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The ARIMA Procedure

Example 7.3: Model for Series J Data from Box and Jenkins

This example uses the Series J data from Box and Jenkins (1976). First the input series, X, is modeled with a univariate ARMA model. Next, the dependent series, Y, is cross correlated with the input series. Since a model has been fit to X, both Y and X are prewhitened by this model before the sample cross correlations are computed. Next, a transfer function model is fit with no structure on the noise term. The residuals from this model are identified by means of the PLOT option; then, the full model, transfer function and noise is fit to the data.

The following statements read Input Gas Rate and Output CO2 from a gas furnace. (Data values are not shown. See "Series J" in Box and Jenkins (1976) for the values.)


   title1 'Gas Furnace Data';
   title2 '(Box and Jenkins, Series J)';
   data seriesj;
      input x y @@;
      label x = 'Input Gas Rate'
            y = 'Output CO2';
   datalines;
   ;

The following statements produce Output 7.3.1 through Output 7.3.5.


   proc arima data=seriesj;
   
      /*--- Look at the input process -------------------*/
      identify var=x nlag=10;
      run;
   
      /*--- Fit a model for the input -------------------*/
      estimate p=3;
      run;
   
      /*--- Crosscorrelation of prewhitened series ------*/
      identify var=y crosscorr=(x) nlag=10;
      run;
   
      /*--- Fit transfer function - look at residuals ---*/
      estimate input=( 3 $ (1,2)/(1,2) x ) plot;
      run;
   
      /*--- Estimate full model -------------------------*/
      estimate p=2 input=( 3 $ (1,2)/(1) x );
      run;
   
   quit;

The results of the first IDENTIFY statement for the input series X are shown in Output 7.3.1.

Output 7.3.1: IDENTIFY Statement Results for X

Gas Furnace Data
(Box and Jenkins, Series J)

The ARIMA Procedure

Name of Variable = x
Mean of Working Series -0.05683
Standard Deviation 1.070952
Number of Observations 296

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 1.146938 1.00000
|                    |********************|
0
1 1.092430 0.95247
|                  . |******************* |
0.058124
2 0.956652 0.83409
|                .   |*****************   |
0.097510
3 0.782051 0.68186
|               .    |**************      |
0.119201
4 0.609291 0.53123
|               .    |***********         |
0.131721
5 0.467380 0.40750
|              .     |********            |
0.138770
6 0.364957 0.31820
|              .     |******              |
0.142756
7 0.298427 0.26019
|              .     |*****.              |
0.145132
8 0.260943 0.22751
|              .     |*****.              |
0.146699
9 0.244378 0.21307
|              .     |**** .              |
0.147887
10 0.238942 0.20833
|              .     |**** .              |
0.148920

"." 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 -0.71090
|      **************| .                  |
2 0.26217
|                  . |*****               |
3 -0.13005
|                 ***| .                  |
4 0.14777
|                  . |***                 |
5 -0.06803
|                  .*| .                  |
6 -0.01147
|                  . | .                  |
7 -0.01649
|                  . | .                  |
8 0.06108
|                  . |*.                  |
9 -0.04490
|                  .*| .                  |
10 0.01100
|                  . | .                  |


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
1 0.95247
|                  . |******************* |
2 -0.78796
|    ****************| .                  |
3 0.33897
|                  . |*******             |
4 0.12121
|                  . |**                  |
5 0.05896
|                  . |*.                  |
6 -0.11147
|                  **| .                  |
7 0.04862
|                  . |*.                  |
8 0.09945
|                  . |**                  |
9 0.01587
|                  . | .                  |
10 -0.06973
|                  .*| .                  |

Autocorrelation Check for White Noise
To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 786.35 6 <.0001 0.952 0.834 0.682 0.531 0.408 0.318


The ESTIMATE statement results for the AR(3) model for the input series X are shown in Output 7.3.2.

Output 7.3.2: Estimates of the AR(3) Model for X

The ARIMA Procedure

Conditional Least Squares Estimation
Parameter Estimate Approx Std Error t Value Pr > |t| Lag
MU -0.12280 0.10902 -1.13 0.2609 0
AR1,1 1.97607 0.05499 35.94 <.0001 1
AR1,2 -1.37499 0.09967 -13.80 <.0001 2
AR1,3 0.34336 0.05502 6.24 <.0001 3

Constant Estimate -0.00682
Variance Estimate 0.035797
Std Error Estimate 0.1892
AIC -141.667
SBC -126.906
Number of Residuals 296

* AIC and SBC do not include log determinant.

Correlations of Parameter Estimates
Parameter MU AR1,1 AR1,2 AR1,3
MU 1.000 -0.017 0.014 -0.016
AR1,1 -0.017 1.000 -0.941 0.790
AR1,2 0.014 -0.941 1.000 -0.941
AR1,3 -0.016 0.790 -0.941 1.000

Autocorrelation Check of Residuals
To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 10.30 3 0.0162 -0.042 0.068 0.056 -0.145 -0.009 0.059
12 19.89 9 0.0186 0.014 0.002 -0.055 0.035 0.143 -0.079
18 27.92 15 0.0221 0.099 0.043 -0.082 0.017 0.066 -0.052
24 31.05 21 0.0729 -0.078 0.024 0.015 0.030 0.045 0.004
30 34.58 27 0.1499 -0.007 -0.004 0.073 -0.038 -0.062 0.003
36 38.84 33 0.2231 0.010 0.002 0.082 0.045 0.056 -0.023
42 41.18 39 0.3753 0.002 0.033 -0.061 -0.003 -0.006 -0.043
48 42.73 45 0.5687 0.018 0.051 -0.012 0.015 -0.027 0.020


The ARIMA Procedure

Model for variable x
Estimated Mean -0.1228

Autoregressive Factors
Factor 1: 1 - 1.97607 B**(1) + 1.37499 B**(2) - 0.34336 B**(3)


The IDENTIFY statement results for the dependent series Y cross correlated with the input series X is shown in Output 7.3.3. Since a model has been fit to X, both Y and X are prewhitened by this model before the sample cross correlations are computed.

Output 7.3.3: IDENTIFY Statement for Y Cross Correlated with X

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
1 0.97076
|                  . |******************* |
2 -0.80388
|    ****************| .                  |
3 0.18833
|                  . |****                |
4 0.25999
|                  . |*****               |
5 0.05949
|                  . |*.                  |
6 -0.06258
|                  .*| .                  |
7 -0.01435
|                  . | .                  |
8 0.05490
|                  . |*.                  |
9 0.00545
|                  . | .                  |
10 0.03141
|                  . |*.                  |

Autocorrelation Check for White Noise
To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 1023.15 6 <.0001 0.971 0.896 0.793 0.680 0.574 0.485


The ARIMA Procedure

Correlation of y and x
Number of Observations 296
Variance of transformed series y 0.131438
Variance of transformed series x 0.035357

Both series have been prewhitened.

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
-10 0.0015683 0.02301
|                  . | .                  |
-9 0.00013502 0.00198
|                  . | .                  |
-8 -0.0060480 -.08872
|                  **| .                  |
-7 -0.0017624 -.02585
|                  .*| .                  |
-6 -0.0080539 -.11814
|                  **| .                  |
-5 -0.0000944 -.00138
|                  . | .                  |
-4 -0.0012802 -.01878
|                  . | .                  |
-3 -0.0031078 -.04559
|                  .*| .                  |
-2 0.00065212 0.00957
|                  . | .                  |
-1 -0.0019166 -.02811
|                  .*| .                  |
0 -0.0003673 -.00539
|                  . | .                  |
1 0.0038939 0.05712
|                  . |*.                  |
2 -0.0016971 -.02489
|                  . | .                  |
3 -0.019231 -.28210
|              ******| .                  |
4 -0.022479 -.32974
|             *******| .                  |
5 -0.030909 -.45341
|           *********| .                  |
6 -0.018122 -.26583
|               *****| .                  |
7 -0.011426 -.16761
|                 ***| .                  |
8 -0.0017355 -.02546
|                  .*| .                  |
9 0.0022590 0.03314
|                  . |*.                  |
10 -0.0035152 -.05156
|                  .*| .                  |

"." marks two standard errors

Crosscorrelation Check Between Series
To Lag Chi-Square DF Pr > ChiSq Crosscorrelations
5 117.75 6 <.0001 -0.005 0.057 -0.025 -0.282 -0.330 -0.453


The ARIMA Procedure

Both variables have been prewhitened by the following filter:

Prewhitening Filter

Autoregressive Factors
Factor 1: 1 - 1.97607 B**(1) + 1.37499 B**(2) - 0.34336 B**(3)


The ESTIMATE statement results for the transfer function model with no structure on the noise term is shown in Output 7.3.4. The PLOT option prints the residual autocorrelation functions from this model.

Output 7.3.4: Estimates of the Transfer Function Model

The ARIMA Procedure

Conditional Least Squares Estimation
Parameter Estimate Approx Std Error t Value Pr > |t| Lag Variable Shift
MU 53.32237 0.04932 1081.24 <.0001 0 y 0
NUM1 -0.62868 0.25385 -2.48 0.0138 0 x 3
NUM1,1 0.47258 0.62253 0.76 0.4484 1 x 3
NUM1,2 0.73660 0.81006 0.91 0.3640 2 x 3
DEN1,1 0.15411 0.90483 0.17 0.8649 1 x 3
DEN1,2 0.27774 0.57345 0.48 0.6285 2 x 3

Constant Estimate 53.32237
Variance Estimate 0.704241
Std Error Estimate 0.839191
AIC 729.7249
SBC 751.7648
Number of Residuals 291

* AIC and SBC do not include log determinant.

Correlations of Parameter Estimates
Variable Parameter y MU x NUM1 x NUM1,1 x NUM1,2 x DEN1,1 x DEN1,2
y MU 1.000 0.013 0.002 -0.002 0.004 -0.006
x NUM1 0.013 1.000 0.755 -0.447 0.089 -0.065
x NUM1,1 0.002 0.755 1.000 0.121 -0.538 0.565
x NUM1,2 -0.002 -0.447 0.121 1.000 -0.892 0.870
x DEN1,1 0.004 0.089 -0.538 -0.892 1.000 -0.998
x DEN1,2 -0.006 -0.065 0.565 0.870 -0.998 1.000


The ARIMA Procedure

Autocorrelation Check of Residuals
To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 496.45 6 <.0001 0.893 0.711 0.502 0.312 0.167 0.064
12 498.58 12 <.0001 -0.003 -0.040 -0.054 -0.040 -0.022 -0.021
18 539.38 18 <.0001 -0.045 -0.083 -0.131 -0.170 -0.196 -0.195
24 561.87 24 <.0001 -0.163 -0.102 -0.026 0.047 0.106 0.142
30 585.90 30 <.0001 0.158 0.156 0.131 0.081 0.013 -0.037
36 592.42 36 <.0001 -0.048 -0.018 0.038 0.070 0.079 0.067
42 593.44 42 <.0001 0.042 0.025 0.013 0.004 0.006 0.019
48 601.94 48 <.0001 0.043 0.068 0.084 0.082 0.061 0.023


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 0.704241 1.00000
|                    |********************|
0
1 0.628846 0.89294
|                  . |******************  |
0.058621
2 0.500490 0.71068
|                .   |**************      |
0.094427
3 0.353404 0.50182
|                .   |**********          |
0.111300
4 0.219895 0.31224
|               .    |******              |
0.118821
5 0.117330 0.16660
|               .    |*** .               |
0.121608
6 0.044967 0.06385
|               .    |*   .               |
0.122390
7 -0.0023551 -.00334
|               .    |    .               |
0.122504
8 -0.028030 -.03980
|               .   *|    .               |
0.122505
9 -0.037891 -.05380
|               .   *|    .               |
0.122549
10 -0.028378 -.04030
|               .   *|    .               |
0.122630

"." 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 -0.57346
|         ***********| .                  |
2 0.02264
|                  . | .                  |
3 0.03631
|                  . |*.                  |
4 0.03941
|                  . |*.                  |
5 -0.01256
|                  . | .                  |
6 -0.01618
|                  . | .                  |
7 0.02680
|                  . |*.                  |
8 -0.05895
|                  .*| .                  |
9 0.07043
|                  . |*.                  |
10 -0.02987
|                  .*| .                  |


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
1 0.89294
|                  . |******************  |
2 -0.42765
|           *********| .                  |
3 -0.13463
|                 ***| .                  |
4 0.02199
|                  . | .                  |
5 0.03891
|                  . |*.                  |
6 -0.02219
|                  . | .                  |
7 -0.02249
|                  . | .                  |
8 0.01538
|                  . | .                  |
9 0.00634
|                  . | .                  |
10 0.07737
|                  . |**                  |

Crosscorrelation Check of Residuals with Input x
To Lag Chi-Square DF Pr > ChiSq Crosscorrelations
5 0.48 2 0.7855 -0.009 -0.005 0.026 0.013 -0.017 -0.022
11 0.93 8 0.9986 -0.006 0.008 0.022 0.023 -0.017 -0.013
17 2.63 14 0.9996 0.012 0.035 0.037 0.039 -0.005 -0.040
23 19.19 20 0.5092 -0.076 -0.108 -0.122 -0.122 -0.094 -0.041
29 20.12 26 0.7857 -0.039 -0.013 0.010 -0.020 -0.031 -0.005
35 24.22 32 0.8363 -0.022 -0.031 -0.074 -0.036 0.014 0.076
41 30.66 38 0.7953 0.108 0.091 0.046 0.018 0.003 0.009
47 31.65 44 0.9180 0.008 -0.011 -0.040 -0.030 -0.002 0.028

Model for variable y
Estimated Intercept 53.32237

Input Number 1
Input Variable x
Shift 3

Numerator Factors
Factor 1: -0.6287 - 0.47258 B**(1) - 0.7366 B**(2)

Denominator Factors
Factor 1: 1 - 0.15411 B**(1) - 0.27774 B**(2)


The ESTIMATE statement results for the final transfer function model with AR(2) noise are shown in Output 7.3.5.

Output 7.3.5: Estimates of the Final Model

The ARIMA Procedure

Conditional Least Squares Estimation
Parameter Estimate Approx Std Error t Value Pr > |t| Lag Variable Shift
MU 53.26307 0.11926 446.63 <.0001 0 y 0
AR1,1 1.53292 0.04754 32.25 <.0001 1 y 0
AR1,2 -0.63297 0.05006 -12.64 <.0001 2 y 0
NUM1 -0.53522 0.07482 -7.15 <.0001 0 x 3
NUM1,1 0.37602 0.10287 3.66 0.0003 1 x 3
NUM1,2 0.51894 0.10783 4.81 <.0001 2 x 3
DEN1,1 0.54842 0.03822 14.35 <.0001 1 x 3

Constant Estimate 5.329371
Variance Estimate 0.058828
Std Error Estimate 0.242544
AIC 8.292811
SBC 34.00607
Number of Residuals 291

* AIC and SBC do not include log determinant.

Correlations of Parameter Estimates
Variable Parameter y MU y AR1,1 y AR1,2 x NUM1 x NUM1,1 x NUM1,2 x DEN1,1
y MU 1.000 -0.063 0.047 -0.008 -0.016 0.017 -0.049
y AR1,1 -0.063 1.000 -0.927 -0.003 0.007 -0.002 0.015
y AR1,2 0.047 -0.927 1.000 0.023 -0.005 0.005 -0.022
x NUM1 -0.008 -0.003 0.023 1.000 0.713 -0.178 -0.013
x NUM1,1 -0.016 0.007 -0.005 0.713 1.000 -0.467 -0.039
x NUM1,2 0.017 -0.002 0.005 -0.178 -0.467 1.000 -0.720
x DEN1,1 -0.049 0.015 -0.022 -0.013 -0.039 -0.720 1.000


The ARIMA Procedure

Autocorrelation Check of Residuals
To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 8.61 4 0.0717 0.024 0.055 -0.073 -0.054 -0.054 0.119
12 15.43 10 0.1172 0.032 0.028 -0.081 0.047 0.022 0.107
18 21.13 16 0.1734 -0.038 0.052 -0.093 -0.013 -0.073 -0.005
24 27.52 22 0.1922 -0.118 -0.002 -0.007 0.076 0.024 -0.004
30 36.94 28 0.1202 0.034 -0.021 0.020 0.094 -0.118 0.065
36 44.26 34 0.1119 -0.025 -0.057 0.113 0.022 0.030 0.065
42 45.62 40 0.2500 -0.017 -0.036 -0.029 -0.013 -0.033 0.017
48 48.60 46 0.3689 0.024 0.069 0.024 0.017 0.022 -0.044

Crosscorrelation Check of Residuals with Input x
To Lag Chi-Square DF Pr > ChiSq Crosscorrelations
5 0.93 3 0.8191 0.008 0.004 0.010 0.008 -0.045 0.030
11 6.60 9 0.6784 0.075 -0.024 -0.019 -0.026 -0.111 0.013
17 13.86 15 0.5365 0.050 0.043 0.014 0.014 -0.141 -0.028
23 18.55 21 0.6142 -0.074 -0.078 0.023 -0.016 0.021 0.060
29 27.99 27 0.4113 -0.071 -0.001 0.038 -0.156 0.031 0.035
35 35.18 33 0.3654 -0.014 0.015 -0.039 0.028 0.046 0.142
41 37.15 39 0.5544 0.031 -0.029 -0.070 -0.006 0.012 -0.004
47 42.42 45 0.5818 0.036 -0.038 -0.053 0.107 0.029 0.021


The ARIMA Procedure

Model for variable y
Estimated Intercept 53.26307

Autoregressive Factors
Factor 1: 1 - 1.53292 B**(1) + 0.63297 B**(2)

Input Number 1
Input Variable x
Shift 3

Numerator Factors
Factor 1: -0.5352 - 0.37602 B**(1) - 0.51894 B**(2)

Denominator Factors
Factor 1: 1 - 0.54842 B**(1)

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