Example 7.5: Using Diagnostics to Identify ARIMA models

The ARIMA Procedure

Example 7.5: Using Diagnostics to Identify ARIMA models

Fitting ARIMA models is as much an art as it is a science. The ARIMA procedure has diagnostic options to help tentatively identify the orders of both stationary and nonstationary ARIMA processes.

Consider the Series A in Box et al (1994), which consists of 197 concentration readings taken every two hours from a chemical process. Let SeriesA be a data set containing these readings in a variable named X. The following SAS statements use the SCAN option of the IDENTIFY statement to generate Output 7.5.1 and Output 7.5.2. See "The SCAN Method" for details of the SCAN method.


   proc arima data=SeriesA;
      identify var=x scan;
   run;

Output 7.5.1: Example of SCAN Tables

SERIES A: Chemical Process Concentration Readings

The ARIMA Procedure

Squared Canonical Correlation Estimates
Lags	MA 0	MA 1	MA 2	MA 3	MA 4	MA 5
AR 0	0.3263	0.2479	0.1654	0.1387	0.1183	0.1417
AR 1	0.0643	0.0012	0.0028	<.0001	0.0051	0.0002
AR 2	0.0061	0.0027	0.0021	0.0011	0.0017	0.0079
AR 3	0.0072	<.0001	0.0007	0.0005	0.0019	0.0021
AR 4	0.0049	0.0010	0.0014	0.0014	0.0039	0.0145
AR 5	0.0202	0.0009	0.0016	<.0001	0.0126	0.0001

SCAN Chi-Square[1] Probability Values
Lags	MA 0	MA 1	MA 2	MA 3	MA 4	MA 5
AR 0	<.0001	<.0001	<.0001	0.0007	0.0037	0.0024
AR 1	0.0003	0.6649	0.5194	0.9235	0.3993	0.8528
AR 2	0.2754	0.5106	0.5860	0.7346	0.6782	0.2766
AR 3	0.2349	0.9812	0.7667	0.7861	0.6810	0.6546
AR 4	0.3297	0.7154	0.7113	0.6995	0.5807	0.2205
AR 5	0.0477	0.7254	0.6652	0.9576	0.2660	0.9168

In Output 7.5.1, there is one (maximal) rectangular region in which all the elements are insignificant with 95% confidence. This region has a vertex at (1,1). Output 7.5.2 gives recommendations based on the significance level specified by the ALPHA=siglevel option.

Output 7.5.2: Example of SCAN Option Tentative Order Selection

The ARIMA Procedure

ARMA(p+d,q) Tentative Order Selection Tests
SCAN
p+d	q
1	1

(5% Significance Level)

Another order identification diagnostic is the extended sample autocorrelation function or ESACF method. See "The ESACF Method" for details of the ESACF method.

The following statements generate Output 7.5.3 and Output 7.5.4.


   proc arima data=SeriesA;
      identify var=x esacf;
   run;

Output 7.5.3: Example of ESACF Tables

The ARIMA Procedure

Extended Sample Autocorrelation Function
Lags	MA 0	MA 1	MA 2	MA 3	MA 4	MA 5
AR 0	0.5702	0.4951	0.3980	0.3557	0.3269	0.3498
AR 1	-0.3907	0.0425	-0.0605	-0.0083	-0.0651	-0.0127
AR 2	-0.2859	-0.2699	-0.0449	0.0089	-0.0509	-0.0140
AR 3	-0.5030	-0.0106	0.0946	-0.0137	-0.0148	-0.0302
AR 4	-0.4785	-0.0176	0.0827	-0.0244	-0.0149	-0.0421
AR 5	-0.3878	-0.4101	-0.1651	0.0103	-0.1741	-0.0231

ESACF Probability Values
Lags	MA 0	MA 1	MA 2	MA 3	MA 4	MA 5
AR 0	<.0001	<.0001	0.0001	0.0014	0.0053	0.0041
AR 1	<.0001	0.5974	0.4622	0.9198	0.4292	0.8768
AR 2	<.0001	0.0002	0.6106	0.9182	0.5683	0.8592
AR 3	<.0001	0.9022	0.2400	0.8713	0.8930	0.7372
AR 4	<.0001	0.8380	0.3180	0.7737	0.8913	0.6213
AR 5	<.0001	<.0001	0.0765	0.9142	0.1038	0.8103

In Output 7.5.3, there are three right-triangular regions in which all elements are insignificant at the 5% level. The triangles have vertices (1,1), (3,1), and (4,1). Since the triangle at (1,1) covers more insignificant terms, it is recommended first. Similarly, the remaining recommendations are ordered by the number of insignificant terms contained in the triangle. Output 7.5.4 gives recommendations based on the significance level specified by the ALPHA=siglevel option.

Output 7.5.4: Example of ESACF Option Tentative Order Selection

The ARIMA Procedure

ARMA(p+d,q) Tentative Order Selection Tests
ESACF
p+d	q
1	1
3	1
4	1

(5% Significance Level)

If you also specify the SCAN option in the same IDENTIFY statement, the two recommendations are printed side by side.


   proc arima data=SeriesA;
      identify var=x scan esacf;
   run;

Output 7.5.5: Example of SCAN and ESACF Option Combined

The ARIMA Procedure

ARMA(p+d,q) Tentative Order Selection Tests
SCAN		ESACF
p+d	q	p+d	q
1	1	1	1
		3	1
		4	1

(5% Significance Level)

From above, the autoregressive and moving average orders are tentatively identified by both SCAN and ESACF tables to be (p+d, q)=(1,1). Because both the SCAN and ESACF indicate a p+d term of 1, a unit root test should be used to determine whether this term is a unit root or an autoregressive term. Since a moving average term appears to be present, a large autoregressive term is appropriate for the Augmented Dickey-Fuller test for a unit root.

Submitting the following code generates Output 7.5.6.


   proc arima data=SeriesA;
      identify var=x stationarity=(adf=(5,6,7,8));
   run;

Output 7.5.6: Example of STATIONARITY Option Output

The ARIMA Procedure

Augmented Dickey-Fuller Unit Root Tests
Type	Lags	Rho	Pr < Rho	Tau	Pr < Tau	F	Pr > F
Zero Mean	5	0.0403	0.6913	0.42	0.8024
	6	0.0479	0.6931	0.63	0.8508
	7	0.0376	0.6907	0.49	0.8200
	8	0.0354	0.6901	0.48	0.8175
Single Mean	5	-18.4550	0.0150	-2.67	0.0821	3.67	0.1367
	6	-10.8939	0.1043	-2.02	0.2767	2.27	0.4931
	7	-10.9224	0.1035	-1.93	0.3172	2.00	0.5605
	8	-10.2992	0.1208	-1.83	0.3650	1.81	0.6108
Trend	5	-18.4360	0.0871	-2.66	0.2561	3.54	0.4703
	6	-10.8436	0.3710	-2.01	0.5939	2.04	0.7694
	7	-10.7427	0.3773	-1.90	0.6519	1.91	0.7956
	8	-10.0370	0.4236	-1.79	0.7081	1.74	0.8293

The preceding test results show that a unit root is very likely and that the series should be differenced. Based on this test and the previous results, an ARIMA(0,1,1) would be a good choice for a tentative model for Series A.

Using the recommendation that the series be differenced, the following statements generate Output 7.5.7.


   proc arima data=SeriesA;
      identify var=x(1) minic;
   run;

Output 7.5.7: Example of MINIC Table

The ARIMA Procedure

Minimum Information Criterion
Lags	MA 0	MA 1	MA 2	MA 3	MA 4	MA 5
AR 0	-2.05761	-2.3497	-2.32358	-2.31298	-2.30967	-2.28528
AR 1	-2.23291	-2.32345	-2.29665	-2.28644	-2.28356	-2.26011
AR 2	-2.23947	-2.30313	-2.28084	-2.26065	-2.25685	-2.23458
AR 3	-2.25092	-2.28088	-2.25567	-2.23455	-2.22997	-2.20769
AR 4	-2.25934	-2.2778	-2.25363	-2.22983	-2.20312	-2.19531
AR 5	-2.2751	-2.26805	-2.24249	-2.21789	-2.19667	-2.17426

The error series is estimated using an AR(7) model, and the minimum of this MINIC table is BIC(0,1). This diagnostic confirms the previous result indicating that an ARIMA(0,1,1) is a tentative model for Series A.

If you also specify the SCAN or MINIC option in the same IDENTIFY statement, the BIC associated with the SCAN table and ESACF table recommendations are listed.


   proc arima data=SeriesA;
      identify var=x(1) minic scan esacf;
   run;

Output 7.5.8: Example of SCAN, ESACF, MINIC Options Combined

The ARIMA Procedure

ARMA(p+d,q) Tentative Order Selection Tests
SCAN			ESACF
p+d	q	BIC	p+d	q	BIC
0	1	-2.3497	0	1	-2.3497
			1	1	-2.32345

(5% Significance Level)

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