Welcome to SHAZAM - Version 9.0 - DEC 2003 SYSTEM=WIN-XP PAR= 2000 CURRENT WORKING DIRECTORY IS: C:\ECONOM~1\Berndt |_FILE PATH CHAP10.DAT/ |_Reference: Chapter 10 of |_* Ernst R. Berndt, The Practice of Econometrics, Addison-Wesley, 1991. |_* Use the TIME command to show annual data starting from 1920 |_TIME 1920 1 |_* Use the decimal point form of the SAMPLE command to show years |_SAMPLE 1920. 1941. |_READ (KLEIN) YEAR CN P W1 I KLAG E W2 G TX / SKIPLINES=1 UNIT 88 IS NOW ASSIGNED TO: CHAP10.DAT\KLEIN 10 VARIABLES AND 22 OBSERVATIONS STARTING AT OBS 1 |_* Exercise 1, p. 557. |_* (a) Variable transformations |_GENR T=YEAR-1931 |_GENR W=W1+W2 |_GENR Y=CN+I+G-TX |_GENR PLAG=LAG(P) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_GENR ELAG=LAG(E) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_PRINT T W Y CN P W1 I KLAG E W2 G TX T W Y CN P & W1 I KLAG E W2 & G TX -11.00000 31.00000 43.70000 39.80000 12.70000 28.80000 2.700000 180.1000 44.90000 2.200000 4.600000 3.400000 -10.00000 28.20000 40.60000 41.90000 12.40000 25.50000 -0.2000000 182.8000 45.60000 2.700000 6.600000 7.700000 -9.000000 32.20000 49.10000 45.00000 16.90000 29.30000 1.900000 182.6000 50.10000 2.900000 6.100000 3.900000 -8.000000 37.00000 55.40000 49.20000 18.40000 34.10000 5.200000 184.5000 57.20000 2.900000 5.700000 4.700000 -7.000000 37.00000 56.40000 50.60000 19.40000 33.90000 3.000000 189.7000 57.10000 3.100000 6.600000 3.800000 -6.000000 38.60000 58.70000 52.60000 20.10000 35.40000 5.100000 192.7000 61.00000 3.200000 6.500000 5.500000 -5.000000 40.70000 60.30000 55.10000 19.60000 37.40000 5.600000 197.8000 64.00000 3.300000 6.600000 7.000000 -4.000000 41.50000 61.30000 56.20000 19.80000 37.90000 4.200000 203.4000 64.40000 3.600000 7.600000 6.700000 -3.000000 42.90000 64.00000 57.30000 21.10000 39.20000 3.000000 207.6000 64.50000 3.700000 7.900000 4.200000 -2.000000 45.30000 67.00000 57.80000 21.70000 41.30000 5.100000 210.6000 67.00000 4.000000 8.100000 4.000000 -1.000000 42.10000 57.70000 55.00000 15.60000 37.90000 1.000000 215.7000 61.20000 4.200000 9.400000 7.700000 0.000000 39.30000 50.70000 50.90000 11.40000 34.50000 -3.400000 216.7000 53.40000 4.800000 10.70000 7.500000 1.000000 34.30000 41.30000 45.60000 7.000000 29.00000 -6.200000 213.3000 44.30000 5.300000 10.20000 8.300000 2.000000 34.10000 45.30000 46.50000 11.20000 28.50000 -5.100000 207.1000 45.10000 5.600000 9.300000 5.400000 3.000000 36.60000 48.90000 48.70000 12.30000 30.60000 -3.000000 202.0000 49.70000 6.000000 10.00000 6.800000 4.000000 39.30000 53.30000 51.30000 14.00000 33.20000 -1.300000 199.0000 54.40000 6.100000 10.50000 7.200000 5.000000 44.20000 61.80000 57.70000 17.60000 36.80000 2.100000 197.7000 62.70000 7.400000 10.30000 8.300000 6.000000 47.70000 65.00000 58.70000 17.30000 41.00000 2.000000 199.8000 65.00000 6.700000 11.00000 6.700000 7.000000 45.90000 61.20000 57.50000 15.30000 38.20000 -1.900000 201.8000 60.90000 7.700000 13.00000 7.400000 8.000000 49.40000 68.40000 61.60000 19.00000 41.60000 1.300000 199.9000 69.50000 7.800000 14.40000 8.900000 9.000000 53.00000 74.10000 65.00000 21.10000 45.00000 3.300000 201.2000 75.70000 8.000000 15.40000 9.600000 10.00000 61.80000 85.30000 69.70000 23.50000 53.30000 4.900000 204.5000 88.40000 8.500000 22.30000 11.60000 |_* (b) OLS regressions |_SAMPLE 1921. 1941. |_OLS CN W P PLAG / RSTAT REQUIRED MEMORY IS PAR= 5 CURRENT PAR= 2000 OLS ESTIMATION 21 OBSERVATIONS DEPENDENT VARIABLE= CN ...NOTE..SAMPLE RANGE SET TO: 2, 22 R-SQUARE = 0.9810 R-SQUARE ADJUSTED = 0.9777 VARIANCE OF THE ESTIMATE-SIGMA**2 = 1.0517 STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.0255 SUM OF SQUARED ERRORS-SSE= 17.879 MEAN OF DEPENDENT VARIABLE = 53.995 LOG OF THE LIKELIHOOD FUNCTION = -28.1086 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 17 DF P-VALUE CORR. COEFFICIENT AT MEANS W 0.79622 0.3994E-01 19.93 0.000 0.979 0.8769 0.6117 P 0.19293 0.9121E-01 2.115 0.049 0.456 0.1187 0.0604 PLAG 0.89885E-01 0.9065E-01 0.9916 0.335 0.234 0.0528 0.0273 CONSTANT 16.237 1.303 12.46 0.000 0.949 0.0000 0.3007 DURBIN-WATSON = 1.3675 VON NEUMANN RATIO = 1.4358 RHO = 0.24630 RESIDUAL SUM = 0.12479E-12 RESIDUAL VARIANCE = 1.0517 SUM OF ABSOLUTE ERRORS= 14.951 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9810 RUNS TEST: 11 RUNS, 10 POS, 0 ZERO, 11 NEG NORMAL STATISTIC = -0.2137 |_OLS I P PLAG KLAG / RSTAT REQUIRED MEMORY IS PAR= 5 CURRENT PAR= 2000 OLS ESTIMATION 21 OBSERVATIONS DEPENDENT VARIABLE= I ...NOTE..SAMPLE RANGE SET TO: 2, 22 R-SQUARE = 0.9313 R-SQUARE ADJUSTED = 0.9192 VARIANCE OF THE ESTIMATE-SIGMA**2 = 1.0190 STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.0094 SUM OF SQUARED ERRORS-SSE= 17.323 MEAN OF DEPENDENT VARIABLE = 1.2667 LOG OF THE LIKELIHOOD FUNCTION = -27.7764 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 17 DF P-VALUE CORR. COEFFICIENT AT MEANS P 0.47964 0.9711E-01 4.939 0.000 0.768 0.5699 6.3957 PLAG 0.33304 0.1009 3.302 0.004 0.625 0.3777 4.3057 KLAG -0.11179 0.2673E-01 -4.183 0.001-0.712 -0.3122 -17.6955 CONSTANT 10.126 5.466 1.853 0.081 0.410 0.0000 7.9940 DURBIN-WATSON = 1.8102 VON NEUMANN RATIO = 1.9007 RHO = 0.08425 RESIDUAL SUM = 0.13467E-12 RESIDUAL VARIANCE = 1.0190 SUM OF ABSOLUTE ERRORS= 13.931 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9313 RUNS TEST: 9 RUNS, 12 POS, 0 ZERO, 9 NEG NORMAL STATISTIC = -1.0460 |_OLS W1 E ELAG T / RSTAT REQUIRED MEMORY IS PAR= 5 CURRENT PAR= 2000 OLS ESTIMATION 21 OBSERVATIONS DEPENDENT VARIABLE= W1 ...NOTE..SAMPLE RANGE SET TO: 2, 22 R-SQUARE = 0.9874 R-SQUARE ADJUSTED = 0.9852 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.58851 STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.76715 SUM OF SQUARED ERRORS-SSE= 10.005 MEAN OF DEPENDENT VARIABLE = 36.362 LOG OF THE LIKELIHOOD FUNCTION = -22.0124 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 17 DF P-VALUE CORR. COEFFICIENT AT MEANS E 0.43948 0.3241E-01 13.56 0.000 0.957 0.7401 0.7259 ELAG 0.14609 0.3742E-01 3.904 0.001 0.688 0.2067 0.2330 T 0.13025 0.3191E-01 4.082 0.001 0.704 0.1282 0.0000 CONSTANT 1.4970 1.270 1.179 0.255 0.275 0.0000 0.0412 DURBIN-WATSON = 1.9584 VON NEUMANN RATIO = 2.0564 RHO = -0.08334 RESIDUAL SUM = 0.82490E-13 RESIDUAL VARIANCE = 0.58851 SUM OF ABSOLUTE ERRORS= 12.096 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9874 RUNS TEST: 10 RUNS, 11 POS, 0 ZERO, 10 NEG NORMAL STATISTIC = -0.6626 |_* The results are the same as the OLS results reported in Table 10.3. |_* While there is nothing in these results that is unreasonable, the OLS estimates |_* are not consistent because of the presence on the right-hand side of the equations |_* of the endogenous variables W, P, and E. |_* |_* (c) 2SLS regressions |_2SLS CN W P PLAG (PLAG KLAG ELAG T TX G W2) / RSTAT TWO STAGE LEAST SQUARES - DEPENDENT VARIABLE = CN 7 EXOGENOUS VARIABLES 3 POSSIBLE ENDOGENOUS VARIABLES 21 OBSERVATIONS R-SQUARE = 0.9767 R-SQUARE ADJUSTED = 0.9726 VARIANCE OF THE ESTIMATE-SIGMA**2 = 1.2897 STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.1357 SUM OF SQUARED ERRORS-SSE= 21.925 MEAN OF DEPENDENT VARIABLE = 53.995 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 17 DF P-VALUE CORR. COEFFICIENT AT MEANS W 0.81018 0.4474E-01 18.11 0.000 0.975 0.8922 0.6224 P 0.17302E-01 0.1312 0.1319 0.897 0.032 0.0106 0.0054 PLAG 0.21623 0.1192 1.814 0.087 0.403 0.1270 0.0656 CONSTANT 16.555 1.468 11.28 0.000 0.939 0.0000 0.3066 DURBIN-WATSON = 1.4851 VON NEUMANN RATIO = 1.5593 RHO = 0.20423 RESIDUAL SUM = -0.17764E-12 RESIDUAL VARIANCE = 1.2897 SUM OF ABSOLUTE ERRORS= 17.866 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9768 RUNS TEST: 9 RUNS, 9 POS, 0 ZERO, 12 NEG NORMAL STATISTIC = -1.0460 |_2SLS I P PLAG KLAG (PLAG KLAG ELAG T TX G W2) / RSTAT TWO STAGE LEAST SQUARES - DEPENDENT VARIABLE = I 7 EXOGENOUS VARIABLES 2 POSSIBLE ENDOGENOUS VARIABLES 21 OBSERVATIONS R-SQUARE = 0.8849 R-SQUARE ADJUSTED = 0.8646 VARIANCE OF THE ESTIMATE-SIGMA**2 = 1.7086 STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.3071 SUM OF SQUARED ERRORS-SSE= 29.047 MEAN OF DEPENDENT VARIABLE = 1.2667 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 17 DF P-VALUE CORR. COEFFICIENT AT MEANS P 0.15022 0.1925 0.7802 0.446 0.186 0.1785 2.0031 PLAG 0.61594 0.1809 3.404 0.003 0.637 0.6985 7.9633 KLAG -0.15779 0.4015E-01 -3.930 0.001-0.690 -0.4406 -24.9755 CONSTANT 20.278 8.383 2.419 0.027 0.506 0.0000 16.0091 DURBIN-WATSON = 2.0853 VON NEUMANN RATIO = 2.1896 RHO = -0.07526 RESIDUAL SUM = 0.24425E-14 RESIDUAL VARIANCE = 1.7086 SUM OF ABSOLUTE ERRORS= 18.613 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.8854 RUNS TEST: 12 RUNS, 12 POS, 0 ZERO, 9 NEG NORMAL STATISTIC = 0.3269 |_2SLS W1 E ELAG T (PLAG KLAG ELAG T TX G W2) / RSTAT TWO STAGE LEAST SQUARES - DEPENDENT VARIABLE = W1 7 EXOGENOUS VARIABLES 2 POSSIBLE ENDOGENOUS VARIABLES 21 OBSERVATIONS R-SQUARE = 0.9874 R-SQUARE ADJUSTED = 0.9852 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.58853 STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.76716 SUM OF SQUARED ERRORS-SSE= 10.005 MEAN OF DEPENDENT VARIABLE = 36.362 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 17 DF P-VALUE CORR. COEFFICIENT AT MEANS E 0.43886 0.3960E-01 11.08 0.000 0.937 0.7391 0.7248 ELAG 0.14667 0.4316E-01 3.398 0.003 0.636 0.2075 0.2339 T 0.13040 0.3239E-01 4.026 0.001 0.699 0.1283 0.0000 CONSTANT 1.5003 1.276 1.176 0.256 0.274 0.0000 0.0413 DURBIN-WATSON = 1.9634 VON NEUMANN RATIO = 2.0616 RHO = -0.08630 RESIDUAL SUM = 0.74607E-13 RESIDUAL VARIANCE = 0.58853 SUM OF ABSOLUTE ERRORS= 12.093 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9874 RUNS TEST: 10 RUNS, 11 POS, 0 ZERO, 10 NEG NORMAL STATISTIC = -0.6626 |_* The set of instruments used is that set of regressors that emerges when the |_* reduced form of the equation system is solved out analytically. |_* These results replicate the 2SLS results reported in Table 10.3. |_* Assuming no serial correlation, the 2SLS estimates are consistent, and asymptotically |_* efficient in the class of single-equation estimators. |_* The coefficients for P and PLAG are quite different from the OLS estimates |_* in both the Consumption and the Investment equations. There is not much difference |_* in the wage equation. |_* The choice between OLS and 2SLS estimates is not clear cut, since the consistency |_* of 2SLS must be offset against the additional variance resulting from the use of |_* (possibly weak) instrumental variables. |_* (d) OLS-AR(1) regressions |_* Using iterative Cochrane-Orcutt (the Hildreth-Lu estimates differ a bit from these, not not to a major extent): |_AUTO CN W P PLAG / RSTAT REQUIRED MEMORY IS PAR= 6 CURRENT PAR= 2000 DEPENDENT VARIABLE = CN ..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS LEAST SQUARES ESTIMATION 21 OBSERVATIONS BY COCHRANE-ORCUTT TYPE PROCEDURE WITH CONVERGENCE = 0.00100 ITERATION RHO LOG L.F. SSE 1 0.00000 -28.1086 17.879 2 0.24630 -27.4320 16.714 3 0.34253 -27.3109 16.473 4 0.39169 -27.2799 16.392 5 0.41949 -27.2715 16.358 6 0.43607 -27.2696 16.342 7 0.44625 -27.2696 16.333 8 0.45263 -27.2700 16.328 9 0.45667 -27.2704 16.325 10 0.45924 -27.2708 16.323 11 0.46089 -27.2711 16.322 12 0.46194 -27.2712 16.321 13 0.46262 -27.2713 16.321 LOG L.F. = -27.2713 AT RHO = 0.46262 ASYMPTOTIC ASYMPTOTIC ASYMPTOTIC ESTIMATE VARIANCE ST.ERROR T-RATIO RHO 0.46262 0.03743 0.19346 2.39129 R-SQUARE = 0.9827 R-SQUARE ADJUSTED = 0.9796 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.96005 STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.97982 SUM OF SQUARED ERRORS-SSE= 16.321 MEAN OF DEPENDENT VARIABLE = 53.995 LOG OF THE LIKELIHOOD FUNCTION = -27.2713 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 17 DF P-VALUE CORR. COEFFICIENT AT MEANS W 0.74108 0.5904E-01 12.55 0.000 0.950 0.8161 0.5693 P 0.24430 0.1021 2.393 0.029 0.502 0.1503 0.0764 PLAG 0.68030E-01 0.9743E-01 0.6982 0.494 0.167 0.0399 0.0206 CONSTANT 17.931 1.741 10.30 0.000 0.928 0.0000 0.3321 DURBIN-WATSON = 1.6613 VON NEUMANN RATIO = 1.7444 RHO = 0.05936 RESIDUAL SUM = 0.38062 RESIDUAL VARIANCE = 0.96858 SUM OF ABSOLUTE ERRORS= 15.254 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9826 RUNS TEST: 11 RUNS, 12 POS, 0 ZERO, 9 NEG NORMAL STATISTIC = -0.1307 DURBIN H STATISTIC (ASYMPTOTIC NORMAL) = 0.58802 MODIFIED FOR AUTO ORDER=1 |_AUTO I P PLAG KLAG / RSTAT REQUIRED MEMORY IS PAR= 6 CURRENT PAR= 2000 DEPENDENT VARIABLE = I ..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS LEAST SQUARES ESTIMATION 21 OBSERVATIONS BY COCHRANE-ORCUTT TYPE PROCEDURE WITH CONVERGENCE = 0.00100 ITERATION RHO LOG L.F. SSE 1 0.00000 -27.7764 17.323 2 0.08425 -27.6980 17.188 3 0.09488 -27.6977 17.186 4 0.09629 -27.6978 17.186 5 0.09648 -27.6978 17.186 LOG L.F. = -27.6978 AT RHO = 0.09648 ASYMPTOTIC ASYMPTOTIC ASYMPTOTIC ESTIMATE VARIANCE ST.ERROR T-RATIO RHO 0.09648 0.04718 0.21720 0.44421 R-SQUARE = 0.9319 R-SQUARE ADJUSTED = 0.9199 VARIANCE OF THE ESTIMATE-SIGMA**2 = 1.0109 STANDARD ERROR OF THE ESTIMATE-SIGMA = 1.0055 SUM OF SQUARED ERRORS-SSE= 17.186 MEAN OF DEPENDENT VARIABLE = 1.2667 LOG OF THE LIKELIHOOD FUNCTION = -27.6978 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 17 DF P-VALUE CORR. COEFFICIENT AT MEANS P 0.49214 0.9716E-01 5.065 0.000 0.776 0.5847 6.5624 PLAG 0.32240 0.1016 3.172 0.006 0.610 0.3656 4.1682 KLAG -0.11130 0.2838E-01 -3.922 0.001-0.689 -0.3108 -17.6168 CONSTANT 9.9853 5.785 1.726 0.102 0.386 0.0000 7.8832 DURBIN-WATSON = 1.9615 VON NEUMANN RATIO = 2.0596 RHO = 0.00834 RESIDUAL SUM = 0.35926E-02 RESIDUAL VARIANCE = 1.0109 SUM OF ABSOLUTE ERRORS= 13.772 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9319 RUNS TEST: 9 RUNS, 11 POS, 0 ZERO, 10 NEG NORMAL STATISTIC = -1.1114 DURBIN H STATISTIC (ASYMPTOTIC NORMAL) = 0.39635 MODIFIED FOR AUTO ORDER=1 |_AUTO W1 E ELAG T / RSTAT REQUIRED MEMORY IS PAR= 6 CURRENT PAR= 2000 DEPENDENT VARIABLE = W1 ..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS LEAST SQUARES ESTIMATION 21 OBSERVATIONS BY COCHRANE-ORCUTT TYPE PROCEDURE WITH CONVERGENCE = 0.00100 ITERATION RHO LOG L.F. SSE 1 0.00000 -22.0124 10.005 2 -0.08334 -21.9017 9.8966 3 -0.11899 -21.8740 9.8671 4 -0.13228 -21.8669 9.8589 5 -0.13694 -21.8649 9.8563 6 -0.13854 -21.8642 9.8555 7 -0.13908 -21.8640 9.8552 LOG L.F. = -21.8640 AT RHO = -0.13908 ASYMPTOTIC ASYMPTOTIC ASYMPTOTIC ESTIMATE VARIANCE ST.ERROR T-RATIO RHO -0.13908 0.04670 0.21610 -0.64362 R-SQUARE = 0.9876 R-SQUARE ADJUSTED = 0.9854 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.57972 STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.76139 SUM OF SQUARED ERRORS-SSE= 9.8552 MEAN OF DEPENDENT VARIABLE = 36.362 LOG OF THE LIKELIHOOD FUNCTION = -21.8640 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 17 DF P-VALUE CORR. COEFFICIENT AT MEANS E 0.42932 0.3021E-01 14.21 0.000 0.960 0.7230 0.7091 ELAG 0.15434 0.3465E-01 4.455 0.000 0.734 0.2183 0.2461 T 0.13001 0.2804E-01 4.637 0.000 0.747 0.1280 0.0000 CONSTANT 1.6321 1.139 1.434 0.170 0.328 0.0000 0.0449 DURBIN-WATSON = 1.8401 VON NEUMANN RATIO = 1.9321 RHO = -0.02927 RESIDUAL SUM = -0.18622 RESIDUAL VARIANCE = 0.58176 SUM OF ABSOLUTE ERRORS= 11.477 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9876 RUNS TEST: 10 RUNS, 12 POS, 0 ZERO, 9 NEG NORMAL STATISTIC = -0.5883 DURBIN H STATISTIC (ASYMPTOTIC NORMAL) = -0.96454 MODIFIED FOR AUTO ORDER=1 |_* The estimate of rho is statistically different from zero only in the consumption equation. |_* The estimate is considerably lower than that reported in Table 10.3, and the parameter |_* estimates are quite different from those reported in the table as well. The other two |_* equations are in closer agreement with Table 10.3. |_* In general, allowing for autocorrelation does not have a substantial effect on the OLS |_* estimates. |_* These estimates are not consistent because of the presence of endogenous variables on |_* the right-hand side of the equations; the autocorrelation equation does not correct this. |_* As well, the estimate of rho is inconsistent because it is based on inconsistent |_* estimates of the equation parameters. |_* However, if the inconsistency is not too serious, the correction for autocorrelation can |_* result in better (reduced variance) estimates than OLS despite the inconsistency. |_* |_* (e) 2SLS-AR(1) regressions |_GENR PLAG2=LAG(PLAG) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_GENR KLAG2=LAG(KLAG) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_GENR ELAG2=LAG(ELAG) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_GENR TXLAG=LAG(TX) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_GENR GLAG=LAG(G) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_GENR W2LAG=LAG(W2) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_* These instrumental variables are now available because accounting for an AR(1) process |_* in the disturbances by partial differencing of the variables in the equations, implicitly |_* adds lagged values of these variables into the equations, and the lagged exogenous variables |_* in the reduced form can serve as instrumental variables. * Note that values of the double-lagged instrumental variables PLAG2, KLAG2 AND ELAG2 are not available * for 1921, which must therefore be dropped from the sample. |_SAMPLE 1922. 1941. |_NL 1 PLAG KLAG ELAG T TX G W2 PLAG2 KLAG2 ELAG2 TXLAG GLAG W2LAG /NCOEF=4 AUTO RSTAT ...NOTE..SAMPLE RANGE SET TO: 3, 22 |_EQ CN=A1+A2*W+A3*P+A4*PLAG |_END 17 VARIABLES IN 1 EQUATIONS WITH 4 COEFFICIENTS WITH 1 AUTOREGRESSIVE COEFFICIENTS NONLINEAR TWO-STAGE LEAST SQUARES: USING 14 INSTRUMENTAL EXOGENOUS VARIABLES ..ALGORITHM USES NUMERIC DERIVATIVES 20 OBSERVATIONS REQUIRED MEMORY IS PAR= 26 CURRENT PAR= 2000 COEFFICIENT STARTING VALUES A1 1.0000 A2 1.0000 A3 1.0000 A4 1.0000 RHO 0.0000 100 MAXIMUM ITERATIONS, CONVERGENCE = 0.100000E-04 INITIAL STATISTICS : TIME = 0.0470 SEC. ITER. NO. 0 FUNCT. EVALUATIONS 6 FUNCTION VALUE= 10542.12 COEFFICIENTS 1.000000 1.000000 1.000000 1.000000 0.000000 GRADIENT 840.7703 37336.64 15460.05 15207.79 -19645.84 INTERMEDIATE STATISTICS : TIME = 0.0470 SEC. ITER. NO. 15 FUNCT. EVALUATIONS 129 FUNCTION VALUE= 8.549030 COEFFICIENTS 20.53541 0.6695750 0.3320647 0.2124869E-01 0.4979840 GRADIENT -1.465398 -63.12443 -25.36566 -24.97530 -0.9625636E-01 TIME = 0.0470 SEC. ITER. NO. 30 FUNCT. EVALUATIONS 290 FUNCTION VALUE= 8.420217 COEFFICIENTS 21.06701 0.6555565 0.3447529 0.2410034E-01 0.5333259 GRADIENT 0.4709388E-06 0.1998997E-04 0.8973764E-05 0.7659677E-05 -0.1071498E-05 FINAL STATISTICS : TIME = 0.062 SEC. ITER. NO. 32 FUNCT. EVALUATIONS 305 FUNCTION VALUE= 8.420217 COEFFICIENTS 21.06701 0.6555565 0.3447528 0.2410037E-01 0.5333260 GRADIENT -0.1423013E-05 -0.5715487E-04 -0.2570523E-04 -0.2190490E-04 0.3319389E-05 ...NOTE..NUMERIC OPTION GIVES GARBAGE FOR COVARIANCES MAXIMUM LIKELIHOOD ESTIMATE OF SIGMA-SQUARED = 0.59833 SUM OF SQUARED ERRORS = 11.967 GTRANSPOSE*INVERSE(H)*G STATISTIC - = 0.0000 COEFFICIENT ST. ERROR T-RATIO A1 21.067 0.0000 0.0000 A2 0.65556 0.0000 0.0000 A3 0.34475 0.0000 0.0000 A4 0.24100E-01 0.0000 0.0000 RHO 0.53333 0.0000 0.0000 DURBIN-WATSON = 2.1297 VON NEUMANN RATIO = 2.2418 RHO = -0.15240 RESIDUAL SUM = -0.64716 RESIDUAL VARIANCE = 0.59833 SUM OF ABSOLUTE ERRORS= 13.079 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9849 RUNS TEST: 9 RUNS, 8 POS, 1 ZERO, 11 NEG NORMAL STATISTIC = -0.6125 |_END | |_* The Investment equation fails to converge to reasonable values at the default starting |_* values -- in particular, rho is almost unity -- so use the 2SLS estimates (including |_* the estimate of rho) as starting values. |_NL 1 PLAG KLAG ELAG T TX G W2 PLAG2 KLAG2 ELAG2 TXLAG GLAG W2LAG /NCOEF=4 AUTO RSTAT ...NOTE..SAMPLE RANGE SET TO: 3, 22 |_EQ I=B1+B2*P+B3*PLAG+B4*KLAG |_COEF B2 .15 B3 .6 B4 -.15 RHO 0.08 16 VARIABLES IN 1 EQUATIONS WITH 4 COEFFICIENTS WITH 1 AUTOREGRESSIVE COEFFICIENTS NONLINEAR TWO-STAGE LEAST SQUARES: USING 14 INSTRUMENTAL EXOGENOUS VARIABLES ..ALGORITHM USES NUMERIC DERIVATIVES 20 OBSERVATIONS REQUIRED MEMORY IS PAR= 26 CURRENT PAR= 2000 |_END COEFFICIENT STARTING VALUES B1 1.0000 B2 0.15000 B3 0.60000 B4 -0.15000 RHO 0.80000E-01 100 MAXIMUM ITERATIONS, CONVERGENCE = 0.100000E-04 INITIAL STATISTICS : TIME = 0.0160 SEC. ITER. NO. 0 FUNCT. EVALUATIONS 6 FUNCTION VALUE= 5123.445 COEFFICIENTS 1.000000 0.1500000 0.6000000 -0.1500000 0.8000000E-01 GRADIENT -568.4961 -9760.479 -9609.263 -115254.9 -11146.79 INTERMEDIATE STATISTICS : TIME = 0.0160 SEC. ITER. NO. 15 FUNCT. EVALUATIONS 134 FUNCTION VALUE= 12.60610 COEFFICIENTS 13.37823 0.3573668 0.4461865 -0.1271280 -0.1647232 GRADIENT -1.111527 -18.52088 -17.93693 -222.4996 -6.652961 TIME = 0.0160 SEC. ITER. NO. 30 FUNCT. EVALUATIONS 268 FUNCTION VALUE= 12.60059 COEFFICIENTS 14.19986 0.3512996 0.4484941 -0.1307165 -0.1564819 GRADIENT 0.4319736 6.759828 6.332737 82.48729 -6.304320 TIME = 0.0160 SEC. ITER. NO. 45 FUNCT. EVALUATIONS 383 FUNCTION VALUE= 11.96578 COEFFICIENTS 12.95575 0.3743006 0.4289742 -0.1250265 0.1850577E-01 GRADIENT -0.2740309 -4.894444 -4.440350 -64.52644 -0.1247999 FINAL STATISTICS : TIME = 0.016 SEC. ITER. NO. 52 FUNCT. EVALUATIONS 476 FUNCTION VALUE= 11.93135 COEFFICIENTS 11.53649 0.3902856 0.4160724 -0.1182649 0.2554447E-01 GRADIENT -0.1534861E-05 -0.1012512E-04 -0.1455310E-04 -0.3210997E-03 0.1806333E-06 ...NOTE..NUMERIC OPTION GIVES GARBAGE FOR COVARIANCES MAXIMUM LIKELIHOOD ESTIMATE OF SIGMA-SQUARED = 0.91017 SUM OF SQUARED ERRORS = 18.203 GTRANSPOSE*INVERSE(H)*G STATISTIC - = 0.0000 COEFFICIENT ST. ERROR T-RATIO B1 11.536 0.0000 0.0000 B2 0.39029 0.0000 0.0000 B3 0.41607 0.0000 0.0000 B4 -0.11826 0.0000 0.0000 RHO 0.25544E-01 0.0000 0.0000 DURBIN-WATSON = 2.0316 VON NEUMANN RATIO = 2.1385 RHO = -0.02007 RESIDUAL SUM = 0.75658 RESIDUAL VARIANCE = 0.91017 SUM OF ABSOLUTE ERRORS= 13.564 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9275 RUNS TEST: 10 RUNS, 9 POS, 1 ZERO, 10 NEG NORMAL STATISTIC = -0.2243 |_END | |_NL 1 PLAG KLAG ELAG T TX G W2 PLAG2 KLAG2 ELAG2 TXLAG GLAG W2LAG /NCOEF=4 AUTO RSTAT ...NOTE..SAMPLE RANGE SET TO: 3, 22 |_EQ W1=C1+C2*E+C3*ELAG+C4*T |_END 16 VARIABLES IN 1 EQUATIONS WITH 4 COEFFICIENTS WITH 1 AUTOREGRESSIVE COEFFICIENTS NONLINEAR TWO-STAGE LEAST SQUARES: USING 14 INSTRUMENTAL EXOGENOUS VARIABLES ..ALGORITHM USES NUMERIC DERIVATIVES 20 OBSERVATIONS REQUIRED MEMORY IS PAR= 26 CURRENT PAR= 2000 COEFFICIENT STARTING VALUES C1 1.0000 C2 1.0000 C3 1.0000 C4 1.0000 RHO 0.0000 100 MAXIMUM ITERATIONS, CONVERGENCE = 0.100000E-04 INITIAL STATISTICS : TIME = 0.0000 SEC. ITER. NO. 0 FUNCT. EVALUATIONS 6 FUNCTION VALUE= 141204.8 COEFFICIENTS 1.000000 1.000000 1.000000 1.000000 0.000000 GRADIENT 3204.654 201776.6 194664.0 5437.029 -270271.7 INTERMEDIATE STATISTICS : TIME = 0.0000 SEC. ITER. NO. 15 FUNCT. EVALUATIONS 175 FUNCTION VALUE= 5.607956 COEFFICIENTS 1.932605 0.4267336 0.1526677 0.1188050 -0.1484861 GRADIENT 0.5017493E-01 5.809122 7.420015 -1.307716 -0.3703687 FINAL STATISTICS : TIME = 0.000 SEC. ITER. NO. 23 FUNCT. EVALUATIONS 260 FUNCTION VALUE= 5.593079 COEFFICIENTS 1.988447 0.4305499 0.1477507 0.1200780 -0.1020291 GRADIENT -0.3712763E-06 -0.1666884E-04 -0.1480722E-04 0.6772360E-06 0.4534151E-06 ...NOTE..NUMERIC OPTION GIVES GARBAGE FOR COVARIANCES MAXIMUM LIKELIHOOD ESTIMATE OF SIGMA-SQUARED = 0.38844 SUM OF SQUARED ERRORS = 7.7688 GTRANSPOSE*INVERSE(H)*G STATISTIC - = 0.0000 COEFFICIENT ST. ERROR T-RATIO C1 1.9884 0.0000 0.0000 C2 0.43055 0.0000 0.0000 C3 0.14775 0.0000 0.0000 C4 0.12008 0.0000 0.0000 RHO -0.10203 0.0000 0.0000 DURBIN-WATSON = 2.0965 VON NEUMANN RATIO = 2.2068 RHO = -0.09359 RESIDUAL SUM = 0.34737 RESIDUAL VARIANCE = 0.38844 SUM OF ABSOLUTE ERRORS= 9.9847 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.9884 RUNS TEST: 11 RUNS, 8 POS, 1 ZERO, 11 NEG NORMAL STATISTIC = 0.3573 |_END | |_* As noted in the Assignment instructions, the numerical algorithm used to compute |_* derivatives does not accurately estimate the covariance matrix of the parameter |_* estimates. |_* |_* The differences between these estimates and those reported in Table 10.3 are greater |_* here than was the case with the other estimators -- this is often the case with |_* nonlinear estimators, which may use different algoithms to converge to a solution. |_* But there are no big discrepancies. |_* When the disturbances are autocorrelated, and lagged endogenous variables appear in |_* an equation, 2SLS estimates of the parameters of this equation are not consistent. |_* 2SLS-AR(1) corrects for this, and is consistent. |_* The results are not very different from traditional 2SLS estimates. Autocorrelation |_* appears to be an issue only in the consumption equation, and even here there are no |_* major differences. |_* | Updated November 16, 2007.