PROC AUTOREG: Missing Values :: SAS/ETS(R) 9.22 User's Guide

The AUTOREG Procedure

Example 8.4 Missing Values

In this example, a pure autoregressive error model with no regressors is used to generate 50 values of a time series. Approximately 15% of the values are randomly chosen and set to missing. The following statements generate the data:

title  'Simulated Time Series with Roots:';
title2 ' (X-1.25)(X**4-1.25)';
title3 'With 15% Missing Values';
data ar;
   do i=1 to 550;
      e = rannor(12345);
      n = sum( e, .8*n1, .8*n4, -.64*n5 );  /* ar process  */
      y = n;
      if ranuni(12345) > .85 then y = .;    /* 15% missing */
      n5=n4; n4=n3; n3=n2; n2=n1; n1=n;     /* set lags    */
      if i>500 then output;
   end;
run;

The model is estimated using maximum likelihood, and the residuals are plotted with 99% confidence limits. The PARTIAL option prints the partial autocorrelations. The following statements fit the model:

proc autoreg data=ar partial;
   model y = / nlag=(1 4 5) method=ml;
   output out=a predicted=p residual=r ucl=u lcl=l alphacli=.01;
run;

The printed output produced by the AUTOREG procedure is shown in Output 8.4.1 and Output 8.4.2. Note: the plot Output 8.4.2 can be viewed in the Autoreg.Model.FitDiagnosticPlots category by selecting View $\text{[math]}$ Results.

Output 8.4.1 Autocorrelation-Corrected Regression Results

Simulated Time Series with Roots:

(X-1.25)(X**4-1.25)

With 15% Missing Values

The AUTOREG Procedure

Dependent Variable	y

Ordinary Least Squares Estimates
SSE	182.972379	DFE	40
MSE	4.57431	Root MSE	2.13876
SBC	181.39282	AIC	179.679248
MAE	1.80469152	AICC	179.781813
MAPE	270.104379	HQC	180.303237
Durbin-Watson	1.3962	Regress R-Square	0.0000
		Total R-Square	0.0000

Parameter Estimates
Variable	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
Intercept	1	-2.2387	0.3340	-6.70	<.0001

Estimates of 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
0	4.4627	1.000000	\| \|********************\|
1	1.4241	0.319109	\| \|****** \|
2	1.6505	0.369829	\| \|******* \|
3	0.6808	0.152551	\| \|*** \|
4	2.9167	0.653556	\| \|************* \|
5	-0.3816	-0.085519	\| **\| \|

Partial Autocorrelations
1	0.319109
4	0.619288
5	-0.821179

Preliminary MSE	0.7609

Estimates of Autoregressive Parameters
Lag	Coefficient	Standard Error	t Value
1	-0.733182	0.089966	-8.15
4	-0.803754	0.071849	-11.19
5	0.821179	0.093818	8.75

Expected Autocorrelations
Lag	Autocorr
0	1.0000
1	0.4204
2	0.2480
3	0.3160
4	0.6903
5	0.0228

Algorithm converged.

Maximum Likelihood Estimates
SSE	48.4396756	DFE	37
MSE	1.30918	Root MSE	1.14419
SBC	146.879013	AIC	140.024725
MAE	0.88786192	AICC	141.135836
MAPE	141.377721	HQC	142.520679
Durbin-Watson	2.9457	Regress R-Square	0.0000
		Total R-Square	0.7353

Parameter Estimates
Variable	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
Intercept	1	-2.2370	0.5239	-4.27	0.0001
AR1	1	-0.6201	0.1129	-5.49	<.0001
AR4	1	-0.7237	0.0914	-7.92	<.0001
AR5	1	0.6550	0.1202	5.45	<.0001

Expected Autocorrelations
Lag	Autocorr
0	1.0000
1	0.4204
2	0.2423
3	0.2958
4	0.6318
5	0.0411

Autoregressive parameters assumed given
Variable	DF	Estimate	Standard Error	t Value	Approx Pr > \|t\|
Intercept	1	-2.2370	0.5225	-4.28	0.0001

Output 8.4.2 Diagnostic Plots

The following statements plot the residuals and confidence limits:

data reshape1;
   set a;
   miss = .;
   if r=. then do;
      miss = p;
      p = .;
   end;
run;

title 'Predicted Values and Confidence Limits';

proc sgplot data=reshape1 NOAUTOLEGEND;
   band x=i upper=u lower=l;
   scatter y=miss x=i/ MARKERATTRS =(symbol=x color=red);
   series y=p x=i/markers MARKERATTRS =(color=blue) lineattrs=(color=blue);
run;

The plot of the predicted values and the upper and lower confidence limits is shown in Output 8.4.3. Note that the confidence interval is wider at the beginning of the series (when there are no past noise values to use in the forecast equation) and after missing values where, again, there is an incomplete set of past residuals.

Output 8.4.3 Plot of Predicted Values and Confidence Interval

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