Welcome to SHAZAM - Version 9.0 - OCT 2003 SYSTEM=WIN-98 PAR= 2000 CURRENT WORKING DIRECTORY IS: C:\ESL\USER |_* Exercise 9.18 |_* |_READ (DATA9-4) YEAR PROFITS SALES UNIT 88 IS NOW ASSIGNED TO: DATA9-4 ...SAMPLE RANGE IS NOW SET TO: 1 21 |_* Estimate the relationship, saving the residuals. |_* Test for AC with a DW test. |_OLS PROFITS SALES / RESID=U DWPVALUE REQUIRED MEMORY IS PAR= 6 CURRENT PAR= 2000 OLS ESTIMATION 21 OBSERVATIONS DEPENDENT VARIABLE= PROFITS ...NOTE..SAMPLE RANGE SET TO: 1, 21 DURBIN-WATSON STATISTIC = 1.07998 DURBIN-WATSON POSITIVE AUTOCORRELATION TEST P-VALUE = 0.005445 NEGATIVE AUTOCORRELATION TEST P-VALUE = 0.994554 R-SQUARE = 0.2463 R-SQUARE ADJUSTED = 0.2067 VARIANCE OF THE ESTIMATE-SIGMA**2 = 976.65 STANDARD ERROR OF THE ESTIMATE-SIGMA = 31.251 SUM OF SQUARED ERRORS-SSE= 18556. MEAN OF DEPENDENT VARIABLE = 91.462 LOG OF THE LIKELIHOOD FUNCTION = -101.030 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 19 DF P-VALUE CORR. COEFFICIENT AT MEANS SALES 0.26544E-01 0.1065E-01 2.492 0.022 0.496 0.4963 0.6281 CONSTANT 34.014 24.04 1.415 0.173 0.309 0.0000 0.3719 |_* Plot the residuals |_PLOT U REQUIRED MEMORY IS PAR= 1 CURRENT PAR= 2000 21 OBSERVATIONS *=U M=MULTIPLE POINT 100.00 | 89.474 | 78.947 | 68.421 | 57.895 | 47.368 | * * 36.842 | 26.316 | * 15.789 | * * 5.2632 | * * * * -5.2632 |* ** * * -15.789 | * * * -26.316 | * -36.842 | * -47.368 | * -57.895 | -68.421 | -78.947 | -89.474 | * -100.00 | ________________________________________ 0.000 6.000 12.000 18.000 24.000 TIME |_* Test for AC using the BG test. |_GENR ULAG=LAG(U) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_OLS U ULAG SALES REQUIRED MEMORY IS PAR= 2 CURRENT PAR= 2000 OLS ESTIMATION 21 OBSERVATIONS DEPENDENT VARIABLE= U ...NOTE..SAMPLE RANGE SET TO: 1, 21 R-SQUARE = 0.1776 R-SQUARE ADJUSTED = 0.0862 VARIANCE OF THE ESTIMATE-SIGMA**2 = 847.81 STANDARD ERROR OF THE ESTIMATE-SIGMA = 29.117 SUM OF SQUARED ERRORS-SSE= 15261. MEAN OF DEPENDENT VARIABLE = 0.0000 LOG OF THE LIKELIHOOD FUNCTION = -98.9771 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 18 DF P-VALUE CORR. COEFFICIENT AT MEANS ULAG 0.47374 0.2403 1.972 0.064 0.421 0.4296 0.0000 SALES 0.38722E-02 0.1012E-01 0.3827 0.706 0.090 0.0834 0.0000 CONSTANT -7.1148 22.69 -0.3136 0.757-0.074 0.0000 0.0000 |_GEN1 LM=$R2*$N ..NOTE..CURRENT VALUE OF $R2 = 0.17761 ..NOTE..CURRENT VALUE OF $N = 21.000 |_DISTRIB LM /TYPE=CHI DF=1 CHI-SQUARE PARAMETERS- DF= 1.0000 MEAN= 1.0000 VARIANCE= 2.0000 MODE= 0.0000 DATA PDF CDF 1-CDF LM ROW 1 3.7297 0.32002E-01 0.94655 0.53452E-01 |_* |_* The DW test is highly significant (p-value=0.005). |_* The BG test is less so (p-value=0.05), but is strictly |_* speaking a large-sample test, and N=21 here. |_* |_* Reestimate the model using the CORC procedure, saving the residuals. |_AUTO PROFITS SALES /RESID=E RSTAT REQUIRED MEMORY IS PAR= 4 CURRENT PAR= 2000 DEPENDENT VARIABLE = PROFITS ..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 -101.030 18556. 2 0.45145 -98.9229 15018. 3 0.50939 -98.9009 14935. 4 0.52870 -98.9067 14924. 5 0.53630 -98.9107 14922. 6 0.53947 -98.9126 14921. 7 0.54082 -98.9135 14921. 8 0.54140 -98.9139 14921. LOG L.F. = -98.9139 AT RHO = 0.54140 ASYMPTOTIC ASYMPTOTIC ASYMPTOTIC ESTIMATE VARIANCE ST.ERROR T-RATIO RHO 0.54140 0.03366 0.18347 2.95087 R-SQUARE = 0.3940 R-SQUARE ADJUSTED = 0.3621 VARIANCE OF THE ESTIMATE-SIGMA**2 = 785.30 STANDARD ERROR OF THE ESTIMATE-SIGMA = 28.023 SUM OF SQUARED ERRORS-SSE= 14921. MEAN OF DEPENDENT VARIABLE = 91.462 LOG OF THE LIKELIHOOD FUNCTION = -98.9139 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 19 DF P-VALUE CORR. COEFFICIENT AT MEANS SALES 0.41307E-01 0.1716E-01 2.408 0.026 0.483 0.7723 0.9774 CONSTANT 4.7307 39.22 0.1206 0.905 0.028 0.0000 0.0517 DURBIN-WATSON = 1.1273 VON NEUMANN RATIO = 1.1836 RHO = 0.40987 RESIDUAL SUM = -5.5001 RESIDUAL VARIANCE = 786.89 SUM OF ABSOLUTE ERRORS= 388.51 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.3981 RUNS TEST: 11 RUNS, 12 POS, 0 ZERO, 9 NEG NORMAL STATISTIC = -0.1307 DURBIN H STATISTIC (ASYMPTOTIC NORMAL) = 3.4693 MODIFIED FOR AUTO ORDER=1 |_DISTRIB $DURH NORMAL DISTRIBUTION - MEAN= 0.0000 VARIANCE= 1.0000 DATA Z PDF CDF 1-CDF $DURH ROW 1 3.4693 3.4693 0.97138E-03 0.99974 0.26095E-03 |_* The DW test is biased here, but the Durbin h-test is highly significant, |_* (p-value=0.0003)indicating the continued presence of autocorrelation. |_* Do a BG test here as well. |_GENR ELAG=LAG(E) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_OLS E ELAG SALES REQUIRED MEMORY IS PAR= 3 CURRENT PAR= 2000 OLS ESTIMATION 21 OBSERVATIONS DEPENDENT VARIABLE= E ...NOTE..SAMPLE RANGE SET TO: 1, 21 R-SQUARE = 0.1273 R-SQUARE ADJUSTED = 0.0303 VARIANCE OF THE ESTIMATE-SIGMA**2 = 724.79 STANDARD ERROR OF THE ESTIMATE-SIGMA = 26.922 SUM OF SQUARED ERRORS-SSE= 13046. MEAN OF DEPENDENT VARIABLE = -0.26191 LOG OF THE LIKELIHOOD FUNCTION = -97.3309 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 18 DF P-VALUE CORR. COEFFICIENT AT MEANS ELAG 0.42715 0.2712 1.575 0.133 0.348 0.3649 5.2298 SALES 0.12334E-02 0.9657E-02 0.1277 0.900 0.030 0.0296 -10.1918 CONSTANT -1.5615 21.46 -0.7275E-01 0.943-0.017 0.0000 5.9620 |_GEN1 LM=$R2*$N ..NOTE..CURRENT VALUE OF $R2 = 0.12731 ..NOTE..CURRENT VALUE OF $N = 21.000 |_DISTRIB LM /TYPE=CHI DF=1 CHI-SQUARE PARAMETERS- DF= 1.0000 MEAN= 1.0000 VARIANCE= 2.0000 MODE= 0.0000 DATA PDF CDF 1-CDF LM ROW 1 2.6736 0.64092E-01 0.89797 0.10203 |_* Here as well the BG test is less decisive (p-value=0.10). |_* |_* Estimate using a HILU procedure. |_AUTO PROFITS SALES /RESID=E GS RSTAT REQUIRED MEMORY IS PAR= 4 CURRENT PAR= 2000 DEPENDENT VARIABLE = PROFITS ..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS LEAST SQUARES ESTIMATION 21 OBSERVATIONS BY GRID SEARCH TO ACCURACY OF .01 ITERATION RHO LOG L.F. SSE 1 -0.90000 -110.584 42590. 2 -0.80000 -109.278 38770. 3 -0.70000 -108.101 35240. 4 -0.60000 -106.974 31997. 5 -0.50000 -105.878 29043. 6 -0.40000 -104.810 26376. 7 -0.30000 -103.776 23995. 8 -0.20000 -102.790 21900. 9 -0.10000 -101.868 20087. 10 0.00000 -101.030 18556. 11 0.10000 -100.301 17303. 12 0.20000 -99.7048 16324. 13 0.30000 -99.2632 15612. 14 0.40000 -98.9926 15157. 15 0.50000 -98.9005 14944. 16 0.60000 -98.9823 14947. 17 0.70000 -99.2217 15127. 18 0.80000 -99.5978 15421. 19 0.90000 -100.141 15753. ITERATION RHO LOG L.F. SSE 20 0.41000 -98.9753 15125. 21 0.42000 -98.9599 15096. 22 0.43000 -98.9462 15068. 23 0.44000 -98.9343 15044. 24 0.45000 -98.9242 15021. 25 0.46000 -98.9159 15001. 26 0.47000 -98.9094 14983. 27 0.48000 -98.9046 14968. 28 0.49000 -98.9017 14955. 29 0.50000 -98.9005 14944. 30 0.51000 -98.9010 14935. 31 0.52000 -98.9033 14928. 32 0.53000 -98.9073 14923. 33 0.54000 -98.9130 14921. 34 0.55000 -98.9204 14920. 35 0.56000 -98.9295 14922. 36 0.57000 -98.9403 14925. 37 0.58000 -98.9527 14931. 38 0.59000 -98.9667 14938. 39 0.55000 -98.9204 14920. LOG L.F. = -98.9204 AT RHO = 0.55000 ASYMPTOTIC ASYMPTOTIC ASYMPTOTIC ESTIMATE VARIANCE ST.ERROR T-RATIO RHO 0.55000 0.03321 0.18225 3.01787 R-SQUARE = 0.3940 R-SQUARE ADJUSTED = 0.3621 VARIANCE OF THE ESTIMATE-SIGMA**2 = 785.28 STANDARD ERROR OF THE ESTIMATE-SIGMA = 28.023 SUM OF SQUARED ERRORS-SSE= 14920. MEAN OF DEPENDENT VARIABLE = 91.462 LOG OF THE LIKELIHOOD FUNCTION = -98.9204 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 19 DF P-VALUE CORR. COEFFICIENT AT MEANS SALES 0.41798E-01 0.1737E-01 2.406 0.026 0.483 0.7815 0.9891 CONSTANT 3.7526 39.72 0.9447E-01 0.926 0.022 0.0000 0.0410 DURBIN-WATSON = 1.1269 VON NEUMANN RATIO = 1.1833 RHO = 0.40961 RESIDUAL SUM = -5.8390 RESIDUAL VARIANCE = 787.08 SUM OF ABSOLUTE ERRORS= 390.14 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.3987 RUNS TEST: 11 RUNS, 12 POS, 0 ZERO, 9 NEG NORMAL STATISTIC = -0.1307 DURBIN H STATISTIC (ASYMPTOTIC NORMAL) = 3.4128 MODIFIED FOR AUTO ORDER=1 |_DISTRIB $DURH NORMAL DISTRIBUTION - MEAN= 0.0000 VARIANCE= 1.0000 DATA Z PDF CDF 1-CDF $DURH ROW 1 3.4128 3.4128 0.11795E-02 0.99968 0.32145E-03 |_GENR ELAG=LAG(E) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_OLS E ELAG SALES REQUIRED MEMORY IS PAR= 3 CURRENT PAR= 2000 OLS ESTIMATION 21 OBSERVATIONS DEPENDENT VARIABLE= E ...NOTE..SAMPLE RANGE SET TO: 1, 21 R-SQUARE = 0.1270 R-SQUARE ADJUSTED = 0.0300 VARIANCE OF THE ESTIMATE-SIGMA**2 = 725.22 STANDARD ERROR OF THE ESTIMATE-SIGMA = 26.930 SUM OF SQUARED ERRORS-SSE= 13054. MEAN OF DEPENDENT VARIABLE = -0.27805 LOG OF THE LIKELIHOOD FUNCTION = -97.3371 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 18 DF P-VALUE CORR. COEFFICIENT AT MEANS ELAG 0.42643 0.2716 1.570 0.134 0.347 0.3642 4.9460 SALES 0.11751E-02 0.9667E-02 0.1216 0.905 0.029 0.0282 -9.1465 CONSTANT -1.4460 21.48 -0.6731E-01 0.947-0.016 0.0000 5.2005 |_GEN1 LM=$R2*$N ..NOTE..CURRENT VALUE OF $R2 = 0.12699 ..NOTE..CURRENT VALUE OF $N = 21.000 |_DISTRIB LM /TYPE=CHI DF=1 CHI-SQUARE PARAMETERS- DF= 1.0000 MEAN= 1.0000 VARIANCE= 2.0000 MODE= 0.0000 DATA PDF CDF 1-CDF LM ROW 1 2.6668 0.64391E-01 0.89754 0.10246 |_* Results are similar as for the CORC results. |_* |_* While the results are unbiased and consistent (there is no lagged dependent |_* variable in the model), the continued presence of autocorrelation |_* suggests that the estimates are not fully efficient. |_* |_* Test an AR(2) process on the original residuals using a BG test. |_GENR ULAG2=LAG(ULAG) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_OLS U ULAG ULAG2 SALES REQUIRED MEMORY IS PAR= 3 CURRENT PAR= 2000 OLS ESTIMATION 21 OBSERVATIONS DEPENDENT VARIABLE= U ...NOTE..SAMPLE RANGE SET TO: 1, 21 R-SQUARE = 0.6472 R-SQUARE ADJUSTED = 0.5849 VARIANCE OF THE ESTIMATE-SIGMA**2 = 385.12 STANDARD ERROR OF THE ESTIMATE-SIGMA = 19.624 SUM OF SQUARED ERRORS-SSE= 6547.0 MEAN OF DEPENDENT VARIABLE = 0.0000 LOG OF THE LIKELIHOOD FUNCTION = -90.0912 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 17 DF P-VALUE CORR. COEFFICIENT AT MEANS ULAG 0.99718 0.1958 5.093 0.000 0.777 0.9043 0.0000 ULAG2 -0.95392 0.2005 -4.757 0.000-0.756 -0.8423 0.0000 SALES 0.10907E-02 0.6844E-02 0.1594 0.875 0.039 0.0235 0.0000 CONSTANT -0.87552 15.35 -0.5705E-01 0.955-0.014 0.0000 0.0000 |_GEN1 LM=$R2*$N ..NOTE..CURRENT VALUE OF $R2 = 0.64718 ..NOTE..CURRENT VALUE OF $N = 21.000 |_DISTRIB LM /TYPE=CHI DF=2 CHI-SQUARE PARAMETERS- DF= 2.0000 MEAN= 2.0000 VARIANCE= 4.0000 MODE= 0.0000 DATA PDF CDF 1-CDF LM ROW 1 13.591 0.55944E-03 0.99888 0.11189E-02 |_* Both lags are significant, and the BG test statistic is significant |_* with p-value=0.002. This is consistent with an AR(2) process. |_* Estimate an AR(2) model. |_AUTO PROFITS SALES /ORDER=2 RSTAT RESID=E REQUIRED MEMORY IS PAR= 4 CURRENT PAR= 2000 DEPENDENT VARIABLE = PROFITS ..NOTE..R-SQUARE,ANOVA,RESIDUALS DONE ON ORIGINAL VARS LEAST SQUARES SECOND-ORDER AUTOCORRELATION BY COCHRANE-ORCUTT TYPE PROCEDURE WITH CONVERGENCE =0.001000 21 OBSERVATIONS ITERATION RHO1 RHO2 SSE SSE/N LOG.L.F. 1 0.00000 0.00000 18556.395 883.63785 -101.03021 2 0.98454 -0.95383 6406.5087 305.07184 -92.415637 3 0.98744 -0.95590 6405.5985 305.02850 -92.459405 4 0.98760 -0.95586 6405.6102 305.02906 -92.458685 ASYMPTOTIC ASYMPTOTIC ASYMPTOTIC ESTIMATE VARIANCE ST.ERROR T-RATIO AUTOCORRELATION RHO1 0.98760 0.00411 0.06412 15.40287 0.50495 RHO2 -0.95586 0.00411 0.06412 -14.90780 -0.45717 COVARIANCE -0.00208 COMPLEX ROOTS - AUTOREGRESSIVE PROCESS DISPLAYS PSEUDO PERIODIC BEHAVIOUR WITH DAMPED SINE WAVE R-SQUARE = 0.7398 R-SQUARE ADJUSTED = 0.7261 VARIANCE OF THE ESTIMATE-SIGMA**2 = 337.14 STANDARD ERROR OF THE ESTIMATE-SIGMA = 18.361 SUM OF SQUARED ERRORS-SSE= 6405.6 MEAN OF DEPENDENT VARIABLE = 91.462 LOG OF THE LIKELIHOOD FUNCTION = -92.4587 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 19 DF P-VALUE CORR. COEFFICIENT AT MEANS SALES 0.25757E-01 0.7559E-02 3.408 0.003 0.616 0.4816 0.6095 CONSTANT 38.058 17.02 2.236 0.038 0.456 0.0000 0.4161 DURBIN-WATSON = 2.3952 VON NEUMANN RATIO = 2.5150 RHO = -0.20105 RESIDUAL SUM = -18.682 RESIDUAL VARIANCE = 347.49 SUM OF ABSOLUTE ERRORS= 300.90 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.7329 RUNS TEST: 14 RUNS, 11 POS, 0 ZERO, 10 NEG NORMAL STATISTIC = 1.1328 |_* Test for the presence of further autocorrelation using a BG test. |_GENR ELAG=LAG(E) ..NOTE.LAG VALUE IN UNDEFINED OBSERVATIONS SET TO ZERO |_OLS E ELAG SALES / DLAG REQUIRED MEMORY IS PAR= 3 CURRENT PAR= 2000 OLS ESTIMATION 21 OBSERVATIONS DEPENDENT VARIABLE= E ...NOTE..SAMPLE RANGE SET TO: 1, 21 R-SQUARE = 0.0482 R-SQUARE ADJUSTED = -0.0576 VARIANCE OF THE ESTIMATE-SIGMA**2 = 348.25 STANDARD ERROR OF THE ESTIMATE-SIGMA = 18.661 SUM OF SQUARED ERRORS-SSE= 6268.5 MEAN OF DEPENDENT VARIABLE = -0.88962 LOG OF THE LIKELIHOOD FUNCTION = -89.6348 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 18 DF P-VALUE CORR. COEFFICIENT AT MEANS ELAG -0.21054 0.2307 -0.9126 0.373-0.210 -0.2105 -0.2182 SALES 0.22326E-02 0.6381E-02 0.3499 0.730 0.082 0.0807 -5.4313 CONSTANT -5.9155 14.41 -0.4104 0.686-0.096 0.0000 6.6495 DURBIN-WATSON = 2.0685 VON NEUMANN RATIO = 2.1719 RHO = -0.03568 RESIDUAL SUM = -0.32419E-13 RESIDUAL VARIANCE = 348.25 SUM OF ABSOLUTE ERRORS= 267.39 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.0482 RUNS TEST: 14 RUNS, 12 POS, 0 ZERO, 9 NEG NORMAL STATISTIC = 1.2421 ...DURBIN H STATISTIC CANNOT BE COMPUTED |_GEN1 LM=$R2*$N ..NOTE..CURRENT VALUE OF $R2 = 0.48165E-01 ..NOTE..CURRENT VALUE OF $N = 21.000 |_DISTRIB LM /TYPE=CHI DF=1 CHI-SQUARE PARAMETERS- DF= 1.0000 MEAN= 1.0000 VARIANCE= 2.0000 MODE= 0.0000 DATA PDF CDF 1-CDF LM ROW 1 1.0115 0.23922 0.68545 0.31455 |_* The test statistic is not significant (p-value=0.31). There is no |_* strong evidence of further autocorrelation. ..INPUT FILE COMPLETED..TYPE A NEW COMMAND OR TYPE: STOP