TSP Version 4.4 Copyright (C) 1997 TSP International ALL RIGHTS RESERVED In case of questions or problems, see your local TSP consultant or send a description of the problem and the associated TSP output to: TSP International P.O. Box 61015, Station A Palo Alto, CA 94306 USA PROGRAM LINE ****************************************************************** 1 OPTIONS CRT NODATE; 2 NAME KLEIN3S '2SLS and 3SLS estimates of Klein Model I' ; 2 ? 2 ? Correct published results for 2SLS and 3SLS can be found in: 2 ? - Henri Theil, "Principles of Econometrics", 1971, p.517 2 ? - A. Zellner and H. Thornber, Econometrica, July 1966, p.728-729 2 ? (use the double precision column - 16 significant figures) 2 ? (Zellner and Thornber do not print estimates of C0, due to 2 ? differing normalization for the time trend variable). 2 ? - Berndt, Ernst, "The Practice of Econometrics", 1st ed., 1991, p.553 2 ? - Greene, William, "Econometric Analysis", 2nd ed., 1993, p.615 2 ? TSP reproduces the above coefficient results with the commands below. 2 ? *Incorrect* published results for 3SLS can be found in: 2 ? - Zellner and Theil, Econometrica, January 1962, (the original 3SLS 2 ? reference). They had problems partly due to some simplifications 2 ? to reduce the calculations, and partly due to the use of 8-bit 2 ? single precision. 2 ? - Pindyck and Rubinfeld "Econometric Models and Economic Forecasts", 2 ? 4th edition, 1997, Example 12.4 (and earlier editions). 2 ? They copied the Zellner and Theil 1962 results. 2 ? 2 LOAD ; 3 SMPL 1 22 ; 4 W = W1+W2 ; 5 TX = YT-Y ; 6 P = CX+I+G-TX-W ; 7 E = Y+TX-W2 ; 8 TM = YEAR-1931 ; 9 SMPL 1 21 ; 10 K = K1(+1); 11 SMPL 2 22 ; 12 LIST IVS C P(-1) K(-1) E(-1) TM W2 G TX ; 13 ? 13 ? 2SLS Estimates 13 ? 13 ? TSP reproduces the coefficients to the precision usually reported 13 ? (4-6 digits). The standard errors do not match, because TSP uses a 13 ? small sample adjustment. If you multiply the TSP standard errors by 13 ? sqrt((@NOB-4)/@NOB), you obtain the published results. 13 ? 13 2SLS(INST=IVS) CX C W P P(-1); 14 2SLS(INST=IVS) I C P P(-1) K(-1); 15 2SLS(INST=IVS) W1 C E E(-1) TM; 16 MAT SEPUB = @SES*SQRT((@NOB-4)/@NOB); 17 PRINT SEPUB; ? Print standard errors of last equation, 18 ? without small sample adjustment 18 ? The small sample adjustment is used so that 2SLS will exactly 18 ? reproduce OLSQ standard errors, when all the RHS variables are 18 ? exogenous. Otherwise, it might appear to the naive student that 18 ? 2SLS outperforms OLSQ in such a model. 18 ? 18 ? 3SLS Estimates 18 ? 18 FRML CONS CX = A0 + A1*W + A2*P + A3*P(-1) ; 19 FRML INV I = B0 + B1*P + B2*P(-1) + B3*K(-1) ; 20 FRML WAGES W1 = C0 + C1*E + C2*E(-1) + C3*TM ; 21 PARAM A0-A3 B0-B3 C0-C3; 22 3SLS(INST=IVS) CONS INV WAGES ; 23 END ; EXECUTION ******************************************************************************* 2 NOPRINT; Current sample: 1 to 22 Current sample: 1 to 22 Current sample: 1 to 21 Current sample: 2 to 22 Equation 1 ============ Method of estimation = Instrumental Variable Endogenous variables: W P Included exogenous variables: C P(-1) Excluded exogenous variables: K(-1) E(-1) TM W2 G TX Dependent variable: CX Current sample: 2 to 22 Number of observations: 21 Mean of dep. var. = 53.9952 R-squared = .976805 Std. dev. of dep. var. = 6.86087 Adjusted R-squared = .972712 Sum of squared residuals = 21.9253 Durbin-Watson = 1.48507 [<.611] Variance of residuals = 1.28972 E'PZ*E = 9.15799 Std. error of regression = 1.13566 Estimated Standard Variable Coefficient Error t-statistic P-value C 16.5548 1.46798 11.2772 [.000] W .810183 .044735 18.1107 [.000] P .017302 .131205 .131872 [.895] P(-1) .216234 .119222 1.81371 [.070] Equation 2 ============ Method of estimation = Instrumental Variable Endogenous variables: P Included exogenous variables: C P(-1) K(-1) Excluded exogenous variables: E(-1) TM W2 G TX Dependent variable: I Current sample: 2 to 22 Number of observations: 21 Mean of dep. var. = 1.26667 R-squared = .885417 Std. dev. of dep. var. = 3.55195 Adjusted R-squared = .865196 Sum of squared residuals = 29.0469 Durbin-Watson = 2.08533 [<.951] Variance of residuals = 1.70864 E'PZ*E = 2.51042 Std. error of regression = 1.30715 Estimated Standard Variable Coefficient Error t-statistic P-value C 20.2782 8.38326 2.41889 [.016] P .150221 .192534 .780233 [.435] P(-1) .615944 .180926 3.40440 [.001] K(-1) -.157788 .040152 -3.92975 [.000] Equation 3 ============ Method of estimation = Instrumental Variable Endogenous variables: E Included exogenous variables: C E(-1) TM Excluded exogenous variables: P(-1) K(-1) W2 G TX Dependent variable: W1 Current sample: 2 to 22 Number of observations: 21 Mean of dep. var. = 36.3619 R-squared = .987414 Std. dev. of dep. var. = 6.30440 Adjusted R-squared = .985193 Sum of squared residuals = 10.0049 Durbin-Watson = 1.96342 [<.916] Variance of residuals = .588526 E'PZ*E = 5.95305 Std. error of regression = .767155 Estimated Standard Variable Coefficient Error t-statistic P-value C 1.50030 1.27569 1.17607 [.240] E .438859 .039603 11.0816 [.000] E(-1) .146674 .043164 3.39807 [.001] TM .130396 .032388 4.02600 [.000] SEPUB 1 1 1.14778 2 0.035632 3 0.038836 4 0.029141 THREE STAGE LEAST SQUARES ========================= EQUATIONS: CONS INV WAGES INSTRUMENTS: C P(-1) K(-1) E(-1) TM W2 G TX MAXIMUM NUMBER OF ITERATIONS ON V-COV MATRIX OF RESIDUALS = 0 NOTE => The model is linear in the parameters. Working space used: 4139 STARTING VALUES A0 A1 A2 A3 VALUE 0.00000 0.00000 0.00000 0.00000 B0 B1 B2 B3 VALUE 0.00000 0.00000 0.00000 0.00000 C0 C1 C2 C3 VALUE 0.00000 0.00000 0.00000 0.00000 F= 90877. FNEW= 17.621 ISQZ= 0 STEP= 1.0000 CRIT= 90859. CONVERGENCE ACHIEVED AFTER 1 ITERATIONS 2 FUNCTION EVALUATIONS. END OF TWO STAGE LEAST SQUARES ITERATIONS (SIGMA=IDENTITY). THREE STAGE LEAST SQUARES ESTIMATES WILL BE OBTAINED USING THIS ESTIMATE OF SIGMA: RESIDUAL COVARIANCE MATRIX CONS INV WAGES CONS 1.04406 INV 0.43785 1.38319 WAGES -0.38523 0.19261 0.47643 WEIGHTING MATRIX CONS INV WAGES CONS 0.97867 -0.38290 1.02814 INV 0.91304 -0.61598 WAGES 2.08638 Working space used: 4139 F= 28.613 FNEW= 24.291 ISQZ= 0 STEP= 1.0000 CRIT= 4.3222 CONVERGENCE ACHIEVED AFTER 1 ITERATIONS 4 FUNCTION EVALUATIONS. THREE STAGE LEAST SQUARES ========================= Residual Covariance Matrix CONS INV WAGES CONS 0.89176 INV 0.41132 2.09305 WAGES -0.39361 0.40305 0.52003 Weighting Matrix CONS INV WAGES CONS 0.97867 -0.38290 1.02814 INV 0.91304 -0.61598 WAGES 2.08638 Covariance Matrix of Transformed Residuals CONS INV WAGES CONS 17.93668 INV 0.70070 33.34776 WAGES -3.24194 0.77980 15.85210 E'HH'E = 24.2910 Number of Observations = 21 Standard Parameter Estimate Error t-statistic P-value A0 16.4408 1.30455 12.6027 [.000] A1 .790081 .037938 20.8256 [.000] A2 .124890 .108129 1.15501 [.248] A3 .163144 .100438 1.62432 [.104] B0 28.1779 6.79378 4.14760 [.000] B1 -.013080 .161896 -.080792 [.936] B2 .755724 .152933 4.94153 [.000] B3 -.194848 .032531 -5.98967 [.000] C0 1.79722 1.11585 1.61062 [.107] C1 .400492 .031813 12.5888 [.000] C2 .181291 .034159 5.30731 [.000] C3 .149674 .027935 5.35790 [.000] Standard Errors computed from quadratic form of analytic first derivatives (Gauss) Equation: CONS Dependent variable: CX Mean of dep. var. = 53.9952 Std. error of regression = .944331 Std. dev. of dep. var. = 6.86087 R-squared = .980168 Sum of squared residuals = 18.7270 Durbin-Watson = 1.42494 [<.557] Variance of residuals = .891760 Equation: INV Dependent variable: I Mean of dep. var. = 1.26667 Std. error of regression = 1.44674 Std. dev. of dep. var. = 3.55195 R-squared = .826207 Sum of squared residuals = 43.9541 Durbin-Watson = 1.99588 [<.927] Variance of residuals = 2.09305 Equation: WAGES Dependent variable: W1 Mean of dep. var. = 36.3619 Std. error of regression = .721128 Std. dev. of dep. var. = 6.30440 R-squared = .986316 Sum of squared residuals = 10.9205 Durbin-Watson = 2.15505 [<.965] Variance of residuals = .520026 ******************************************************************************* END OF OUTPUT FOR USER KLEIN3S MEMORY USAGE: ITEM: DATA ARRAY TOTAL MEMORY UNITS: (4-BYTE WORDS) (MEGABYTES) MEMORY ALLOCATED : 500000 4.0 MEMORY ACTUALLY REQUIRED : 6619 2.1 CURRENT VARIABLE STORAGE : 2480