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The X12 Procedure

Example 34.4 RegARIMA Automatic Model Selection

This example demonstrates regARIMA modeling and TRAMO-based automatic model selection, which is available with the AUTOMDL statement. ODS SELECT statements are used to limit the displayed output to the model selection and estimation stages. The same data set is used as in the previous examples. 

title 'TRAMO Automatic Model Identification';
ods select ModelEstimation.AutoModel.UnitRootTestModel
           ModelEstimation.AutoModel.UnitRootTest
           ModelEstimation.AutoModel.AutoChoiceModel
           ModelEstimation.AutoModel.Best5Model
           ModelEstimation.AutoModel.AutomaticModelChoice
           ModelEstimation.AutoModel.FinalModelChoice
           ModelEstimation.AutoModel.AutomdlNote;
proc x12 data=sales date=date;
   var sales;
   transform function=log;
   regression predefined=td;
   automdl maxorder=(1,1)
           print=unitroottest unitroottestmdl autochoicemdl best5model;
   estimate;
   x11;
   output out=out(obs=23) a1 a2 a6 b1 c17 c20 d1 d7 d8 d9 d10
                  d11 d12 d13 d16 d18;
run;

proc print data=out(obs=23);
   title 'Output Variables Related to Trading Day Regression';
run;

The automatic model selection output is shown in Output 34.4.1, Output 34.4.2, and Output 34.4.3. The first table, "ARIMA Estimate for Unit Root Identification," gives details of the method that TRAMO uses to automatically select the orders of differencing. The second table, "Results of Unit Root Test for Identifying Orders of Differencing," shows that a regular difference order of 1 and a seasonal difference order of 1 has been determined by TRAMO. The third table, "Models estimated by Automatic ARIMA Model Selection procedure," shows all the models examined by the TRAMO-based method. The fourth table, "Best Five ARIMA Models Chosen by Automatic Modeling," shows the top five models in order of rank and their BIC2 statistic. The fifth table, "Comparison of Automatically Selected Model and Default Model," compares the model selected by the TRAMO model to the default X-12-ARIMA model. The sixth table, "Final Automatic Model Selection," shows which model was actually selected.

Output 34.4.1 Output from the AUTOMDL Statement
TRAMO Automatic Model Identification

The X12 Procedure

ARIMA Estimates for Unit Root Identification
For Variable sales
Model Number   Estimation Method       ARMA  
Estimated Model Parameter   Estimate  
1   H-R   ( 2, 0, 0) ( 1, 0, 0)   NS_AR_1   0.67540    
    H-R   ( 2, 0, 0) ( 1, 0, 0)   NS_AR_2   0.28425    
    H-R   ( 2, 0, 0) ( 1, 0, 0)   S_AR_12   0.91963    
2   H-R   ( 1, 1, 1) ( 1, 0, 1)   NS_AR_1   0.13418    
    H-R   ( 1, 1, 1) ( 1, 0, 1)   S_AR_12   0.98500    
    H-R   ( 1, 1, 1) ( 1, 0, 1)   NS_MA_1   0.47884    
    H-R   ( 1, 1, 1) ( 1, 0, 1)   S_MA_12   0.51726    
3   H-R   ( 1, 1, 1) ( 1, 1, 1)   NS_AR_1   -0.39269    
    H-R   ( 1, 1, 1) ( 1, 1, 1)   S_AR_12   0.06223    
    H-R   ( 1, 1, 1) ( 1, 1, 1)   NS_MA_1   -0.09570    
    H-R   ( 1, 1, 1) ( 1, 1, 1)   S_MA_12   0.58536    

Results of Unit Root Test
for Identifying Orders
of Differencing
For Variable sales
Regular difference
order		 Seasonal
difference
order		 Mean Significant
1 1 no

Output 34.4.2 Output from the AUTOMDL Statement
Models estimated by Automatic ARIMA Model Selection procedure
For Variable sales
Model Number     ARMA   Statistics of Fit
  Estimated Model Parameter   Estimate BIC   BIC2
1     ( 3, 1, 0) ( 0, 1, 0)   NS_AR_1   -0.33524        
      ( 3, 1, 0) ( 0, 1, 0)   NS_AR_2   -0.05558        
      ( 3, 1, 0) ( 0, 1, 0)   NS_AR_3   -0.15649        
      ( 3, 1, 0) ( 0, 1, 0)           1024.469   -3.40549
2     ( 3, 1, 0) ( 0, 1, 1)   NS_AR_1   -0.33186        
      ( 3, 1, 0) ( 0, 1, 1)   NS_AR_2   -0.05823        
      ( 3, 1, 0) ( 0, 1, 1)   NS_AR_3   -0.15200        
      ( 3, 1, 0) ( 0, 1, 1)   S_MA_12   0.55279        
      ( 3, 1, 0) ( 0, 1, 1)           993.7880   -3.63970
3     ( 3, 1, 0) ( 1, 1, 0)   NS_AR_1   -0.38673        
      ( 3, 1, 0) ( 1, 1, 0)   NS_AR_2   -0.08768        
      ( 3, 1, 0) ( 1, 1, 0)   NS_AR_3   -0.18143        
      ( 3, 1, 0) ( 1, 1, 0)   S_AR_12   -0.47336        
      ( 3, 1, 0) ( 1, 1, 0)           1000.224   -3.59057
4     ( 3, 1, 0) ( 1, 1, 1)   NS_AR_1   -0.34352        
      ( 3, 1, 0) ( 1, 1, 1)   NS_AR_2   -0.06504        
      ( 3, 1, 0) ( 1, 1, 1)   NS_AR_3   -0.15728        
      ( 3, 1, 0) ( 1, 1, 1)   S_AR_12   -0.12163        
      ( 3, 1, 0) ( 1, 1, 1)   S_MA_12   0.47073        
      ( 3, 1, 0) ( 1, 1, 1)           998.0548   -3.60713
5     ( 0, 1, 0) ( 0, 1, 1)   S_MA_12   0.60446        
      ( 0, 1, 0) ( 0, 1, 1)           996.8560   -3.61628
6     ( 0, 1, 1) ( 0, 1, 1)   NS_MA_1   0.36272        
      ( 0, 1, 1) ( 0, 1, 1)   S_MA_12   0.55599        
      ( 0, 1, 1) ( 0, 1, 1)           986.6405   -3.69426
7     ( 1, 1, 0) ( 0, 1, 1)   NS_AR_1   -0.32734        
      ( 1, 1, 0) ( 0, 1, 1)   S_MA_12   0.55834        
      ( 1, 1, 0) ( 0, 1, 1)           987.1500   -3.69037
8     ( 1, 1, 1) ( 0, 1, 1)   NS_AR_1   0.17833        
      ( 1, 1, 1) ( 0, 1, 1)   NS_MA_1   0.52867        
      ( 1, 1, 1) ( 0, 1, 1)   S_MA_12   0.56212        
      ( 1, 1, 1) ( 0, 1, 1)           991.2363   -3.65918
9     ( 0, 1, 1) ( 0, 1, 0)   NS_MA_1   0.36005        
      ( 0, 1, 1) ( 0, 1, 0)           1017.770   -3.45663

Output 34.4.3 Output from the AUTOMDL Statement
TRAMO Automatic Model Identification

The X12 Procedure


Automatic ARIMA Model Selection
Methodology based on research by Gomez and Maravall (2000).

Best Five ARIMA Models Chosen by Automatic
Modeling
For Variable sales
Rank   Estimated Model   BIC2
1   ( 0, 1, 1) ( 0, 1, 1)   -3.69426
2   ( 1, 1, 0) ( 0, 1, 1)   -3.69037
3   ( 1, 1, 1) ( 0, 1, 1)   -3.65918
4   ( 0, 1, 0) ( 0, 1, 1)   -3.61628
5   ( 0, 1, 1) ( 0, 1, 0)   -3.45663

Comparison of Automatically Selected Model and Default Model
For Variable sales
Source of Candidate Models       Statistics of Fit
Estimated Model Plbox   Rvr   Number of
Outliers
Automatic Model Choice   ( 0, 1, 1) ( 0, 1, 1)   0.62560   0.03546   0
Airline Model (Default)   ( 0, 1, 1) ( 0, 1, 1)   0.62561   0.03546   0

Final Automatic Model Selection
For Variable sales
Source of Model Estimated Model
Automatic Model Choice ( 0, 1, 1) ( 0, 1, 1)


Table 34.11 and Output 34.4.4 illustrate the regARIMA modeling method. Table 34.11 shows the relationship between the output variables in PROC X12 that results from a regARIMA model. Note that some of these formulas apply only to this example. Output 34.4.4 shows the values of these variables for the first 23 observations in the example.

Table 34.11 regARIMA Output Variables and Descriptions

Table

Title

Type

Formula

A1

Time series data (for the span analyzed)

Data

Input

A2

Prior-adjustment factors

Factor

Calculated from regression

 

leap year (from trading day regression)

   
 

adjustments

   

A6

RegARIMA trading day component

Factor

Calculated from regression

 

leap year prior adjustments included

   
 

from Table A2

   

B1

Original series (prior adjusted)

Data

*

 

(adjusted for regARIMA factors)

 

* because only TD specified

C17

Final weights for irregular component

Factor

Calculated using moving

     

standard deviation

C20

Final extreme value adjustment factors

Factor

Calculated using C16 and C17

D1

Modified original data, D iteration

Data

**

     

     

** C19=B1 in this example

D7

Preliminary trend cycle, D iteration

Data

Calculated using Henderson

     

moving average

D8

Final unmodified SI ratios

Factor

***

     

     

*** TD specified in regression

D9

Final replacement values for SI ratios

Factor

If C17 shows extreme values,

     

;

     

otherwise

D10

Final seasonal factors

Factor

Calculated using moving averages

D11

Final seasonally adjusted data

Data

****

 

(also adjusted for trading day)

  

     

**** for this example

D12

Final trend cycle

Data

Calculated using Henderson

     

moving average

D13

Final irregular component

Factor

D16

Combined adjustment factors

Factor

 

(includes seasonal, trading day factors)

   

D18

Combined calendar adjustment factors

Factor

 

(includes trading day factors)

 

*****

     

***** regression TD is the only

     

calendar adjustment factor

     

in this example

Output 34.4.4 Output Variables Related to Trading Day Regression
Output Variables Related to Trading Day Regression

Obs DATE sales_A1 sales_A2 sales_A6 sales_B1 sales_C17 sales_C20 sales_D1 sales_D7 sales_D8 sales_D9 sales_D10 sales_D11 sales_D12 sales_D13 sales_D16 sales_D18
1 SEP78 112 1.00000 1.01328 110.532 1.00000 1.00000 110.532 124.138 0.89040 . 0.90264 122.453 124.448 0.98398 0.91463 1.01328
2 OCT78 118 1.00000 0.99727 118.323 1.00000 1.00000 118.323 124.905 0.94731 . 0.94328 125.438 125.115 1.00258 0.94070 0.99727
3 NOV78 132 1.00000 0.98960 133.388 1.00000 1.00000 133.388 125.646 1.06161 . 1.06320 125.459 125.723 0.99790 1.05214 0.98960
4 DEC78 129 1.00000 1.00957 127.777 1.00000 1.00000 127.777 126.231 1.01225 . 0.99534 128.375 126.205 1.01720 1.00487 1.00957
5 JAN79 121 1.00000 0.99408 121.721 1.00000 1.00000 121.721 126.557 0.96179 . 0.97312 125.083 126.479 0.98896 0.96735 0.99408
6 FEB79 135 0.99115 0.99115 136.205 1.00000 1.00000 136.205 126.678 1.07521 . 1.05931 128.579 126.587 1.01574 1.04994 0.99115
7 MAR79 148 1.00000 1.00966 146.584 1.00000 1.00000 146.584 126.825 1.15580 . 1.17842 124.391 126.723 0.98160 1.18980 1.00966
8 APR79 148 1.00000 0.99279 149.075 1.00000 1.00000 149.075 127.038 1.17347 . 1.18283 126.033 126.902 0.99315 1.17430 0.99279
9 MAY79 136 1.00000 0.99406 136.813 1.00000 1.00000 136.813 127.433 1.07360 . 1.06125 128.916 127.257 1.01303 1.05495 0.99406
10 JUN79 119 1.00000 1.01328 117.440 1.00000 1.00000 117.440 127.900 0.91822 . 0.91663 128.121 127.747 1.00293 0.92881 1.01328
11 JUL79 104 1.00000 0.99727 104.285 1.00000 1.00000 104.285 128.499 0.81156 . 0.81329 128.226 128.421 0.99848 0.81107 0.99727
12 AUG79 118 1.00000 0.99678 118.381 1.00000 1.00000 118.381 129.253 0.91589 . 0.91135 129.897 129.316 1.00449 0.90841 0.99678
13 SEP79 115 1.00000 1.00229 114.737 0.98630 0.99964 114.778 130.160 0.88151 0.88182 0.90514 126.761 130.347 0.97249 0.90722 1.00229
14 OCT79 126 1.00000 0.99408 126.751 0.88092 1.00320 126.346 131.238 0.96581 0.96273 0.93820 135.100 131.507 1.02732 0.93264 0.99408
15 NOV79 141 1.00000 1.00366 140.486 1.00000 1.00000 140.486 132.699 1.05869 . 1.06183 132.306 132.937 0.99525 1.06571 1.00366
16 DEC79 135 1.00000 0.99872 135.173 1.00000 1.00000 135.173 134.595 1.00429 . 0.99339 136.072 134.720 1.01004 0.99212 0.99872
17 JAN80 125 1.00000 0.99406 125.747 0.00000 0.95084 132.248 136.820 0.91906 0.96658 0.97481 128.996 136.763 0.94321 0.96902 0.99406
18 FEB80 149 1.02655 1.03400 144.100 1.00000 1.00000 144.100 139.215 1.03509 . 1.06153 135.748 138.996 0.97663 1.09762 1.03400
19 MAR80 170 1.00000 0.99872 170.217 1.00000 1.00000 170.217 141.559 1.20245 . 1.17965 144.295 141.221 1.02177 1.17814 0.99872
20 APR80 170 1.00000 0.99763 170.404 1.00000 1.00000 170.404 143.777 1.18520 . 1.18499 143.802 143.397 1.00283 1.18218 0.99763
21 MAY80 158 1.00000 1.00966 156.489 1.00000 1.00000 156.489 145.925 1.07239 . 1.06005 147.624 145.591 1.01397 1.07028 1.00966
22 JUN80 133 1.00000 0.99279 133.966 1.00000 1.00000 133.966 148.133 0.90436 . 0.91971 145.662 147.968 0.98442 0.91307 0.99279
23 JUL80 114 1.00000 0.99406 114.681 0.00000 0.94057 121.927 150.682 0.76108 0.80917 0.81275 141.103 150.771 0.93588 0.80792 0.99406

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