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Comparing the ordinary least square regression with the instrumental variable estimator.
data example16_1;
input Year Q P L NptCost CPI Income;
cards;
1960 72 51 24 46 88.7 6036
1961 70 52 25 46 89.6 6113
1962 71 54 26 47 90.6 6271
1963 74 55 27 47 91.7 6378
1964 72 55 29 47 92.9 6727
1965 76 53 31 48 94.5 7027
1966 73 55 33 50 97.2 7280
1967 77 52 35 50 100.0 7513
1968 79 52 38 50 104.2 7728
1969 80 50 40 52 109.8 7891
1970 77 52 42 54 116.3 8134
1971 86 56 43 57 121.3 8322
1972 87 60 47 61 125.3 8562
1973 92 91 53 73 133.1 9042
1974 84 117 66 83 147.7 8867
1975 93 105 75 91 161.2 8944
1976 92 102 86 97 170.5 9175
1977 100 100 100 100 181.5 9381
1978 102 105 109 108 195.4 9735
1979 113 116 125 125 217.4 9829
1980 101 125 145 138 246.8 9722
1981 117 134 158 148 272.4 9769
1982 117 121 157 150 289.1 9725
1983 88 128 148 153 298.4 9930
1984 111 139 146 155 311.1 10421
1985 117 120 128 151 322.2 10563
1986 108 106 112 146 328.4 10780
;
run;
proc reg data=example16_1;
model q=p;
run;
quit;
The REG Procedure
Model: MODEL1
Dependent Variable: Q
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 4918.74252 4918.74252 71.57 <.0001
Error 25 1718.22044 68.72882
Corrected Total 26 6636.96296
Root MSE 8.29028 R-Square 0.7411
Dependent Mean 89.96296 Adj R-Sq 0.7308
Coeff Var 9.21522
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 54.13198 4.52600 11.96 <.0001
P 1 0.41953 0.04959 8.46 <.0001
proc model data=example16_1;
q = intercept + P_co * p;
fit q /n2sls vardef=N;
instruments income ;
run;
quit;
The MODEL Procedure
Nonlinear 2SLS Summary of Residual Errors
DF DF Adj
Equation Model Error SSE MSE Root MSE R-Square R-Sq
Q 2 25 1916.7 70.9877 8.4254 0.7112 0.6997
Nonlinear 2SLS Parameter Estimates
Approx Approx
Parameter Estimate Std Err t Value Pr > |t|
intercept 46.93491 5.1954 9.03 <.0001
P_co 0.503798 0.0578 8.72 <.0001
Number of Observations Statistics for System
Used 27 Objective 5.971E-26
Missing 0 Objective*N 1.612E-24
On page 687 with four difference estimates for Klein's model.
data klein;
input Year C P Wp I Klag X Wg G T ;
cards;
1920 39.8 12.7 28.8 2.7 180.1 44.9 2.2 2.4 3.4
1921 41.9 12.4 25.5 -0.2 182.8 45.6 2.7 3.9 7.7
1922 45.0 16.9 29.3 1.9 182.6 50.1 2.9 3.2 3.9
1923 49.2 18.4 34.1 5.2 184.5 57.2 2.9 2.8 4.7
1924 50.6 19.4 33.9 3.0 189.7 57.1 3.1 3.5 3.8
1925 52.6 20.1 35.4 5.1 192.7 61.0 3.2 3.3 5.5
1926 55.1 19.6 37.4 5.6 197.8 64.0 3.3 3.3 7.0
1927 56.2 19.8 37.9 4.2 203.4 64.4 3.6 4.0 6.7
1928 57.3 21.1 39.2 3.0 207.6 64.5 3.7 4.2 4.2
1929 57.8 21.7 41.3 5.1 210.6 67.0 4.0 4.1 4.0
1930 55.0 15.6 37.9 1.0 215.7 61.2 4.2 5.2 7.7
1931 50.9 11.4 34.5 -3.4 216.7 53.4 4.8 5.9 7.5
1932 45.6 7.0 29.0 -6.2 213.3 44.3 5.3 4.9 8.3
1933 46.5 11.2 28.5 -5.1 207.1 45.1 5.6 3.7 5.4
1934 48.7 12.3 30.6 -3.0 202.0 49.7 6.0 4.0 6.8
1935 51.3 14.0 33.2 -1.3 199.0 54.4 6.1 4.4 7.2
1936 57.7 17.6 36.8 2.1 197.7 62.7 7.4 2.9 8.3
1937 58.7 17.3 41.0 2.0 199.8 65.0 6.7 4.3 6.7
1938 57.5 15.3 38.2 -1.9 201.8 60.9 7.7 5.3 7.4
1939 61.6 19.0 41.6 1.3 199.9 69.5 7.8 6.6 8.9
1940 65.0 21.1 45.0 3.3 201.2 75.7 8.0 7.4 9.6
1941 69.7 23.5 53.3 4.9 204.5 88.4 8.5 13.8 11.6
;
run;
data klein;
set klein;
W=Wp+Wg;
A=Year-1931;
Plag=lag(P);
Xlag=lag(X);
Y=C + I + G - T;
run;
/*OLS Estimate*/
proc reg data=klein;
model C = P Plag W;
run;
quit;
The REG Procedure
Model: MODEL1
Dependent Variable: C
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 923.55008 307.85003 292.71 <.0001
Error 17 17.87945 1.05173
Corrected Total 20 941.42952
Root MSE 1.02554 R-Square 0.9810
Dependent Mean 53.99524 Adj R-Sq 0.9777
Coeff Var 1.89932
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 16.23660 1.30270 12.46 <.0001
P 1 0.19293 0.09121 2.12 0.0495
Plag 1 0.08988 0.09065 0.99 0.3353
W 1 0.79622 0.03994 19.93 <.0001
/* 2sls Estimate*/
proc model data=klein;
C = cons_c + P_c * P + Plag_c * Plag + W_c*W;
ENDOGENOUS W P X;
EXOGENOUS T Wg G;
fit C /n2sls vardef=N;
instruments G T A Wg Plag Klag Xlag ;
run;
quit;
The MODEL Procedure
Nonlinear 2SLS Summary of Residual Errors
DF DF Adj
Equation Model Error SSE MSE Root MSE R-Square R-Sq
C 4 17 21.9252 1.0441 1.0218 0.9767 0.9726
Nonlinear 2SLS Parameter Estimates
Approx Approx
Parameter Estimate Std Err t Value Pr > |t|
cons_c 16.55476 1.3208 12.53 <.0001
P_c 0.017302 0.1180 0.15 0.8852
Plag_c 0.216234 0.1073 2.02 0.0599
W_c 0.810183 0.0402 20.13 <.0001
Number of Observations Statistics for System
Used 21 Objective 0.4361
Missing 1 Objective*N 9.1580
/* GMM(H2sls)*/
proc model data=klein;
C = cons_c + P_c * P + Plag_c * Plag + W_c*W;
fit C /n2sls outv=vdata;
instruments G T A Wg Plag Klag Xlag ;
run;
quit;
data vblkdiag;
set vdata;
if (eq_row ^= eq_col) then value=0; /* create block-diagonal V */
run;
/*use the block-diagonal in gmm*/
proc model data=klein OUTPARMS=parm1;
C = cons_c + P_c * P + Plag_c * Plag + W_c*W;
fit C / gmm vdata=vblkdiag;
instruments G T A Wg Plag Klag Xlag ;
run;
quit;
/*use the parameter estimate from the previous model as the initial estimate*/
proc model data=klein;
C = cons_c + P_c * P + Plag_c * Plag + W_c*W;
fit C /estdata=parm1 no2sls gmm kernel=(bart,0,) vardef=N;
instruments G T A Wg Plag Klag Xlag ;
run;
quit;
The MODEL Procedure
Nonlinear GMM Summary of Residual Errors
DF DF Adj
Equation Model Error SSE MSE Root MSE R-Square R-Sq
C 4 17 21.7278 1.0347 1.0172 0.9769 0.9728
Nonlinear GMM Parameter Estimates
Approx Approx
Parameter Estimate Std Err t Value Pr > |t|
cons_c 14.35319 0.8984 15.98 <.0001
P_c 0.091284 0.0626 1.46 0.1627
Plag_c 0.141792 0.0658 2.16 0.0457
W_c 0.862988 0.0291 29.62 <.0001
Number of Observations Statistics for System
Used 21 Objective 0.1810
Missing 1 Objective*N 3.8004
/*LIML*/
proc syslin data=klein liml vardef=N;
endogenous C P Wp I X W Klag Y ;
instruments Klag Plag Xlag Wg G T A;
consume: model c = P Plag W;
run;
The SYSLIN Procedure
Limited-Information Maximum Likelihood Estimation
Model CONSUME
Dependent Variable C
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 854.3541 284.7847 118.42 <.0001
Error 17 40.88419 2.404952
Corrected Total 20 941.4295
Root MSE 1.55079 R-Square 0.95433
Dependent Mean 53.99524 Adj R-Sq 0.94627
Coeff Var 2.87209
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 17.14765 1.840295 9.32 <.0001
P 1 -0.22251 0.201748 -1.10 0.2854
Plag 1 0.396027 0.173598 2.28 0.0357
W 1 0.822559 0.055378 14.85 <.0001
NOTE: K-Class Estimation with K=1.4987455056
Limited-Information Estimates vs. Full-Information estimates
/* 2SLS */
proc model data=klein;
C = cons_c + P_c * P + Plag_c * Plag + W_c*W;
I = cons_i + P_i * P + Plag_i * Plag +Klag_i * Klag;
Wp = cons_w + X_w * X + Xlag_w * Xlag + A_w *A;
ENDOGENOUS W P X;
EXOGENOUS T Wg G;
fit C I Wp /n2sls vardef=N;
instruments G T A Wg Plag Klag Xlag ;
run;
quit;
The MODEL Procedure
Nonlinear 2SLS Summary of Residual Errors
DF DF Adj
Equation Model Error SSE MSE Root MSE R-Square R-Sq
C 4 17 21.9252 1.0441 1.0218 0.9767 0.9726
I 4 17 29.0469 1.3832 1.1761 0.8849 0.8646
Wp 4 17 10.0050 0.4764 0.6902 0.9874 0.9852
Nonlinear 2SLS Parameter Estimates
Approx Approx
Parameter Estimate Std Err t Value Pr > |t|
cons_c 16.55476 1.3208 12.53 <.0001
P_c 0.017302 0.1180 0.15 0.8852
Plag_c 0.216234 0.1073 2.02 0.0599
W_c 0.810183 0.0402 20.13 <.0001
cons_i 20.27821 7.5427 2.69 0.0155
P_i 0.150222 0.1732 0.87 0.3979
Plag_i 0.615944 0.1628 3.78 0.0015
Klag_i -0.15779 0.0361 -4.37 0.0004
cons_w 1.500297 1.1478 1.31 0.2086
X_w 0.438859 0.0356 12.32 <.0001
Xlag_w 0.146674 0.0388 3.78 0.0015
A_w 0.130396 0.0291 4.47 0.0003
Number of Observations Statistics for System
Used 21 Objective 0.8391
Missing 1 Objective*N 17.6215
/* LIML */
/*The standard errors for the second and third equation are different from the book*/
proc syslin data=klein liml vardef=N;
endogenous C P Wp I X W Klag Y ;
instruments Klag Plag Xlag Wg G T A;
consume: model c = P Plag W;
invest: model i = P Plag Klag;
labor: model wp = X Xlag A;
run;
The SYSLIN Procedure
Limited-Information Maximum Likelihood Estimation
Model CONSUME
Dependent Variable C
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 854.3541 284.7847 118.42 <.0001
Error 17 40.88419 2.404952
Corrected Total 20 941.4295
Root MSE 1.55079 R-Square 0.95433
Dependent Mean 53.99524 Adj R-Sq 0.94627
Coeff Var 2.87209
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 17.14765 1.840295 9.32 <.0001
P 1 -0.22251 0.201748 -1.10 0.2854
Plag 1 0.396027 0.173598 2.28 0.0357
W 1 0.822559 0.055378 14.85 <.0001
NOTE: K-Class Estimation with K=1.4987455056
The SYSLIN Procedure
Limited-Information Maximum Likelihood Estimation
Model INVEST
Dependent Variable I
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 210.3790 70.12634 34.06 <.0001
Error 17 34.99649 2.058617
Corrected Total 20 252.3267
Root MSE 1.43479 R-Square 0.85738
Dependent Mean 1.26667 Adj R-Sq 0.83221
Coeff Var 113.27274
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 22.59083 8.545818 2.64 0.0171
P 1 0.075185 0.202181 0.37 0.7146
Plag 1 0.680386 0.188175 3.62 0.0021
Klag 1 -0.16826 0.040798 -4.12 0.0007
NOTE: K-Class Estimation with K=1.0859528454
The SYSLIN Procedure
Limited-Information Maximum Likelihood Estimation
Model LABOR
Dependent Variable Wp
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 696.1485 232.0495 393.62 <.0001
Error 17 10.02192 0.589525
Corrected Total 20 794.9095
Root MSE 0.76781 R-Square 0.98581
Dependent Mean 36.36190 Adj R-Sq 0.98330
Coeff Var 2.11156
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 1.526187 1.188405 1.28 0.2163
X 1 0.433941 0.067937 6.39 <.0001
Xlag 1 0.151321 0.067054 2.26 0.0375
A 1 0.131593 0.032386 4.06 0.0008
NOTE: K-Class Estimation with K=2.4685825667
/* OLS Estimate */
proc reg data=klein;
model C= P Plag W;
model I = P Plag Klag;
model Wp= X Xlag A;
run;
quit;
The REG Procedure
Model: MODEL1
Dependent Variable: C
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 923.55008 307.85003 292.71 <.0001
Error 17 17.87945 1.05173
Corrected Total 20 941.42952
Root MSE 1.02554 R-Square 0.9810
Dependent Mean 53.99524 Adj R-Sq 0.9777
Coeff Var 1.89932
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 16.23660 1.30270 12.46 <.0001
P 1 0.19293 0.09121 2.12 0.0495
Plag 1 0.08988 0.09065 0.99 0.3353
W 1 0.79622 0.03994 19.93 <.0001
The REG Procedure
Model: MODEL2
Dependent Variable: I
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 235.00396 78.33465 76.88 <.0001
Error 17 17.32270 1.01898
Corrected Total 20 252.32667
Root MSE 1.00945 R-Square 0.9313
Dependent Mean 1.26667 Adj R-Sq 0.9192
Coeff Var 79.69315
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 10.12579 5.46555 1.85 0.0814
P 1 0.47964 0.09711 4.94 0.0001
Plag 1 0.33304 0.10086 3.30 0.0042
Klag 1 -0.11179 0.02673 -4.18 0.0006
The REG Procedure
Model: MODEL3
Dependent Variable: Wp
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 3 784.90477 261.63492 444.57 <.0001
Error 17 10.00475 0.58851
Corrected Total 20 794.90952
Root MSE 0.76715 R-Square 0.9874
Dependent Mean 36.36190 Adj R-Sq 0.9852
Coeff Var 2.10976
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 1.49704 1.27003 1.18 0.2547
X 1 0.43948 0.03241 13.56 <.0001
Xlag 1 0.14609 0.03742 3.90 0.0011
A 1 0.13025 0.03191 4.08 0.0008
/* 3SLS */
proc model data=klein;
C = cons_c + P_c * P + Plag_c * Plag + W_c*W;
I = cons_i + P_i * P + Plag_i * Plag +Klag_i * Klag;
Wp = cons_w + X_w * X + Xlag_w * Xlag + A_w *A;
ENDOGENOUS W P X ;
EXOGENOUS T Wg G;
fit C I Wp /n3sls vardef=N;
instruments G T A Wg Plag Klag Xlag ;
run;
quit;
The MODEL Procedure
Nonlinear 3SLS Summary of Residual Errors
DF DF Adj
Equation Model Error SSE MSE Root MSE R-Square R-Sq
C 4 17 18.7270 0.8918 0.9443 0.9801 0.9766
I 4 17 43.9540 2.0930 1.4467 0.8258 0.7951
Wp 4 17 10.9205 0.5200 0.7211 0.9863 0.9838
Nonlinear 3SLS Parameter Estimates
Approx Approx
Parameter Estimate Std Err t Value Pr > |t|
cons_c 16.44079 1.3045 12.60 <.0001
P_c 0.12489 0.1081 1.16 0.2641
Plag_c 0.163144 0.1004 1.62 0.1227
W_c 0.790081 0.0379 20.83 <.0001
cons_i 28.17785 6.7938 4.15 0.0007
P_i -0.01308 0.1619 -0.08 0.9366
Plag_i 0.755724 0.1529 4.94 0.0001
Klag_i -0.19485 0.0325 -5.99 <.0001
cons_w 1.797216 1.1159 1.61 0.1257
X_w 0.400492 0.0318 12.59 <.0001
Xlag_w 0.181291 0.0342 5.31 <.0001
A_w 0.149674 0.0279 5.36 <.0001
Number of Observations Statistics for System
Used 21 Objective 1.1567
Missing 1 Objective*N 24.2910
/* FIML */
proc syslin data=klein fiml;
endogenous C P Wp I X W Y ;
instruments Klag Plag Xlag Wg G T A;
consume: model C = P Plag W;
invest: model I = P Plag Klag;
labor: model Wp = X Xlag A;
product: identity X = C + I + G;
profit: identity P = Y - Wp;
wage: identity W = Wg + Wp;
income: identity Y = C + I + G - T;
run;
Model CONSUME
Dependent Variable C
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 18.34865 2.488480 7.37 <.0001
P 1 -0.23308 0.312439 -0.75 0.4659
Plag 1 0.385971 0.217618 1.77 0.0940
W 1 0.801879 0.035898 22.34 <.0001
Model INVEST
Dependent Variable I
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 27.26501 7.939025 3.43 0.0032
P 1 -0.80179 0.491990 -1.63 0.1216
Plag 1 1.052094 0.352749 2.98 0.0084
Klag 1 -0.14806 0.029844 -4.96 0.0001
Model LABOR
Dependent Variable Wp
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 5.797016 1.804906 3.21 0.0051
X 1 0.234023 0.048829 4.79 0.0002
Xlag 1 0.284728 0.045214 6.30 <.0001
A 1 0.234878 0.034506 6.81 <.0001
/* I3SLS */
proc syslin data=klein i3sls vardef=N;
endogenous C P Wp I X W Klag Y;
instruments Klag Plag Xlag Wg G T A;
consume: model c = P Plag W;
invest: model i = P Plag Klag;
labor: model wp = X Xlag A;
run;
The SYSLIN Procedure
Iterative Three-Stage Least Squares Estimation
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 16.55898 1.224394 13.52 <.0001
P 1 0.164505 0.096198 1.71 0.1054
Plag 1 0.176562 0.090100 1.96 0.0666
W 1 0.765804 0.034760 22.03 <.0001
Model INVEST
Dependent Variable I
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 42.89425 10.59316 4.05 0.0008
P 1 -0.35649 0.260138 -1.37 0.1884
Plag 1 1.011268 0.248757 4.07 0.0008
Klag 1 -0.26019 0.050866 -5.12 <.0001
Model LABOR
Dependent Variable Wp
Parameter Estimates
Parameter Standard
Variable DF Estimate Error t Value Pr > |t|
Intercept 1 2.624646 1.195543 2.20 0.0423
X 1 0.374782 0.031103 12.05 <.0001
Xlag 1 0.193650 0.032402 5.98 <.0001
A 1 0.167923 0.028929 5.80 <.0001
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