Stepwise Autoregression

The AUTOREG Procedure

Stepwise Autoregression

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 high-order 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:


   proc autoreg data=a;
      model y = time / method=ml nlag=5 backstep;
   run;

The results are shown in Figure 8.9.

The AUTOREG Procedure

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
Regress R-Square	0.8200	Total R-Square	0.8200
Durbin-Watson	0.4752

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

Figure 8.9: Stepwise Autoregression

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 second-order model shown previously in Figure 8.4. By default, retained autoregressive parameters must be significant at the .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 Yule-Walker method. The maximum likelihood estimates are produced after the order of the model is determined from the significance tests of the preliminary Yule-Walker 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.

Subset and Factored Models

In the previous example, the BACKSTEP option dropped lags 3, 4, and 5, leaving an order 2 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 smaller 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 first-order model for recent effects with a twelfth-order subset model for the seasonality, with a single parameter at lag 12. This results in an order 13 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 first-order process for recent effects with a fourth-order seasonal process:


   proc autoreg data=a;
      model y = time / nlag=(1 4 5);
   run;

The MODEL statement specifies the following fifth-order autoregressive error model:

$y_{t} = a + b t + {\nu}_{t}$

${\nu}_{t} = - {\varphi}_{1}{\nu}_{t-1} - {\varphi}_{4}{\nu}_{t-4} - {\varphi}_{5}{\nu}_{t-5} + {\epsilon}_{t}$

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