Once you determine that autocorrelation correction is needed, you must select the order of the autoregressive error model to use. One way to select the order of the autoregressive error model is stepwise autoregression. The stepwise autoregression method initially fits a highorder model with many autoregressive lags and then sequentially removes autoregressive parameters until all remaining autoregressive parameters have significant t tests.
To use stepwise autoregression, specify the BACKSTEP option, and specify a large order with the NLAG= option. The following statements show the stepwise feature, using an initial order of 5:
/* stepwise autoregression */ proc autoreg data=a; model y = time / method=ml nlag=5 backstep; run;
The results are shown in Figure 8.9.
Figure 8.9: Stepwise Autoregression
Forecasting Autocorrelated Time Series 
Dependent Variable  y 

Ordinary Least Squares Estimates  

SSE  214.953429  DFE  34 
MSE  6.32216  Root MSE  2.51439 
SBC  173.659101  AIC  170.492063 
MAE  2.01903356  AICC  170.855699 
MAPE  12.5270666  HQC  171.597444 
DurbinWatson  0.4752  Regress RSquare  0.8200 
Total RSquare  0.8200 
Parameter Estimates  

Variable  DF  Estimate  Standard Error 
t Value  Approx Pr > t 
Intercept  1  8.2308  0.8559  9.62  <.0001 
time  1  0.5021  0.0403  12.45  <.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  5.9709  1.000000   ******************** 
1  4.5169  0.756485   ***************  
2  2.0241  0.338995   *******  
3  0.4402  0.073725   *  
4  2.1175  0.354632   *******  
5  2.8534  0.477887   **********  
Backward Elimination of Autoregressive Terms 


Lag  Estimate  t Value  Pr > t 
4  0.052908  0.20  0.8442 
3  0.115986  0.57  0.5698 
5  0.131734  1.21  0.2340 
The estimates of the autocorrelations are shown for 5 lags. The backward elimination of autoregressive terms report shows that the autoregressive parameters at lags 3, 4, and 5 were insignificant and eliminated, resulting in the secondorder model shown previously in Figure 8.4. By default, retained autoregressive parameters must be significant at the 0.05 level, but you can control this with the SLSTAY= option. The remainder of the output from this example is the same as that in Figure 8.3 and Figure 8.4, and it is not repeated here.
The stepwise autoregressive process is performed using the YuleWalker method. The maximum likelihood estimates are produced after the order of the model is determined from the significance tests of the preliminary YuleWalker estimates.
When using stepwise autoregression, it is a good idea to specify an NLAG= option value larger than the order of any potential seasonality, since seasonality produces autocorrelation at the seasonal lag. For example, for monthly data use NLAG=13, and for quarterly data use NLAG=5.
In the previous example, the BACKSTEP option dropped lags 3, 4, and 5, leaving a secondorder model. However, in other cases a parameter at a longer lag may be kept while some smaller lags are dropped. For example, the stepwise autoregression method might drop lags 2, 3, and 5 but keep lags 1 and 4. This is called a subset model, since the number of estimated autoregressive parameters is lower than the order of the model.
Subset models are common for seasonal data and often correspond to factored autoregressive models. A factored model is the product of simpler autoregressive models. For example, the best model for seasonal monthly data may be the combination of a firstorder model for recent effects with a 12thorder subset model for the seasonality, with a single parameter at lag 12. This results in a 13thorder subset model with nonzero parameters at lags 1, 12, and 13. See Chapter 7: The ARIMA Procedure, for further discussion of subset and factored autoregressive models.
You can specify subset models with the NLAG= option. List the lags to include in the autoregressive model within parentheses. The following statements show an example of specifying the subset model resulting from the combination of a firstorder process for recent effects with a fourthorder seasonal process:
/* specifying the lags */ proc autoreg data=a; model y = time / nlag=(1 4 5); run;
The MODEL statement specifies the following fifthorder autoregressive error model:

