Example 7.2: Seasonal Model for the Airline Series

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

Chapter Contents
Previous
Next
Top