0.002163
954.3673 25.13146
-12.4821 0.331598
0.303414 0.087992
7.96384 0.226367
0.050763 0.007529
0.095351 0.017317
SAS Program
data 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;
STATA Program
insheet 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