The SAS System 08:49 Friday, October 14, 2005 1 The MEANS Procedure Variable Label N Mean Std Dev Minimum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ CPI U.S. Consumer Price index 234 0.0023712 0.0025709 -0.0038500 OBSNUM Observation Number 234 117.5000000 67.6941652 1.0000000 INT1 1 for Sept-Nov 1971 234 0.0128205 0.1127407 0 INT2 1 Dec 1971 - end of period 234 0.0555556 0.2295524 0 CONSTANT 234 1.0000000 0 1.0000000 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Variable Label Maximum ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ CPI U.S. Consumer Price index 0.0096300 OBSNUM Observation Number 234.0000000 INT1 1 for Sept-Nov 1971 1.0000000 INT2 1 Dec 1971 - end of period 1.0000000 CONSTANT 1.0000000 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ The SAS System 08:49 Friday, October 14, 2005 2 The ARIMA Procedure Name of Variable = CPI Period(s) of Differencing 1 Mean of Working Series 8.412E-6 Standard Deviation 0.00268 Number of Observations 233 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 7.18334E-6 1.00000 | |********************| 0 1 -3.1178E-6 -.43403 | *********| . | 0.065512 2 1.1771E-7 0.01639 | . | . | 0.076869 3 -1.2589E-6 -.17525 | ****| . | 0.076884 4 9.77855E-7 0.13613 | . |*** | 0.078580 5 -1.4537E-8 -.00202 | . | . | 0.079585 6 -2.8606E-7 -.03982 | . *| . | 0.079586 7 -6.5536E-7 -.09123 | .**| . | 0.079671 8 6.99626E-7 0.09740 | . |**. | 0.080118 9 1.42287E-7 0.01981 | . | . | 0.080625 10 2.84787E-7 0.03965 | . |* . | 0.080646 11 -1.2765E-6 -.17770 | ****| . | 0.080729 12 1.58677E-6 0.22090 | . |**** | 0.082391 13 -5.4336E-7 -.07564 | .**| . | 0.084895 14 -6.5098E-7 -.09062 | .**| . | 0.085183 15 4.51515E-7 0.06286 | . |* . | 0.085596 16 1.70983E-7 0.02380 | . | . | 0.085794 17 2.82936E-7 0.03939 | . |* . | 0.085822 18 -6.8061E-8 -.00947 | . | . | 0.085900 19 -1.0736E-6 -.14946 | ***| . | 0.085904 20 6.49032E-7 0.09035 | . |**. | 0.087013 21 2.61326E-7 0.03638 | . |* . | 0.087415 22 -6.1011E-7 -.08493 | .**| . | 0.087480 23 1.17278E-9 0.00016 | . | . | 0.087833 24 1.14934E-6 0.16000 | . |***. | 0.087833 "." marks two standard errors The SAS System 08:49 Friday, October 14, 2005 3 The ARIMA Procedure 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 0.78043 | . |**************** | 2 0.65558 | . |************* | 3 0.59978 | . |************ | 4 0.44689 | . |********* | 5 0.36797 | . |******* | 6 0.30512 | . |****** | 7 0.20986 | . |**** | 8 0.13065 | . |*** | 9 0.02648 | . |* . | 10 -0.02030 | . | . | 11 -0.00181 | . | . | 12 -0.05853 | . *| . | 13 -0.03739 | . *| . | 14 -0.01789 | . | . | 15 -0.05532 | . *| . | 16 -0.03273 | . *| . | 17 -0.02220 | . | . | 18 0.00694 | . | . | 19 0.05173 | . |* . | 20 0.01318 | . | . | 21 -0.00579 | . | . | 22 0.00318 | . | . | 23 -0.04580 | . *| . | 24 -0.04750 | . *| . | 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 -0.43403 | *********| . | 2 -0.21192 | ****| . | 3 -0.33361 | *******| . | 4 -0.15516 | ***| . | 5 -0.07814 | .**| . | 6 -0.13030 | ***| . | 7 -0.22673 | *****| . | 8 -0.12753 | ***| . | 9 -0.08379 | .**| . | 10 -0.01178 | . | . | 11 -0.19631 | ****| . | 12 0.06992 | . |* . | 13 0.05530 | . |* . | 14 -0.16282 | ***| . | 15 0.01371 | . | . | 16 0.05350 | . |* . | 17 0.05634 | . |* . | The SAS System 08:49 Friday, October 14, 2005 4 The ARIMA Procedure 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 18 0.13646 | . |*** | 19 -0.03704 | . *| . | 20 -0.01694 | . | . | 21 0.03707 | . |* . | 22 -0.15964 | ***| . | 23 -0.07601 | .**| . | 24 0.15399 | . |*** | Autocorrelation Check for White Noise To Chi- Pr > Lag Square DF ChiSq --------------------Autocorrelations-------------------- 6 56.65 6 <.0001 -0.434 0.016 -0.175 0.136 -0.002 -0.040 12 81.33 12 <.0001 -0.091 0.097 0.020 0.040 -0.178 0.221 18 86.36 18 <.0001 -0.076 -0.091 0.063 0.024 0.039 -0.009 24 103.10 24 <.0001 -0.149 0.090 0.036 -0.085 0.000 0.160 Variable INT1 has been differenced. Correlation of CPI and INT1 Period(s) of Differencing 1 Variance of input = 0.008584 Number of Observations 233 Observation(s) eliminated by differencing 1 Crosscorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 -24 8.58369E-8 0.00035 | . | . | -23 7.897E-6 0.03180 | . |* . | -22 1.28755E-7 0.00052 | . | . | -21 3.9485E-6 0.01590 | . | . | -20 -0.0000157 -.06326 | . *| . | -19 0.00001562 0.06291 | . |* . | -18 -4.2918E-8 -.00017 | . | . | -17 7.63948E-6 0.03077 | . |* . | -16 -2.1459E-7 -.00086 | . | . | -15 -0.0000114 -.04598 | . *| . | -14 -0.0000114 -.04580 | . *| . | -13 7.51073E-6 0.03025 | . |* . | -12 1.28755E-7 0.00052 | . | . | The SAS System 08:49 Friday, October 14, 2005 5 The ARIMA Procedure Crosscorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 -11 0.00002991 0.12047 | . |**. | -10 -0.0000226 -.09109 | .**| . | -9 7.51073E-6 0.03025 | . |* . | -8 -0.0000261 -.10509 | .**| . | -7 1.28755E-7 0.00052 | . | . | -6 1.71674E-7 0.00069 | . | . | -5 0.00001837 0.07398 | . |* . | -4 0.00001099 0.04425 | . |* . | -3 0.00001077 0.04338 | . |* . | -2 -0.0000219 -.08832 | .**| . | -1 -8.5837E-8 -.00035 | . | . | 0 -0.0000181 -.07277 | . *| . | 1 0.00001807 0.07277 | . |* . | 2 -0.0000179 -.07225 | . *| . | 3 0.00001803 0.07259 | . |* . | 4 -7.2103E-6 -.02904 | . *| . | 5 0.00001077 0.04338 | . |* . | 6 -7.1674E-6 -.02886 | . *| . | 7 -7.1674E-6 -.02886 | . *| . | 8 0.00001069 0.04304 | . |* . | 9 -0.0000106 -.04252 | . *| . | 10 0.00001060 0.04269 | . |* . | 11 -7.0815E-6 -.02852 | . *| . | 12 0.00001056 0.04252 | . |* . | 13 -7.1176E-6 -.02866 | . *| . | 14 -7.9022E-8 -.00032 | . | . | 15 -7.9022E-8 -.00032 | . | . | 16 0 0.00000 | . | . | 17 0 0.00000 | . | . | 18 0 0.00000 | . | . | 19 0 0.00000 | . | . | 20 0 0.00000 | . | . | 21 0 0.00000 | . | . | 22 0 0.00000 | . | . | 23 0 0.00000 | . | . | 24 0 0.00000 | . | . | "." marks two standard errors Variable INT2 has been differenced. Correlation of CPI and INT2 Period(s) of Differencing 1 Variance of input = 0.004273 Number of Observations 233 The SAS System 08:49 Friday, October 14, 2005 6 The ARIMA Procedure Correlation of CPI and INT2 Observation(s) eliminated by differencing 1 Crosscorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 -24 3.72742E-6 0.02127 | . | . | -23 -0.0000120 -.06832 | . *| . | -22 7.69778E-6 0.04394 | . |* . | -21 -1.7198E-7 -.00098 | . | . | -20 3.67518E-6 0.02098 | . | . | -19 -7.9869E-6 -.04559 | . *| . | -18 -1.9067E-7 -.00109 | . | . | -17 -3.9471E-6 -.02253 | . | . | -16 -7.7079E-6 -.04399 | . *| . | -15 0.00001134 0.06470 | . |* . | -14 7.4428E-6 0.04248 | . |* . | -13 -0.0000152 -.08677 | .**| . | -12 0.00001115 0.06362 | . |* . | -11 -0.0000225 -.12816 | ***| . | -10 7.35464E-6 0.04198 | . |* . | -9 3.65183E-6 0.02084 | . | . | -8 3.62473E-6 0.02069 | . | . | -7 7.25766E-6 0.04142 | . |* . | -6 3.48118E-6 0.01987 | . | . | -5 -0.0000148 -.08432 | .**| . | -4 -3.7133E-6 -.02119 | . | . | -3 -7.3205E-6 -.04178 | . *| . | -2 7.17352E-6 0.04094 | . |* . | -1 -3.6416E-6 -.02078 | . | . | 0 0.00001078 0.06152 | . |* . | 1 -0.0000108 -.06191 | . *| . | 2 0.00001432 0.08174 | . |**. | 3 -7.2471E-6 -.04136 | . *| . | 4 -3.7791E-6 -.02157 | . | . | 5 3.47801E-6 0.01985 | . | . | 6 -3.2796E-8 -.00019 | . | . | 7 3.41121E-6 0.01947 | . | . | 8 -7.2327E-6 -.04128 | . *| . | 9 0.00001043 0.05953 | . |* . | 10 -7.072E-6 -.04036 | . *| . | 11 -8.091E-8 -.00046 | . | . | 12 -8.1065E-8 -.00046 | . | . | 13 -7.2931E-8 -.00042 | . | . | 14 -9.6664E-8 -.00055 | . | . | 15 -4.9663E-8 -.00028 | . | . | 16 -2.4792E-9 -.00001 | . | . | 17 -9.7313E-8 -.00056 | . | . | The SAS System 08:49 Friday, October 14, 2005 7 The ARIMA Procedure Crosscorrelations Lag Covariance Correlation -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1 18 -2.6551E-8 -.00015 | . | . | 19 -2.6706E-8 -.00015 | . | . | 20 -2.6861E-8 -.00015 | . | . | 21 -5.0777E-8 -.00029 | . | . | 22 -2.7171E-8 -.00016 | . | . | 23 2.01979E-8 0.00012 | . | . | 24 2.00429E-8 0.00011 | . | . | "." marks two standard errors Preliminary Estimation Initial Moving Average Estimates Estimate 1 0.43403 White Noise Variance Est 6.045E-6 Conditional Least Squares Estimation Iteration SSE MA1,1 NUM1 NUM2 Lambda R Crit 0 0.001259 0.43403 -0.00211 0.00252 0.00001 1 1 0.001107 0.70035 -0.00267 -0.00118 1E-6 0.274867 2 0.001065 0.83506 -0.00259 -0.00068 1E-7 0.162191 3 0.001064 0.85132 -0.00265 -0.00091 1E-8 0.029882 4 0.001064 0.85232 -0.00265 -0.00091 1E-9 0.001921 Maximum Likelihood Estimation Iter Loglike MA1,1 NUM1 NUM2 Lambda R Crit 0 1101.4 0.85232 -0.00265 -0.00091 0.00001 1 1 1101.4 0.85262 -0.00265 -0.00091 1E-6 0.00056 ARIMA Estimation Optimization Summary Estimation Method Maximum Likelihood Parameters Estimated 3 The SAS System 08:49 Friday, October 14, 2005 8 The ARIMA Procedure ARIMA Estimation Optimization Summary Termination Criteria Maximum Relative Change in Estimates Iteration Stopping Value 0.001 Criteria Value 0.000353 Alternate Criteria Relative Change in Objective Function Alternate Criteria Value 1.994E-8 Maximum Absolute Value of Gradient 1.115E-6 R-Square Change from Last Iteration 0.00056 Objective Function Log Gaussian Likelihood Objective Function Value 1101.449 Marquardt's Lambda Coefficient 1E-6 Numerical Derivative Perturbation Delta 0.001 Iterations 1 Maximum Likelihood Estimation Standard Approx Parameter Estimate Error t Value Pr > |t| Lag Variable Shift MA1,1 0.85262 0.03537 24.11 <.0001 1 CPI 0 NUM1 -0.0026466 0.0014335 -1.85 0.0649 0 INT1 0 NUM2 -0.0009089 0.0012594 -0.72 0.4705 0 INT2 0 Variance Estimate 4.621E-6 Std Error Estimate 0.00215 AIC -2196.9 SBC -2186.54 Number of Residuals 233 Correlations of Parameter Estimates Variable CPI INT1 INT2 Parameter MA1,1 NUM1 NUM2 CPI MA1,1 1.000 -0.089 -0.092 INT1 NUM1 -0.089 1.000 0.438 INT2 NUM2 -0.092 0.438 1.000 The SAS System 08:49 Friday, October 14, 2005 9 The ARIMA Procedure Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq --------------------Autocorrelations-------------------- 6 8.80 5 0.1174 0.096 -0.014 -0.135 0.053 -0.007 -0.081 12 22.61 11 0.0201 -0.086 0.079 0.081 0.053 -0.041 0.177 18 27.33 17 0.0535 -0.019 -0.073 0.047 0.055 0.022 -0.085 24 41.68 23 0.0099 -0.178 -0.010 0.001 -0.055 0.043 0.137 30 54.14 29 0.0031 -0.058 -0.114 -0.032 -0.041 -0.043 -0.160 36 75.46 35 <.0001 -0.122 0.077 0.171 -0.077 -0.073 0.128 42 80.33 41 0.0002 0.025 0.012 0.012 0.010 -0.095 -0.083 Model for variable CPI Period(s) of Differencing 1 No mean term in this model. Moving Average Factors Factor 1: 1 - 0.85262 B**(1) Input Number 1 Input Variable INT1 Period(s) of Differencing 1 Overall Regression Factor -0.00265 Input Number 2 Input Variable INT2 Period(s) of Differencing 1 Overall Regression Factor -0.00091 Preliminary Estimation The SAS System 08:49 Friday, October 14, 2005 10 The ARIMA Procedure Initial Moving Average Estimates Estimate 1 0.43403 White Noise Variance Est 6.045E-6 Conditional Least Squares Estimation Iteration SSE MA1,1 NUM1 NUM2 Lambda R Crit 0 0.001259 0.43403 -0.00211 0.00252 0.00001 1 1 0.001107 0.70035 -0.00267 -0.00118 1E-6 0.274867 2 0.001065 0.83506 -0.00259 -0.00068 1E-7 0.162191 3 0.001064 0.85132 -0.00265 -0.00091 1E-8 0.029882 4 0.001064 0.85232 -0.00265 -0.00091 1E-9 0.001921 Unconditional Least Squares Estimation Iteration SSE MA1,1 NUM1 NUM2 Lambda R Crit 0 0.001063 0.85232 -0.00265 -0.00091 0.00001 1 1 0.001063 0.85650 -0.00266 -0.00092 1E-6 0.007778 2 0.001063 0.85687 -0.00266 -0.00092 1E-7 0.00071 ARIMA Estimation Optimization Summary Estimation Method Unconditional Least Squares Parameters Estimated 3 Termination Criteria Maximum Relative Change in Estimates Iteration Stopping Value 0.001 Criteria Value 0.000432 Alternate Criteria Relative Change in Objective Function Alternate Criteria Value 4.181E-7 Maximum Absolute Value of Gradient 3.899E-6 R-Square Change from Last Iteration 0.00071 Objective Function Sum of Squared Residuals Objective Function Value 0.001063 Marquardt's Lambda Coefficient 1E-7 Numerical Derivative Perturbation Delta 0.001 Iterations 2 The SAS System 08:49 Friday, October 14, 2005 11 The ARIMA Procedure Unconditional Least Squares Estimation Standard Approx Parameter Estimate Error t Value Pr > |t| Lag Variable Shift MA1,1 0.85687 0.03461 24.76 <.0001 1 CPI 0 NUM1 -0.0026605 0.0014320 -1.86 0.0645 0 INT1 0 NUM2 -0.0009224 0.0012442 -0.74 0.4592 0 INT2 0 Variance Estimate 4.62E-6 Std Error Estimate 0.002149 AIC -2196.88 SBC -2186.53 Number of Residuals 233 Correlations of Parameter Estimates Variable CPI INT1 INT2 Parameter MA1,1 NUM1 NUM2 CPI MA1,1 1.000 -0.091 -0.094 INT1 NUM1 -0.091 1.000 0.433 INT2 NUM2 -0.094 0.433 1.000 Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq --------------------Autocorrelations-------------------- 6 8.70 5 0.1218 0.099 -0.010 -0.131 0.055 -0.005 -0.079 12 22.54 11 0.0205 -0.084 0.080 0.082 0.054 -0.040 0.178 18 27.22 17 0.0549 -0.018 -0.072 0.047 0.055 0.022 -0.085 24 41.45 23 0.0105 -0.178 -0.011 0.001 -0.056 0.042 0.136 30 54.05 29 0.0032 -0.059 -0.115 -0.032 -0.042 -0.044 -0.160 36 75.17 35 <.0001 -0.123 0.076 0.169 -0.077 -0.073 0.127 42 80.04 41 0.0003 0.025 0.012 0.012 0.009 -0.095 -0.083 Model for variable CPI Period(s) of Differencing 1 No mean term in this model. The SAS System 08:49 Friday, October 14, 2005 12 The ARIMA Procedure Moving Average Factors Factor 1: 1 - 0.85687 B**(1) Input Number 1 Input Variable INT1 Period(s) of Differencing 1 Overall Regression Factor -0.00266 Input Number 2 Input Variable INT2 Period(s) of Differencing 1 Overall Regression Factor -0.00092 Preliminary Estimation Initial Moving Average Estimates Estimate 1 0.43403 White Noise Variance Est 6.045E-6 Conditional Least Squares Estimation Iteration SSE MA1,1 NUM1 NUM2 Lambda R Crit 0 0.001259 0.43403 -0.00211 0.00252 0.00001 1 1 0.001107 0.70035 -0.00267 -0.00118 1E-6 0.274867 2 0.001065 0.83506 -0.00259 -0.00068 1E-7 0.162191 3 0.001064 0.85132 -0.00265 -0.00091 1E-8 0.029882 4 0.001064 0.85232 -0.00265 -0.00091 1E-9 0.001921 5 0.001064 0.85243 -0.00265 -0.00091 1E-10 0.000203 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 0.000124 The SAS System 08:49 Friday, October 14, 2005 13 The ARIMA Procedure ARIMA Estimation Optimization Summary Alternate Criteria Relative Change in Objective Function Alternate Criteria Value 4.186E-9 Maximum Absolute Value of Gradient 1.024E-6 R-Square Change from Last Iteration 0.000203 Objective Function Sum of Squared Residuals Objective Function Value 0.001064 Marquardt's Lambda Coefficient 1E-10 Numerical Derivative Perturbation Delta 0.001 Iterations 5 Conditional Least Squares Estimation Standard Approx Parameter Estimate Error t Value Pr > |t| Lag Variable Shift MA1,1 0.85243 0.03458 24.65 <.0001 1 CPI 0 NUM1 -0.0026459 0.0014381 -1.84 0.0671 0 INT1 0 NUM2 -0.0009082 0.0012634 -0.72 0.4730 0 INT2 0 Variance Estimate 4.626E-6 Std Error Estimate 0.002151 AIC -2197.9 SBC -2187.55 Number of Residuals 233 * AIC and SBC do not include log determinant. Correlations of Parameter Estimates Variable CPI INT1 INT2 Parameter MA1,1 NUM1 NUM2 CPI MA1,1 1.000 -0.087 -0.090 INT1 NUM1 -0.087 1.000 0.438 INT2 NUM2 -0.090 0.438 1.000 The SAS System 08:49 Friday, October 14, 2005 14 The ARIMA Procedure Autocorrelation Check of Residuals To Chi- Pr > Lag Square DF ChiSq --------------------Autocorrelations-------------------- 6 9.03 5 0.1080 0.094 -0.017 -0.137 0.053 -0.009 -0.085 12 22.67 11 0.0197 -0.085 0.083 0.085 0.051 -0.047 0.172 18 27.43 17 0.0521 -0.023 -0.075 0.044 0.054 0.022 -0.086 24 41.77 23 0.0097 -0.178 -0.008 0.005 -0.052 0.044 0.138 30 54.69 29 0.0027 -0.059 -0.119 -0.037 -0.045 -0.044 -0.159 36 76.98 35 <.0001 -0.116 0.089 0.182 -0.071 -0.073 0.128 42 81.62 41 0.0002 0.027 0.011 0.010 0.010 -0.092 -0.081 Model for variable CPI Period(s) of Differencing 1 No mean term in this model. Moving Average Factors Factor 1: 1 - 0.85243 B**(1) Input Number 1 Input Variable INT1 Period(s) of Differencing 1 Overall Regression Factor -0.00265 Input Number 2 Input Variable INT2 Period(s) of Differencing 1 Overall Regression Factor -0.00091