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 The ARIMA Procedure

## Example 7.2: Seasonal Model for the Airline Series

The airline passenger data, given as Series G in Box and Jenkins (1976), has been used in time series analysis literature as an example of a nonstationary seasonal time series. This example uses PROC ARIMA to fit the "airline model," ARIMA(0,1,1)×(0,1,1)12, to Box and Jenkins' Series G.

The following statements read the data and log transform the series. The PROC GPLOT step plots the series, as shown in Output 7.2.1.

```
title1 'International Airline Passengers';
title2 '(Box and Jenkins Series-G)';
data seriesg;
input x @@;
xlog = log( x );
date = intnx( 'month', '31dec1948'd, _n_ );
format date monyy.;
datalines;
112 118 132 129 121 135 148 148 136 119 104 118
115 126 141 135 125 149 170 170 158 133 114 140
145 150 178 163 172 178 199 199 184 162 146 166
171 180 193 181 183 218 230 242 209 191 172 194
196 196 236 235 229 243 264 272 237 211 180 201
204 188 235 227 234 264 302 293 259 229 203 229
242 233 267 269 270 315 364 347 312 274 237 278
284 277 317 313 318 374 413 405 355 306 271 306
315 301 356 348 355 422 465 467 404 347 305 336
340 318 362 348 363 435 491 505 404 359 310 337
360 342 406 396 420 472 548 559 463 407 362 405
417 391 419 461 472 535 622 606 508 461 390 432
;

symbol1 i=join  v=dot;
proc gplot data=seriesg;
plot x * date = 1 / haxis= '1jan49'd to '1jan61'd by year;
run;
```

Output 7.2.1: Plot of Data

The following PROC ARIMA step fits an ARIMA(0,1,1)×(0,1,1)12 model without a mean term to the logarithms of the airline passengers series. The model is forecast, and the results stored in the data set B.

```
proc arima data=seriesg;
identify var=xlog(1,12) nlag=15;
run;
estimate q=(1)(12) noconstant method=uls;
run;
forecast out=b lead=24 id=date interval=month noprint;
quit;
```

The printed output from the IDENTIFY statement is shown in Output 7.2.2. The autocorrelation plots shown are for the twice differenced series (1-B)(1-B12)X. Note that the autocorrelation functions have the pattern characteristic of a first-order moving average process combined with a seasonal moving average process with lag 12.

Output 7.2.2: IDENTIFY Statement Output

 The ARIMA Procedure

 Name of Variable = xlog Period(s) of Differencing 1,12 Mean of Working Series 0.000291 Standard Deviation 0.045673 Number of Observations 131 Observation(s) eliminated by differencing 13

 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 0.0020860 1.00000 `| |********************|` 0 1 -0.0007116 -.34112 `| *******| . |` 0.087370 2 0.00021913 0.10505 `| . |** . |` 0.097006 3 -0.0004217 -.20214 `| ****| . |` 0.097870 4 0.00004456 0.02136 `| . | . |` 0.101007 5 0.00011610 0.05565 `| . |* . |` 0.101042 6 0.00006426 0.03080 `| . |* . |` 0.101275 7 -0.0001159 -.05558 `| . *| . |` 0.101347 8 -1.5867E-6 -.00076 `| . | . |` 0.101579 9 0.00036791 0.17637 `| . |**** |` 0.101579 10 -0.0001593 -.07636 `| . **| . |` 0.103891 11 0.00013431 0.06438 `| . |* . |` 0.104318 12 -0.0008065 -.38661 `| ********| . |` 0.104621 13 0.00031624 0.15160 `| . |*** . |` 0.115011 14 -0.0001202 -.05761 `| . *| . |` 0.116526 15 0.00031200 0.14957 `| . |*** . |` 0.116744

 "." 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 0.41027 `| . |******** |` 2 0.12711 `| . |*** |` 3 0.10189 `| . |**. |` 4 0.01978 `| . | . |` 5 -0.10310 `| .**| . |` 6 -0.11886 `| .**| . |` 7 -0.04088 `| . *| . |` 8 -0.05086 `| . *| . |` 9 -0.06022 `| . *| . |` 10 0.06460 `| . |* . |` 11 0.19907 `| . |**** |` 12 0.31709 `| . |****** |` 13 0.12434 `| . |**. |` 14 0.06583 `| . |* . |` 15 0.01515 `| . | . |`

 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 -0.34112 `| *******| . |` 2 -0.01281 `| . | . |` 3 -0.19266 `| ****| . |` 4 -0.12503 `| ***| . |` 5 0.03309 `| . |* . |` 6 0.03468 `| . |* . |` 7 -0.06019 `| . *| . |` 8 -0.02022 `| . | . |` 9 0.22558 `| . |***** |` 10 0.04307 `| . |* . |` 11 0.04659 `| . |* . |` 12 -0.33869 `| *******| . |` 13 -0.10918 `| .**| . |` 14 -0.07684 `| .**| . |` 15 -0.02175 `| . | . |`

 The ARIMA Procedure

 Autocorrelation Check for White Noise To Lag Chi-Square DF Pr > ChiSq Autocorrelations 6 23.27 6 0.0007 -0.341 0.105 -0.202 0.021 0.056 0.031 12 51.47 12 <.0001 -0.056 -0.001 0.176 -0.076 0.064 -0.387

The results of the ESTIMATE statement are shown in Output 7.2.3.

Output 7.2.3: ESTIMATE Statement Output

 The ARIMA Procedure

 Unconditional Least Squares Estimation Parameter Estimate Approx Std Error t Value Pr > |t| Lag MA1,1 0.39594 0.08149 4.86 <.0001 1 MA2,1 0.61331 0.07961 7.70 <.0001 12

 Variance Estimate 0.001363 Std Error Estimate 0.036921 AIC -484.755 SBC -479.005 Number of Residuals 131

 Correlations of Parameter Estimates Parameter MA1,1 MA2,1 MA1,1 1.000 -0.055 MA2,1 -0.055 1.000

 Autocorrelation Check of Residuals To Lag Chi-Square DF Pr > ChiSq Autocorrelations 6 5.56 4 0.2349 0.022 0.024 -0.125 -0.129 0.057 0.065 12 8.49 10 0.5816 -0.065 -0.042 0.102 -0.060 0.023 0.007 18 13.23 16 0.6560 0.022 0.039 0.045 -0.162 0.035 0.001 24 24.99 22 0.2978 -0.106 -0.104 -0.037 -0.027 0.219 0.040

 Model for variable xlog Period(s) of Differencing 1,12

 Moving Average Factors Factor 1: 1 - 0.39594 B**(1) Factor 2: 1 - 0.61331 B**(12)

The following statements retransform the forecast values to get forecasts in the original scales. See the section "Forecasting Log Transformed Data" earlier in this chapter for more information.

```
data c;
set b;
x        = exp( xlog );
forecast = exp( forecast + std*std/2 );
l95      = exp( l95 );
u95      = exp( u95 );
run;
```

The forecasts and their confidence limits are plotted using the following PROC GPLOT step. The plot is shown in Output 7.2.4.

```
symbol1 i=none  v=star;
symbol2 i=join  v=circle;
symbol3 i=join  v=none l=3;
proc gplot data=c;
where date >= '1jan58'd;
plot x * date = 1 forecast * date = 2
l95 * date = 3 u95 * date = 3 /
overlay haxis= '1jan58'd to '1jan62'd by year;
run;
```

Output 7.2.4: Plot of the Forecast for the Original Series

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