This example shows how to use the OUTLIER statement to detect changes in the dynamics of the time series being modeled. The time series used here is discussed in de Jong and Penzer (1998). The data consist of readings of the annual flow volume of the Nile River at Aswan from 1871 to 1970. These data have also been studied by Cobb (1978). These studies indicate that river flow levels in the years 1877 and 1913 are strong candidates for additive outliers and that there was a shift in the flow levels starting from the year 1899. This shift in 1899 is attributed partly to the weather changes and partly to the start of construction work for a new dam at Aswan. The following DATA step statements create the input data set.
data nile; input level @@; year = intnx( 'year', '1jan1871'd, _n_-1 ); format year year4.; datalines; 1120 1160 963 1210 1160 1160 813 1230 1370 1140 995 935 1110 994 1020 960 1180 799 958 1140 1100 1210 1150 1250 1260 1220 1030 1100 774 840 ... more lines ...
The following program fits an ARIMA model, ARIMA(0,1,1), similar to the structural model suggested in de Jong and Penzer (1998). This model is also suggested by the usual correlation analysis of the series. By default, the OUTLIER statement requests detection of additive outliers and level shifts, assuming that the series follows the estimated model.
/*-- ARIMA(0, 1, 1) Model --*/ proc arima data=nile; identify var=level(1); estimate q=1 noint method=ml; outlier maxnum= 5 id=year; run;
The outlier detection output is shown in Output 7.6.1.
Output 7.6.1: ARIMA(0, 1, 1) Model
SERIES A: Chemical Process Concentration Readings |
Outlier Detection Summary | |
---|---|
Maximum number searched | 5 |
Number found | 5 |
Significance used | 0.05 |
Outlier Details | |||||
---|---|---|---|---|---|
Obs | Time ID | Type | Estimate | Chi-Square | Approx Prob>ChiSq |
29 | 1899 | Shift | -315.75346 | 13.13 | 0.0003 |
43 | 1913 | Additive | -403.97105 | 11.83 | 0.0006 |
7 | 1877 | Additive | -335.49351 | 7.69 | 0.0055 |
94 | 1964 | Additive | 305.03568 | 6.16 | 0.0131 |
18 | 1888 | Additive | -287.81484 | 6.00 | 0.0143 |
Note that the first three outliers detected are indeed the ones discussed earlier. You can include the shock signatures that correspond to these three outliers in the Nile data set as follows:
data nile; set nile; AO1877 = ( year = '1jan1877'd ); AO1913 = ( year = '1jan1913'd ); LS1899 = ( year >= '1jan1899'd ); run;
Now you can refine the earlier model by including these outliers. After examining the parameter estimates and residuals (not shown) of the ARIMA(0,1,1) model with these regressors, the following stationary MA1 model (with regressors) appears to fit the data well:
/*-- MA1 Model with Outliers --*/ proc arima data=nile; identify var=level crosscorr=( AO1877 AO1913 LS1899 ); estimate q=1 input=( AO1877 AO1913 LS1899 ) method=ml; outlier maxnum=5 alpha=0.01 id=year; run;
The relevant outlier detection process output is shown in Output 7.6.2. No outliers, at significance level 0.01, were detected.
Output 7.6.2: MA1 Model with Outliers
SERIES A: Chemical Process Concentration Readings |
Outlier Detection Summary | |
---|---|
Maximum number searched | 5 |
Number found | 0 |
Significance used | 0.01 |