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 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)TentativeOrder SelectionTests 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)TentativeOrder SelectionTests 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) TentativeOrder SelectionTests 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;
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 SelectionTests 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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