The X12 Procedure

Example 37.5 Automatic Outlier Detection

This example demonstrates the use of the OUTLIER statement to automatically detect and remove outliers from a time series to be seasonally adjusted. The data set is the same as in the section Basic Seasonal Adjustment and the previous examples. Adding the OUTLIER statement to Example 37.3 requests that outliers be detected by using the default critical value as described in the section OUTLIER Statement. The tables associated with outlier detection for this example are shown in Output 37.5.1. The first table shows the critical values; the second table shows that a single potential outlier was identified; the third table shows the estimates for the ARMA parameters. Since no outliers are included in the regression model, the "Regression Model Parameter Estimates" table is not displayed. Because only a potential outlier was identified, and not an actual outlier, in this case the A1 series and the B1 series are identical.

title 'Automatic Outlier Identification';
proc x12 data=sales date=date;
   var sales;
   transform function=log;
   arima model=( (0,1,1)(0,1,1) );
   outlier;
   estimate;
   x11;
   output out=nooutlier a1 b1 d10;
run ;

Output 37.5.1: PROC X12 Output When Potential Outliers Are Identified

Automatic Outlier Identification

The X12 Procedure

Critical Values to use in Outlier
Detection
For Variable sales
Begin SEP1978
End AUG1990
Observations 144
Method Add One
AO Critical Value 3.889838
LS Critical Value 3.889838


Note: The following time series values might later be identified as outliers when data are added or revised. They were not identified as outliers in this run either because their test t-statistics were slightly below the critical value or because they were eliminated during the backward deletion step of the identification procedure, when a non-robust t-statistic is used.

Potential Outliers
For Variable sales
Type of Outlier Date t Value for AO t Value for LS
AO NOV1989 -3.48 -1.51

Exact ARMA Maximum Likelihood Estimation
For Variable sales
Parameter Lag Estimate Standard
Error
t Value Pr > |t|
Nonseasonal MA 1 0.40181 0.07887 5.09 <.0001
Seasonal MA 12 0.55695 0.07626 7.30 <.0001



In the next example, reducing the critical value to 3.3 causes the outlier identification routine to more aggressively identify outliers as shown in Output 37.5.2. The first table shows the critical values. The second table shows that three additive outliers and a level-shift have been included in the regression model. The third table shows how the inclusion of outliers in the model affects the ARMA parameters.

proc x12 data=sales date=date;
   var sales;
   transform function=log;
   arima model=((0,1,1) (0,1,1));
   outlier cv=3.3;
   estimate;
   x11;
   output out=outlier(obs=50) a1 a8  a8ao a8ls b1 d10;
run;
proc print data=outlier(obs=50);
run;

Output 37.5.2: PROC X12 Output When Outliers Are Identified

Automatic Outlier Identification

The X12 Procedure

Critical Values to use in Outlier
Detection
For Variable sales
Begin SEP1978
End AUG1990
Observations 144
Method Add One
AO Critical Value 3.3
LS Critical Value 3.3

Regression Model Parameter Estimates
For Variable sales
Type Parameter NoEst Estimate Standard
Error
t Value Pr > |t|
Automatically Identified AO JAN1981 Est 0.09590 0.02168 4.42 <.0001
  LS FEB1983 Est -0.09673 0.02488 -3.89 0.0002
  AO OCT1983 Est -0.08032 0.02146 -3.74 0.0003
  AO NOV1989 Est -0.10323 0.02480 -4.16 <.0001

Exact ARMA Maximum Likelihood Estimation
For Variable sales
Parameter Lag Estimate Standard
Error
t Value Pr > |t|
Nonseasonal MA 1 0.33205 0.08239 4.03 <.0001
Seasonal MA 12 0.49647 0.07676 6.47 <.0001



The first 50 observations of the A1, A8, A8AO, A8LS, B1, and D10 series are displayed in Output 37.5.3. You can confirm the following relationships from the data:

\[  A8 = A8AO \times A8LS  \]
\[  B1 = A1/A8  \]

The seasonal factors are stored in the variable sales_D10.

Output 37.5.3: PROC X12 Output Series Related to Outlier Detection

Automatic Outlier Identification

Obs DATE sales_A1 sales_A8 sales_A8AO sales_A8LS sales_B1 sales_D10
1 SEP78 112 1.10156 1.00000 1.10156 101.674 0.90496
2 OCT78 118 1.10156 1.00000 1.10156 107.121 0.94487
3 NOV78 132 1.10156 1.00000 1.10156 119.830 1.04711
4 DEC78 129 1.10156 1.00000 1.10156 117.107 1.00119
5 JAN79 121 1.10156 1.00000 1.10156 109.844 0.94833
6 FEB79 135 1.10156 1.00000 1.10156 122.553 1.06817
7 MAR79 148 1.10156 1.00000 1.10156 134.355 1.18679
8 APR79 148 1.10156 1.00000 1.10156 134.355 1.17607
9 MAY79 136 1.10156 1.00000 1.10156 123.461 1.07565
10 JUN79 119 1.10156 1.00000 1.10156 108.029 0.91844
11 JUL79 104 1.10156 1.00000 1.10156 94.412 0.81206
12 AUG79 118 1.10156 1.00000 1.10156 107.121 0.91602
13 SEP79 115 1.10156 1.00000 1.10156 104.397 0.90865
14 OCT79 126 1.10156 1.00000 1.10156 114.383 0.94131
15 NOV79 141 1.10156 1.00000 1.10156 128.000 1.04496
16 DEC79 135 1.10156 1.00000 1.10156 122.553 0.99766
17 JAN80 125 1.10156 1.00000 1.10156 113.475 0.94942
18 FEB80 149 1.10156 1.00000 1.10156 135.263 1.07172
19 MAR80 170 1.10156 1.00000 1.10156 154.327 1.18663
20 APR80 170 1.10156 1.00000 1.10156 154.327 1.18105
21 MAY80 158 1.10156 1.00000 1.10156 143.433 1.07383
22 JUN80 133 1.10156 1.00000 1.10156 120.738 0.91930
23 JUL80 114 1.10156 1.00000 1.10156 103.490 0.81385
24 AUG80 140 1.10156 1.00000 1.10156 127.093 0.91466
25 SEP80 145 1.10156 1.00000 1.10156 131.632 0.91302
26 OCT80 150 1.10156 1.00000 1.10156 136.171 0.93086
27 NOV80 178 1.10156 1.00000 1.10156 161.589 1.03965
28 DEC80 163 1.10156 1.00000 1.10156 147.972 0.99440
29 JAN81 172 1.21243 1.10065 1.10156 141.864 0.95136
30 FEB81 178 1.10156 1.00000 1.10156 161.589 1.07981
31 MAR81 199 1.10156 1.00000 1.10156 180.653 1.18661
32 APR81 199 1.10156 1.00000 1.10156 180.653 1.19097
33 MAY81 184 1.10156 1.00000 1.10156 167.036 1.06905
34 JUN81 162 1.10156 1.00000 1.10156 147.064 0.92446
35 JUL81 146 1.10156 1.00000 1.10156 132.539 0.81517
36 AUG81 166 1.10156 1.00000 1.10156 150.695 0.91148
37 SEP81 171 1.10156 1.00000 1.10156 155.234 0.91352
38 OCT81 180 1.10156 1.00000 1.10156 163.405 0.91632
39 NOV81 193 1.10156 1.00000 1.10156 175.206 1.03194
40 DEC81 181 1.10156 1.00000 1.10156 164.312 0.98879
41 JAN82 183 1.10156 1.00000 1.10156 166.128 0.95699
42 FEB82 218 1.10156 1.00000 1.10156 197.901 1.09125
43 MAR82 230 1.10156 1.00000 1.10156 208.795 1.19059
44 APR82 242 1.10156 1.00000 1.10156 219.688 1.20448
45 MAY82 209 1.10156 1.00000 1.10156 189.731 1.06355
46 JUN82 191 1.10156 1.00000 1.10156 173.391 0.92897
47 JUL82 172 1.10156 1.00000 1.10156 156.142 0.81476
48 AUG82 194 1.10156 1.00000 1.10156 176.114 0.90667
49 SEP82 196 1.10156 1.00000 1.10156 177.930 0.91200
50 OCT82 196 1.10156 1.00000 1.10156 177.930 0.89970



From the two previous examples, you can examine how outlier detection affects the seasonally adjusted series. Output 37.5.4 shows a plot of A1 versus B1 in the series where outliers are detected. B1 has been adjusted for the additive outliers and the level-shift.

proc sgplot data=outlier;
   series x=date y=sales_A1 / name='A1' markers
                              markerattrs=(color=red symbol='circle')
                              lineattrs=(color=red);
   series x=date y=sales_B1 / name='B1' markers
                              markerattrs=(color=black symbol='asterisk')
                              lineattrs=(color=black);
   yaxis label='Original and Outlier Adjusted Time Series';
run;

Output 37.5.4: Original Series and Outlier Adjusted Series

Original Series and Outlier Adjusted Series


Output 37.5.5 compares the seasonal factors (table D10) of the series unadjusted for outliers to the series adjusted for outliers. The seasonal factors are based on the B1 series.

data both;
   merge nooutlier(rename=(sales_D10=unadj_D10)) outlier;
run;

title 'Results of Outlier Identification on Final Seasonal Factors';
proc sgplot data=both;
   series x=date y=unadj_D10 / name='unadjusted' markers
                               markerattrs=(color=red symbol='circle')
                               lineattrs=(color=red)
                               legendlabel='Unadjusted for Outliers';
   series x=date y=sales_D10 / name='adjusted' markers
                               markerattrs=(color=blue symbol='asterisk')
                               lineattrs=(color=blue)
                               legendlabel='Adjusted for Outliers';
   yaxis label='Final Seasonal Factors';
run;

Output 37.5.5: Seasonal Factors Based on Original and Outlier Adjusted Series

Seasonal Factors Based on Original and Outlier Adjusted Series