Diagnosing and Modeling Autocorrelation

Specialized Control Charts

Diagnosing and Modeling Autocorrelation

You can diagnose autocorrelation with an autocorrelation plot created with the ARIMA procedure.

   proc arima data=chemical;
      identify var = xt;
   run;

Refer to SAS/ETS User's Guide for details on the ARIMA procedure. The plot, shown in Figure 49.2, indicates that the data are highly autocorrelated with a lag 1 autocorrelation of 0.83.

The ARIMA Procedure

Autocorrelations
Lag	Covariance	Correlation	-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1	Std Error
0	48.348400	1.00000	\| \|********************\|	0
1	40.141884	0.83026	\| . \|***************** \|	0.100000
2	34.732168	0.71837	\| . \|************** \|	0.154229
3	29.950852	0.61948	\| . \|************ \|	0.184683
4	24.739536	0.51169	\| . \|********** \|	0.204409
5	20.594420	0.42596	\| . \|********* \|	0.216840
6	18.427704	0.38114	\| . \|********. \|	0.225052
7	17.400188	0.35989	\| . \|******* . \|	0.231417
8	17.621272	0.36446	\| . \|******* . \|	0.236948
9	18.363756	0.37982	\| . \|******** . \|	0.242489
10	16.754040	0.34653	\| . \|******* . \|	0.248367
11	16.844924	0.34841	\| . \|******* . \|	0.253156
12	17.137208	0.35445	\| . \|******* . \|	0.257906
13	16.884092	0.34922	\| . \|******* . \|	0.262732
14	17.927976	0.37081	\| . \|******* . \|	0.267334
15	16.801860	0.34752	\| . \|******* . \|	0.272429
16	17.076544	0.35320	\| . \|******* . \|	0.276826
17	17.815028	0.36847	\| . \|******* . \|	0.281296
18	16.501312	0.34130	\| . \|******* . \|	0.286082
19	14.662196	0.30326	\| . \|****** . \|	0.290126
20	12.612280	0.26086	\| . \|***** . \|	0.293278
21	11.105364	0.22969	\| . \|***** . \|	0.295590
22	8.891648	0.18391	\| . \|**** . \|	0.297369
23	6.794132	0.14052	\| . \|*** . \|	0.298504
24	4.732816	0.09789	\| . \|** . \|	0.299165

"." marks two standard errors

Partial Autocorrelations
Lag	Correlation	-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
1	0.83026	\| . \|***************** \|
2	0.09346	\| . \|** . \|
3	0.00385	\| . \| . \|
4	-0.07340	\| . *\| . \|
5	-0.00278	\| . \| . \|
6	0.09013	\| . \|** . \|
7	0.08781	\| . \|** . \|
8	0.10327	\| . \|** . \|
9	0.07240	\| . \|* . \|
10	-0.11637	\| . **\| . \|
11	0.08210	\| . \|** . \|
12	0.07580	\| . \|** . \|
13	0.04429	\| . \|* . \|
14	0.11661	\| . \|** . \|
15	-0.10446	\| . **\| . \|
16	0.07703	\| . \|** . \|
17	0.07376	\| . \|* . \|
18	-0.07080	\| . *\| . \|
19	-0.02814	\| . *\| . \|
20	-0.08559	\| . **\| . \|
21	0.01962	\| . \| . \|
22	-0.04599	\| . *\| . \|
23	-0.07878	\| . **\| . \|
24	-0.02303	\| . \| . \|

Figure 49.2: Autocorrelation Plots for Chemical Data

The partial autocorrelation plot in Figure 49.2 suggests that the data can be modeled with a first-order autoregressive model, commonly referred to as an AR(1) model.

You can fit this model with the ARIMA procedure. The results in Figure 49.3 show that the equation of the fitted model is $\tilde{x}_{t} = 13.05 + 0.847 \tilde{x}_{t-1}$ .

   proc arima data=chemical;
      identify var=xt;
      estimate p=1 method=ml;
   run;

The ARIMA Procedure

Maximum Likelihood Estimation
Parameter	Estimate	Standard Error	t Value	Approx Pr > \|t\|	Lag
MU	85.28375	2.32973	36.61	<.0001	0
AR1,1	0.84694	0.05221	16.22	<.0001	1

Constant Estimate	13.05329
Variance Estimate	14.27676
Std Error Estimate	3.77846
AIC	552.8942
SBC	558.1045
Number of Residuals	100

Figure 49.3: Fitted AR(1) Model

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