Regression nonlinéaire 17:10 Wednesday, March 25, 2009 1 The NLIN Procedure Dependent Variable prod Method: Gauss-Newton Iterative Phase Sum of Iter a b c Squares 0 3.4000 0.0800 -0.00017 3.4357E8 1 -5.1294 0.2965 -0.00146 60735639 2 -8.3679 0.3587 -0.00176 26595742 3 -11.3523 0.4224 -0.00209 23815296 4 -12.7718 0.4527 -0.00226 23322804 5 -13.3308 0.4647 -0.00232 23252423 6 -13.5349 0.4690 -0.00234 23243331 7 -13.6073 0.4706 -0.00235 23242202 8 -13.6326 0.4711 -0.00235 23242064 9 -13.6415 0.4713 -0.00235 23242047 10 -13.6446 0.4714 -0.00235 23242045 11 -13.6456 0.4714 -0.00235 23242045 12 -13.6460 0.4714 -0.00235 23242045 13 -13.6461 0.4714 -0.00235 23242045 NOTE: Convergence criterion met. Estimation Summary Method Gauss-Newton Iterations 13 R 3.769E-6 PPC(a) 3.339E-6 RPC(a) 9.581E-6 Object 1.58E-10 Objective 23242045 Observations Read 27 Observations Used 27 Observations Missing 0 NOTE: An intercept was not specified for this model. Somme des Carré Valeur Approx Source DF carrés moyen F Pr > F Model 3 3.7893E9 1.2631E9 1304.30 <.0001 Error 24 23242045 968419 Uncorrected Total 27 3.8126E9 Regression nonlinéaire 17:10 Wednesday, March 25, 2009 2 The NLIN Procedure Erreur std Paramètre Estimation approchée Limites de confiance approchées 95% a -13.6461 2.5234 -18.8542 -8.4381 b 0.4714 0.0544 0.3591 0.5837 c -0.00235 0.000293 -0.00296 -0.00175 Matrice de corrélation approchée a b c a 1.0000000 -0.9988772 0.9957959 b -0.9988772 1.0000000 -0.9990081 c 0.9957959 -0.9990081 1.0000000 Série chronologique 17:10 Wednesday, March 25, 2009 3 Etape identification The ARIMA Procedure Name of Variable = residus Mean of Working Series 335.3768 Standard Deviation 865.0658 Number of Observations 27 Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error 0 748339 1.00000 | |********************| 0 1 506595 0.67696 | . |************** | 0.192450 2 263067 0.35153 | . |******* . | 0.266427 3 117612 0.15716 | . |*** . | 0.283085 4 -117588 -.15713 | . ***| . | 0.286298 5 -291143 -.38905 | . ********| . | 0.289475 6 -333280 -.44536 | . *********| . | 0.308233 "." marks two standard errors Inverse Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 -0.61030 | ************| . | 2 0.34379 | . |*******. | 3 -0.32988 | .*******| . | 4 0.17497 | . |*** . | 5 -0.01034 | . | . | 6 0.05007 | . |* . | Partial Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 0.67696 | . |************** | 2 -0.19704 | . ****| . | 3 0.01092 | . | . | 4 -0.42227 | ********| . | 5 -0.12442 | . **| . | 6 -0.09492 | . **| . | Série chronologique 17:10 Wednesday, March 25, 2009 4 Etape identification The ARIMA Procedure Autocorrelation Check for White Noise To Chi- Pr > Lag Square DF Khi2 --------------------Auto-corrélations------------------- 6 32.10 6 <.0001 0.677 0.352 0.157 -0.157 -0.389 -0.445 Augmented Dickey-Fuller Unit Root Tests Type Retards Rho Pr < Rho Tau Pr < Tau F Pr > F Zero Mean 0 -6.6373 0.0652 -1.88 0.0583 1 -8.0707 0.0397 -1.86 0.0609 2 -9.7260 0.0217 -1.80 0.0678 3 80.0546 0.9999 -2.90 0.0056 4 -768.509 0.0001 -2.00 0.0456 5 -8.9606 0.0270 -1.26 0.1837 6 -2.8108 0.2371 -0.84 0.3407 Single Mean 0 -7.5520 0.1998 -1.94 0.3124 1.89 0.5996 1 -10.5008 0.0824 -1.97 0.2962 1.97 0.5947 2 -13.7658 0.0277 -1.93 0.3133 1.89 0.6133 3 43.1258 0.9999 -3.15 0.0361 5.08 0.0549 4 33.8151 0.9999 -2.24 0.2002 2.61 0.4473 5 -35.1596 <.0001 -1.41 0.5588 1.04 0.8112 6 -4.1437 0.4885 -0.72 0.8206 0.35 0.9801 Trend 0 -8.4144 0.4838 -2.12 0.5110 2.48 0.6908 1 -10.9779 0.2852 -2.12 0.5106 2.52 0.6914 2 -15.8649 0.0771 -2.16 0.4867 2.59 0.6800 3 35.0087 0.9999 -3.99 0.0241 8.44 0.0287 4 20.7475 0.9999 -3.48 0.0673 6.66 0.0719 5 16.5380 0.9999 -2.99 0.1567 4.91 0.2743 6 15.8196 0.9999 -2.58 0.2910 3.91 0.4499 Série chronologique 17:10 Wednesday, March 25, 2009 5 Etape identification The ARIMA Procedure Name of Variable = residus Period(s) of Differencing 1 Mean of Working Series -24.915 Standard Deviation 680.3908 Number of Observations 26 Observation(s) eliminated by differencing 1 Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error 0 462932 1.00000 | |********************| 0 1 -26982.444 -.05829 | . *| . | 0.196116 2 -21474.358 -.04639 | . *| . | 0.196781 3 109790 0.23716 | . |***** . | 0.197201 4 -104673 -.22611 | . *****| . | 0.207882 5 -92579.243 -.19998 | . ****| . | 0.217135 6 -133508 -.28840 | . ******| . | 0.224107 "." marks two standard errors Inverse Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 0.14843 | . |*** . | 2 0.06167 | . |* . | 3 -0.27458 | . *****| . | 4 0.20192 | . |**** . | 5 0.20843 | . |**** . | 6 0.30706 | . |****** . | Partial Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 -0.05829 | . *| . | 2 -0.04995 | . *| . | 3 0.23278 | . |***** . | 4 -0.21531 | . ****| . | 5 -0.21392 | . ****| . | 6 -0.43184 | *********| . | Série chronologique 17:10 Wednesday, March 25, 2009 6 Etape identification The ARIMA Procedure Autocorrelation Check for White Noise To Chi- Pr > Lag Square DF Khi2 --------------------Auto-corrélations------------------- 6 8.05 6 0.2344 -0.058 -0.046 0.237 -0.226 -0.200 -0.288 Augmented Dickey-Fuller Unit Root Tests Type Retards Rho Pr < Rho Tau Pr < Tau F Pr > F Zero Mean 0 -26.4485 <.0001 -5.06 <.0001 1 -29.3166 <.0001 -3.33 0.0019 2 -9.1443 0.0260 -1.59 0.1034 3 -101.120 0.0001 -2.16 0.0320 4 19.0369 0.9999 -2.96 0.0051 5 10.7056 0.9999 -3.43 0.0017 6 8.6942 0.9999 -2.89 0.0062 Single Mean 0 -26.5447 0.0001 -4.96 0.0005 12.34 0.0010 1 -29.1900 <.0001 -3.26 0.0286 5.32 0.0468 2 -8.8987 0.1302 -1.53 0.5003 1.27 0.7561 3 -88.9063 <.0001 -2.09 0.2491 2.32 0.5141 4 19.4094 0.9999 -2.84 0.0700 4.17 0.0977 5 10.7744 0.9999 -3.31 0.0283 5.64 0.0397 6 8.6445 0.9999 -2.84 0.0713 4.17 0.0975 Trend 0 -27.4032 0.0009 -4.94 0.0029 12.28 0.0010 1 -29.9085 0.0002 -3.27 0.0955 5.36 0.1953 2 -9.0859 0.4168 -1.56 0.7772 1.40 0.8884 3 -107.058 0.0001 -2.15 0.4930 2.52 0.6916 4 19.8822 0.9999 -2.81 0.2086 4.17 0.4034 5 10.6447 0.9999 -3.35 0.0867 5.76 0.1256 6 8.1931 0.9998 -2.92 0.1787 4.26 0.3876 Série chronologique 17:10 Wednesday, March 25, 2009 7 Etape identification The ARIMA Procedure Name of Variable = residus Period(s) of Differencing 1,1 Mean of Working Series 28.63501 Standard Deviation 996.5792 Number of Observations 25 Observation(s) eliminated by differencing 2 Autocorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error 0 993170 1.00000 | |********************| 0 1 -420871 -.42377 | ********| . | 0.200000 2 -185314 -.18659 | . ****| . | 0.233166 3 309520 0.31165 | . |****** . | 0.239064 4 -162359 -.16348 | . ***| . | 0.254797 5 336.512 0.00034 | . | . | 0.258958 6 -105986 -.10671 | . **| . | 0.258958 "." marks two standard errors Inverse Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 0.57011 | . |*********** | 2 0.38509 | . |******** | 3 0.13844 | . |*** . | 4 0.24621 | . |***** . | 5 0.18435 | . |**** . | 6 0.15953 | . |*** . | Partial Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 -0.42377 | ********| . | 2 -0.44631 | *********| . | 3 0.01241 | . | . | 4 -0.07732 | . **| . | 5 0.01423 | . | . | 6 -0.27849 | . ******| . | Série chronologique 17:10 Wednesday, March 25, 2009 8 Etape identification The ARIMA Procedure Autocorrelation Check for White Noise To Chi- Pr > Lag Square DF Khi2 --------------------Auto-corrélations------------------- 6 10.32 6 0.1120 -0.424 -0.187 0.312 -0.163 0.000 -0.107 Augmented Dickey-Fuller Unit Root Tests Type Retards Rho Pr < Rho Tau Pr < Tau F Pr > F Zero Mean 0 -39.5465 <.0001 -7.53 <.0001 1 -211.408 0.0001 -7.36 <.0001 2 -33.4405 <.0001 -2.45 0.0166 3 -26.0793 <.0001 -1.94 0.0514 4 -78.1783 <.0001 -1.76 0.0737 5 14.8660 0.9999 -2.27 0.0255 6 6.7050 0.9999 -4.33 0.0002 Single Mean 0 -39.6468 <.0001 -7.34 0.0003 27.17 0.0010 1 -218.786 0.0001 -7.22 0.0003 26.14 0.0010 2 -33.7420 <.0001 -2.42 0.1474 3.00 0.3585 3 -27.4008 <.0001 -1.92 0.3170 1.90 0.6127 4 -76.9623 <.0001 -1.73 0.4031 1.57 0.6872 5 15.2842 0.9999 -2.19 0.2144 2.50 0.4735 6 6.7438 0.9999 -4.12 0.0060 8.61 0.0010 Trend 0 -39.8176 <.0001 -7.09 0.0002 25.78 0.0010 1 -237.087 0.0001 -7.06 0.0002 25.33 0.0010 2 -33.7744 <.0001 -2.40 0.3667 3.01 0.6073 3 -29.6909 <.0001 -1.92 0.6086 1.92 0.7974 4 -68.9282 <.0001 -1.68 0.7226 1.49 0.8728 5 15.4260 0.9999 -2.08 0.5221 2.21 0.7463 6 6.3737 0.9998 -4.42 0.0133 9.83 0.0106 Série chronologique 17:10 Wednesday, March 25, 2009 9 Etape estimation The ARIMA Procedure Conditional Least Squares Estimation Erreur Pr. Paramètre Estimation standard Valeur du test t Approx. > |t| Retard MU -18.10763 108.83867 -0.17 0.8693 0 AR1,1 -0.65183 0.22003 -2.96 0.0070 1 Constant Estimate -29.9107 Variance Estimate 781380.9 Std Error Estimate 883.9575 AIC 412.0828 SBC 414.5206 Number of Residuals 25 * AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter MU AR1,1 MU 1.000 0.089 AR1,1 0.089 1.000 Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF Khi2 --------------------Auto-corrélations------------------- 6 9.09 5 0.1054 -0.299 -0.179 0.305 -0.167 -0.071 -0.219 12 12.95 11 0.2964 -0.017 0.098 -0.121 0.018 0.231 -0.070 18 13.41 17 0.7082 0.026 0.027 -0.048 0.030 -0.017 -0.037 24 13.44 23 0.9418 -0.001 0.009 -0.003 0.001 -0.002 0.004 Autocorrelation Plot of Residuals Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error 0 781381 1.00000 | |********************| 0 1 -233492 -.29882 | . ******| . | 0.200000 2 -139801 -.17892 | . ****| . | 0.217125 3 238529 0.30527 | . |****** . | 0.222945 4 -130738 -.16732 | . ***| . | 0.239080 5 -55605.573 -.07116 | . *| . | 0.243719 6 -171501 -.21949 | . ****| . | 0.244549 Série chronologique 17:10 Wednesday, March 25, 2009 10 Etape estimation The ARIMA Procedure "." marks two standard errors Inverse Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 0.31529 | . |****** . | 2 0.19033 | . |**** . | 3 -0.11960 | . **| . | 4 0.24847 | . |***** . | 5 0.23529 | . |***** . | 6 0.29487 | . |****** . | Partial Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 -0.29882 | . ******| . | 2 -0.29451 | . ******| . | 3 0.17801 | . |**** . | 4 -0.06508 | . *| . | 5 -0.05400 | . *| . | 6 -0.44539 | *********| . | Model for variable residus Estimated Mean -18.1076 Period(s) of Differencing 1,1 Autoregressive Factors Factor 1: 1 + 0.65183 B**(1) AVERTISSEMENT: The model defined by the new estimates is unstable. The iteration process has been terminated. AVERTISSEMENT: Estimates may not have converged. ARIMA Estimation Optimization Summary Estimation Method Conditional Least Squares Parameters Estimated 2 Termination Criteria Maximum Relative Change in Estimates Iteration Stopping Value 0.001 Criteria Value 2.763713 Maximum Absolute Value of Gradient 2318555 Série chronologique 17:10 Wednesday, March 25, 2009 11 Etape estimation The ARIMA Procedure ARIMA Estimation Optimization Summary R-Square Change from Last Iteration 0.444401 Objective Function Sum of Squared Residuals Objective Function Value 12570335 Marquardt's Lambda Coefficient 1E-6 Numerical Derivative Perturbation Delta 0.001 Iterations 12 Warning Message Estimates may not have converged. Conditional Least Squares Estimation Erreur Pr. Paramètre Estimation standard Valeur du test t Approx. > |t| Retard MU -18.51736 24.23081 -0.76 0.4525 0 MA1,1 1.00000 0.16392 6.10 <.0001 1 Constant Estimate -18.5174 Variance Estimate 546536.3 Std Error Estimate 739.281 AIC 403.1463 SBC 405.584 Number of Residuals 25 * AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter MU MA1,1 MU 1.000 0.912 MA1,1 0.912 1.000 Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF Khi2 --------------------Auto-corrélations------------------- 6 6.21 5 0.2862 -0.045 0.029 0.309 -0.175 -0.117 -0.222 12 9.65 11 0.5622 -0.103 0.037 -0.061 0.035 0.240 -0.003 18 10.71 17 0.8714 0.082 0.085 0.009 0.053 0.010 -0.011 24 10.76 23 0.9855 0.012 0.013 0.000 0.003 -0.001 0.004 Série chronologique 17:10 Wednesday, March 25, 2009 12 Etape estimation The ARIMA Procedure Autocorrelation Plot of Residuals Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error 0 546536 1.00000 | |********************| 0 1 -24647.971 -.04510 | . *| . | 0.200000 2 15689.260 0.02871 | . |* . | 0.200406 3 168894 0.30903 | . |****** . | 0.200571 4 -95592.790 -.17491 | . ***| . | 0.218788 5 -64127.999 -.11734 | . **| . | 0.224312 6 -121362 -.22206 | . ****| . | 0.226754 "." marks two standard errors Inverse Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 0.10026 | . |** . | 2 -0.08287 | . **| . | 3 -0.41905 | ********| . | 4 0.09396 | . |** . | 5 0.15498 | . |*** . | 6 0.26398 | . |***** . | Partial Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 -0.04510 | . *| . | 2 0.02673 | . |* . | 3 0.31235 | . |****** . | 4 -0.16205 | . ***| . | 5 -0.16908 | . ***| . | 6 -0.36132 | .*******| . | Model for variable residus Estimated Mean -18.5174 Period(s) of Differencing 1,1 Moving Average Factors Factor 1: 1 - 1 B**(1) AVERTISSEMENT: The model defined by the new estimates is unstable. The iteration process has been terminated. Série chronologique 17:10 Wednesday, March 25, 2009 13 Etape estimation The ARIMA Procedure AVERTISSEMENT: Estimates may not have converged. ARIMA Estimation Optimization Summary Estimation Method Conditional Least Squares Parameters Estimated 3 Termination Criteria Maximum Relative Change in Estimates Iteration Stopping Value 0.001 Criteria Value 0.863891 Maximum Absolute Value of Gradient 8377175 R-Square Change from Last Iteration 0.639475 Objective Function Sum of Squared Residuals Objective Function Value 16874762 Marquardt's Lambda Coefficient 0.00001 Numerical Derivative Perturbation Delta 0.001 Iterations 5 Warning Message Estimates may not have converged. Conditional Least Squares Estimation Erreur Pr. Paramètre Estimation standard Valeur du test t Approx. > |t| Retard MU -4.65000 27.80232 -0.17 0.8687 0 MA1,1 0.99999 0.56033 1.78 0.0881 1 AR1,1 0.59258 0.53952 1.10 0.2839 1 Constant Estimate -1.89452 Variance Estimate 767034.6 Std Error Estimate 875.8051 AIC 412.5083 SBC 416.1649 Number of Residuals 25 * AIC and SBC do not include log determinant. Correlations of Parameter Estimates Parameter MU MA1,1 AR1,1 MU 1.000 0.254 0.289 MA1,1 0.254 1.000 0.872 AR1,1 0.289 0.872 1.000 Série chronologique 17:10 Wednesday, March 25, 2009 14 Etape estimation The ARIMA Procedure Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF Khi2 --------------------Auto-corrélations------------------- 6 10.17 4 0.0377 -0.405 -0.173 0.315 -0.184 -0.016 -0.121 12 14.58 10 0.1481 -0.037 0.138 -0.076 -0.035 0.229 -0.122 18 14.95 16 0.5283 0.002 0.026 -0.055 0.024 -0.009 -0.028 24 15.00 22 0.8620 0.010 0.014 -0.001 0.005 -0.002 0.004 Autocorrelation Plot of Residuals Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 Std Error 0 767035 1.00000 | |********************| 0 1 -310850 -.40526 | ********| . | 0.200000 2 -132864 -.17322 | . ***| . | 0.230519 3 241734 0.31515 | . |****** . | 0.235668 4 -141307 -.18423 | . ****| . | 0.251963 5 -11898.060 -.01551 | . | . | 0.257294 6 -93092.716 -.12137 | . **| . | 0.257332 "." marks two standard errors Inverse Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 0.50847 | . |********** | 2 0.32007 | . |****** . | 3 0.08029 | . |** . | 4 0.27084 | . |***** . | 5 0.22824 | . |***** . | 6 0.20206 | . |**** . | Partial Autocorrelations Lag Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 1 -0.40526 | ********| . | 2 -0.40377 | ********| . | 3 0.07574 | . |** . | 4 -0.06665 | . *| . | 5 -0.01822 | . | . | 6 -0.33426 | .*******| . | Série chronologique 17:10 Wednesday, March 25, 2009 15 Etape estimation The ARIMA Procedure Model for variable residus Estimated Mean -4.65 Period(s) of Differencing 1,1 Autoregressive Factors Factor 1: 1 - 0.59258 B**(1) Moving Average Factors Factor 1: 1 - 0.99999 B**(1)