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 The ARIMA Procedure

Example 7.1: Simulated IMA Model

This example illustrates the ARIMA procedure results for a case where the true model is known. An integrated moving average model is used for this illustration.

The following DATA step generates a pseudo-random sample of 100 periods from the ARIMA(0,1,1) process ut = ut-1 + at - .8at-1, at  iid   N(0,1).


title1 'Simulated IMA(1,1) Series';
data a;
u1 = 0.9; a1 = 0;
do i = -50 to 100;
a = rannor( 32565 );
u = u1 + a - .8 * a1;
if i > 0 then output;
a1 = a;
u1 = u;
end;
run;


The following ARIMA procedure statements identify and estimate the model.


proc arima data=a;
identify var=u nlag=15;
run;
identify var=u(1) nlag=15;
run;
estimate q=1 ;
run;
quit;


The results of the first IDENTIFY statement are shown in Output 7.1.1. The output shows the behavior of the sample autocorrelation function when the process is nonstationary. Note that in this case the estimated autocorrelations are not very high, even at small lags. Nonstationarity is reflected in a pattern of significant autocorrelations that do not decline quickly with increasing lag, not in the size of the autocorrelations.

Output 7.1.1: Output from the First IDENTIFY Statement

 Simulated IMA(1,1) Series

 The ARIMA Procedure

 Name of Variable = u Mean of Working Series 0.099637 Standard Deviation 1.115604 Number of Observations 100

 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 1.244572 1.00000 | |********************| 0 1 0.547457 0.43988 | . |********* | 0.100000 2 0.534787 0.42970 | . |********* | 0.117770 3 0.569849 0.45787 | . |********* | 0.132524 4 0.384428 0.30888 | . |****** | 0.147497 5 0.405137 0.32552 | . |******* | 0.153830 6 0.253617 0.20378 | . |**** . | 0.160571 7 0.321830 0.25859 | . |***** . | 0.163136 8 0.363871 0.29237 | . |******. | 0.167185 9 0.271180 0.21789 | . |**** . | 0.172222 10 0.419208 0.33683 | . |******* | 0.174957 11 0.298127 0.23954 | . |***** . | 0.181326 12 0.186460 0.14982 | . |*** . | 0.184463 13 0.313270 0.25171 | . |***** . | 0.185676 14 0.314594 0.25277 | . |***** . | 0.189057 15 0.156329 0.12561 | . |*** . | 0.192407

 "." marks two standard errors

 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 -0.12382 | . **| . | 2 -0.17396 | .***| . | 3 -0.19966 | ****| . | 4 -0.01476 | . | . | 5 -0.02895 | . *| . | 6 0.20612 | . |**** | 7 0.01258 | . | . | 8 -0.09616 | . **| . | 9 0.00025 | . | . | 10 -0.16879 | .***| . | 11 0.05680 | . |* . | 12 0.14306 | . |***. | 13 -0.02466 | . | . | 14 -0.15549 | .***| . | 15 0.08247 | . |** . |

 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 0.43988 | . |********* | 2 0.29287 | . |****** | 3 0.26499 | . |***** | 4 -0.00728 | . | . | 5 0.06473 | . |* . | 6 -0.09926 | . **| . | 7 0.10048 | . |** . | 8 0.12872 | . |***. | 9 0.03286 | . |* . | 10 0.16034 | . |***. | 11 -0.03794 | . *| . | 12 -0.14469 | .***| . | 13 0.06415 | . |* . | 14 0.15482 | . |***. | 15 -0.10989 | . **| . |

 The ARIMA Procedure

 Autocorrelation Check for White Noise To Lag Chi-Square DF Pr > ChiSq Autocorrelations 6 87.22 6 <.0001 0.440 0.430 0.458 0.309 0.326 0.204 12 131.39 12 <.0001 0.259 0.292 0.218 0.337 0.240 0.150

The second IDENTIFY statement differences the series. The results of the second IDENTIFY statement are shown in Output 7.1.2. This output shows autocorrelation, inverse autocorrelation, and partial autocorrelation functions typical of MA(1) processes.

Output 7.1.2: Output from the Second IDENTIFY Statement

 The ARIMA Procedure

 Name of Variable = u Period(s) of Differencing 1 Mean of Working Series 0.019752 Standard Deviation 1.160921 Number of Observations 99 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 1.347737 1.00000 | |********************| 0 1 -0.699404 -.51895 | **********| . | 0.100504 2 -0.036142 -.02682 | . *| . | 0.124666 3 0.245093 0.18186 | . |****. | 0.124724 4 -0.234167 -.17375 | . ***| . | 0.127374 5 0.181778 0.13488 | . |*** . | 0.129746 6 -0.184601 -.13697 | . ***| . | 0.131155 7 0.0088659 0.00658 | . | . | 0.132592 8 0.146372 0.10861 | . |** . | 0.132595 9 -0.241579 -.17925 | .****| . | 0.133490 10 0.240512 0.17846 | . |****. | 0.135900 11 0.031005 0.02301 | . | . | 0.138247 12 -0.250954 -.18620 | . ****| . | 0.138285 13 0.095295 0.07071 | . |* . | 0.140795 14 0.194110 0.14403 | . |*** . | 0.141153 15 -0.219688 -.16300 | . ***| . | 0.142630

 "." marks two standard errors

 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 0.72538 | . |*************** | 2 0.48987 | . |********** | 3 0.35415 | . |******* | 4 0.34169 | . |******* | 5 0.33466 | . |******* | 6 0.34003 | . |******* | 7 0.24192 | . |***** | 8 0.12899 | . |***. | 9 0.06597 | . |* . | 10 0.01654 | . | . | 11 0.06434 | . |* . | 12 0.08659 | . |** . | 13 0.02485 | . | . | 14 -0.03545 | . *| . | 15 -0.00113 | . | . |

 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 -0.51895 | **********| . | 2 -0.40526 | ********| . | 3 -0.07862 | . **| . | 4 -0.14588 | .***| . | 5 0.02735 | . |* . | 6 -0.13782 | .***| . | 7 -0.16741 | .***| . | 8 -0.06041 | . *| . | 9 -0.18372 | ****| . | 10 -0.01478 | . | . | 11 0.14277 | . |***. | 12 -0.04345 | . *| . | 13 -0.19959 | ****| . | 14 0.08302 | . |** . | 15 0.00278 | . | . |

 The ARIMA Procedure

 Autocorrelation Check for White Noise To Lag Chi-Square DF Pr > ChiSq Autocorrelations 6 38.13 6 <.0001 -0.519 -0.027 0.182 -0.174 0.135 -0.137 12 50.62 12 <.0001 0.007 0.109 -0.179 0.178 0.023 -0.186

The ESTIMATE statement fits an ARIMA(0,1,1) model to the simulated data. Note that in this case the parameter estimates are reasonably close to the values used to generate the simulated data. ()

The ESTIMATE statement results are shown in Output 7.1.3.

Output 7.1.3: Output from Fitting ARIMA(0,1,1) Model

 The ARIMA Procedure

 Conditional Least Squares Estimation Parameter Estimate Approx Std Error t Value Pr > |t| Lag MU 0.02056 0.01972 1.04 0.2997 0 MA1,1 0.79142 0.06474 12.22 <.0001 1

 Constant Estimate 0.020558 Variance Estimate 0.819807 Std Error Estimate 0.905432 AIC 263.259 SBC 268.45 Number of Residuals 99

 * AIC and SBC do not include log determinant.

 Correlations of Parameter Estimates Parameter MU MA1,1 MU 1.000 -0.124 MA1,1 -0.124 1.000

 Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq Autocorrelations 6 6.48 5 0.2623 -0.033 0.030 0.153 -0.096 0.013 -0.163 12 13.11 11 0.2862 -0.048 0.046 -0.086 0.159 0.027 -0.145 18 20.12 17 0.2680 0.069 0.130 -0.099 0.006 0.164 -0.013 24 24.73 23 0.3645 0.064 0.032 0.076 -0.077 -0.075 0.114

 Model for variable u Estimated Mean 0.020558 Period(s) of Differencing 1

 Moving Average Factors Factor 1: 1 - 0.79142 B**(1)

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