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, Jenkins, and Reinsel (1994), which consists of 197 concentration readings taken every two hours from a chemical process. Let Series A be a data set that contains 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.

```/*-- Order Identification Diagnostic with 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
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:

```/*-- Order Identification Diagnostic with ESACF Method --*/
proc arima data=SeriesA;
identify var=x esacf;
run;
```

Output 7.5.3 Example of ESACF Tables
 SERIES A: Chemical Process Concentration Readings

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

(5% Significance Level)

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

```/*-- Combination of SCAN and ESACF Methods --*/
proc arima data=SeriesA;
identify var=x scan esacf;
run;
```

Output 7.5.5 shows the results.

Output 7.5.5 Example of SCAN and ESACF Option Combined
 SERIES A: Chemical Process Concentration Readings

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 Output 7.5.5, the autoregressive and moving-average orders are tentatively identified by both SCAN and ESACF tables to be ()=(1,1). Because both the SCAN and ESACF indicate a term of 1, a unit root test should be used to determine whether this autoregressive term is a unit root. 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 statements generates Output 7.5.6:

```/*-- Augmented Dickey-Fuller Unit Root Tests --*/
proc arima data=SeriesA;
run;
```

Output 7.5.6 Example of STATIONARITY Option Output
 SERIES A: Chemical Process Concentration Readings

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 given that none of the p-values are small enough to cause you to reject the null hypothesis that the series has a unit root. Based on this test and the previous results, the series should be differenced, and 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:

```/*-- Minimum Information Criterion --*/
proc arima data=SeriesA;
identify var=x(1) minic;
run;
```

Output 7.5.7 Example of MINIC Table
 SERIES A: Chemical Process Concentration Readings

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 by using an AR(7) model, and the minimum of this MINIC table is . This diagnostic confirms the previous result which indicates 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 as follows, the BIC associated with the SCAN table and ESACF table recommendations is listed. Output 7.5.8 shows the results.

```/*-- Combination of MINIC, SCAN, and ESACF Options --*/
proc arima data=SeriesA;
identify var=x(1) minic scan esacf;
run;
```

Output 7.5.8 Example of SCAN, ESACF, MINIC Options Combined
 SERIES A: Chemical Process Concentration Readings

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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