Example 3.1: Nile Data

The ARIMA Procedure

Example 3.1: Nile Data

This example 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 levels in the years 1877 and 1913, that is, the 7th and 43rd measurements, are strong candidates for additive outliers, and that there was a shift in the flow levels starting from the year 1899. The year 1899 corresponds to the 29th observation. 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.

   data nile;
      input level @@;
      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
   874   694   940  833   701   916   692  1020   1050  969
   831   726   456  824   702   1120  1100 832    764   821
   768   845   864  862   698   845   744  796    1040  759
   781   865   845  944   984   897   822  1010   771   676
   649   846   812  742   801   1040  860  874    848   890
   744   749   838  1050  918   986   797  923    975   815
   1020  906   901  1170  912   746   919  718    714   740
   ;

You can start the modeling process with the ARIMA(0, 1, 1) model, an ARIMA model close to the Structural model suggested in de Jong and Penzer (1998), and examine the parameter estimates, the residual autocorrelations, and the outliers.

   proc arima data=nile;
      identify var=level(1) noprint;
      estimate q=1 noint method=ml plot;
      outlier maxnum=5;
   run;

A portion of the estimation and the outlier detection output is shown in Output 3.1.1.

Output 3.1.1: Output from Fitting ARIMA(0, 1, 1) Model

The ARIMA Procedure

Maximum Likelihood Estimation
Parameter	Estimate	Standard Error	t Value	Approx Pr > \|t\|	Lag
MA1,1	0.73271	0.07132	10.27	<.0001	1

Variance Estimate	20810.22
Std Error Estimate	144.2575
AIC	1267.091
SBC	1269.686
Number of Residuals	99

Model for variable level
Period(s) of Differencing	1

No mean term in this model.

Outlier Summary
Maximum number searched	5
Number found	5
Significance used	0.05

Outlier Table
Obs	Type	Estimate	Chi-Square	Approx Prob>ChiSq
29	Shift	-315.75346	13.13	0.0003
43	Additive	-403.97105	11.83	0.0006
7	Additive	-335.49351	7.69	0.0055
94	Additive	305.03568	6.16	0.0131
18	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 corresponding to these three outliers in the Nile data set.

   data nile;
      set nile;
         if _n_ = 7 then AO7 = 1.0;
         else AO7 = 0.0;
         if _n_ = 43 then AO43 = 1.0;
         else AO43 = 0.0;
         if _n_ >= 29 then LS29 = 1.0;
         else LS29 = 0.0;
   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.

   proc arima data=nile;
      identify var=level crosscorr=( AO7 AO43 LS29 ) noprint;
      estimate q=1 noint 
         input=(AO7 AO43 LS29) method=ml plot;
      outlier maxnum=5 alpha=0.01;
   run;

The relevant estimation output is shown in Output 3.1.2. No outliers, at significance level 0.01, were detected.

Output 3.1.2: MA1 Model with Outliers

The ARIMA Procedure

Maximum Likelihood Estimation
Parameter	Estimate	Standard Error	t Value	Approx Pr > \|t\|	Lag	Variable	Shift
MU	1109.9	26.65625	41.64	<.0001	0	level	0
MA1,1	-0.19584	0.10148	-1.93	0.0536	1	level	0
NUM1	-319.63689	116.50628	-2.74	0.0061	0	AO7	0
NUM2	-376.48708	116.11662	-3.24	0.0012	0	AO43	0
NUM3	-255.20007	31.32609	-8.15	<.0001	0	LS29	0

Constant Estimate	1109.861
Variance Estimate	13761.25
Std Error Estimate	117.3083
AIC	1241.659
SBC	1254.685
Number of Residuals	100

Autocorrelation Check of Residuals
To Lag	Chi-Square	DF	Pr > ChiSq	Autocorrelations
6	3.43	5	0.6334	-0.000	0.007	0.074	-0.100	-0.068	-0.109
12	9.73	11	0.5552	-0.070	0.100	-0.107	-0.123	-0.083	-0.087
18	14.36	17	0.6417	0.079	-0.071	-0.025	0.132	-0.007	0.094
24	17.22	23	0.7983	0.008	-0.058	0.055	-0.049	-0.073	-0.087

Outlier Summary
Maximum number searched	5
Number found	0
Significance used	0.01

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