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The AUTOREG Procedure

Example 8.1: Analysis of Real Output Series

In this example, the annual real output series is analyzed over the period 1901 to 1983 (Gordon 1986, pp 781-783). With the DATA step, the original data is transformed using the natural logarithm, and the differenced series DY is created for further analysis. The log of real output is plotted in Output 8.1.1.


   title 'Analysis of Real GNP';
   data gnp;
      date = intnx( 'year', '01jan1901'd, _n_-1 );
      format date year4.;
      input x @@;
      y  = log(x);
      dy = dif(y);
      t  = _n_;
      label y  = 'Real GNP'
            dy = 'First Difference of Y'
            t  = 'Time Trend';
   datalines;
   ... datalines omitted ...
   ;
   
   proc gplot data=gnp;
      plot y * date /
          haxis='01jan1901'd '01jan1911'd '01jan1921'd '01jan1931'd
                '01jan1941'd '01jan1951'd '01jan1961'd '01jan1971'd
                '01jan1981'd '01jan1991'd;
      symbol i=join;
   run;

Output 8.1.1: Real Output Series: 1901 - 1983
autex01a.gif (3200 bytes)

The (linear) trend-stationary process is estimated using the following form:

y_{t} = {\beta}_{0} + {\beta}_{1} t + {\nu}_{t}

where

{\nu}_{t} = {\epsilon}_{t} - {\varphi}_{1}
 {\nu}_{t-1} - {\varphi}_{2} {\nu}_{t-2}
{\epsilon}_{t} {\sim} \rm{IN}(0, {\sigma}_{{\epsilon}})

The preceding trend-stationary model assumes that uncertainty over future horizons is bounded since the error term, {{\nu}_{t}}, has a finite variance. The maximum likelihood AR estimates are shown in Output 8.1.2.


   proc autoreg data=gnp;
      model y = t / nlag=2 method=ml;
   run;

Output 8.1.2: Estimating the Linear Trend Model

The AUTOREG Procedure

Maximum Likelihood Estimates
SSE 0.23954331 DFE 79
MSE 0.00303 Root MSE 0.05507
SBC -230.39355 AIC -240.06891
Regress R-Square 0.8645 Total R-Square 0.9947
Durbin-Watson 1.9935    

Variable DF Estimate Standard Error t Value Approx
Pr > |t|
Variable Label
Intercept 1 4.8206 0.0661 72.88 <.0001  
t 1 0.0302 0.001346 22.45 <.0001 Time Trend
AR1 1 -1.2041 0.1040 -11.58 <.0001  
AR2 1 0.3748 0.1039 3.61 0.0005  

Autoregressive parameters assumed given.
Variable DF Estimate Standard Error t Value Approx
Pr > |t|
Variable Label
Intercept 1 4.8206 0.0661 72.88 <.0001  
t 1 0.0302 0.001346 22.45 <.0001 Time Trend


Nelson and Plosser (1982) failed to reject the hypothesis that macroeconomic time series are nonstationary and have no tendency to return to a trend line. In this context, the simple random walk process can be used as an alternative process:

y_{t} = {\beta}_{0} + y_{t-1} + {\nu}_{t}

where { {\nu}_{t}= {\epsilon}_{t}} and y0=0. In general, the difference-stationary process is written as

{\phi}(L)(1-L) y_{t} = {\beta}_{0}{\phi}(1) + {\theta}(L)
 {\epsilon}_{t}

where L is the lag operator. You can observe that the class of a difference-stationary process should have at least one unit root in the AR polynomial {{\phi}(L)(1-L)}.

The Dickey-Fuller procedure is used to test the null hypothesis that the series has a unit root in the AR polynomial. Consider the following equation for the augmented Dickey-Fuller test:

{\Delta} y_{t} = {\beta}_{0} + {\delta}t +
 {\beta}_{1} y_{t-1} + \sum_{i=1}^m{{\gamma}_{i}{\Delta} y_{t-i}} + {\epsilon}_{t}

where {{\Delta}=1-L}. The test statistic { {\tau}_{{\tau}}} is the usual t ratio for the parameter estimate { \hat{{\beta}}_{1}}, but the { {\tau}_{{\tau}}} does not follow a t distribution.

The %DFTEST macro computes the test statistic { {\tau}_{{\tau}}} and its p value to perform the Dickey-Fuller test. The default value of m is 3, but you can specify m with the AR= option. The option TREND=2 implies that the Dickey-Fuller test equation contains linear time trend. See Chapter 4, "SAS Macros and Functions," for details.


   %dftest(gnp,y,trend=2,outstat=stat1)
   
   proc print data=stat1;
   run;

The augmented Dickey-Fuller test indicates that the output series may have a difference-stationary process. The statistic _TAU_ has a value of -2.61903 and its p-value is 0.29104. See Output 8.1.3.

Output 8.1.3: Augmented Dickey-Fuller Test Results

Obs _TYPE_ _STATUS_ _DEPVAR_ _NAME_ _MSE_ Intercept AR_V time DLAG_V AR_V1 AR_V2 AR_V3 _NOBS_ _TAU_ _TREND_ _DLAG_ _PVALUE_
1 OLS 0 Converged AR_V   .003198469 0.76919 -1 0.004816233 -0.15629 0.37194 0.025483 -0.082422 79 -2.61903 2 1 0.27321
2 COV 0 Converged AR_V Intercept .003198469 0.08085 . 0.000513286 -0.01695 0.00549 0.008422 0.010556 79 -2.61903 2 1 0.27321
3 COV 0 Converged AR_V time .003198469 0.00051 . 0.000003387 -0.00011 0.00004 0.000054 0.000068 79 -2.61903 2 1 0.27321
4 COV 0 Converged AR_V DLAG_V .003198469 -0.01695 . -.000108543 0.00356 -0.00120 -0.001798 -0.002265 79 -2.61903 2 1 0.27321
5 COV 0 Converged AR_V AR_V1 .003198469 0.00549 . 0.000035988 -0.00120 0.01242 -0.003455 0.002095 79 -2.61903 2 1 0.27321
6 COV 0 Converged AR_V AR_V2 .003198469 0.00842 . 0.000054197 -0.00180 -0.00346 0.014238 -0.002910 79 -2.61903 2 1 0.27321
7 COV 0 Converged AR_V AR_V3 .003198469 0.01056 . 0.000067710 -0.00226 0.00209 -0.002910 0.013538 79 -2.61903 2 1 0.27321


The AR(1) model for the differenced series DY is estimated using the maximum likelihood method for the period 1902 to 1983. The difference-stationary process is written

{\Delta} y_{t} = {\beta}_{0} + {\nu}_{t}
{\nu}_{t} = {\epsilon}_{t}
- {\varphi}_{1} {\nu}_{t-1}

The estimated value of {\varphi}_{1} is -0.297 and that of {{\beta}_{0}} is 0.0293. All estimated values are statistically significant.


   proc autoreg data=gnp;
      model dy = / nlag=1 method=ml;
   run;

Output 8.1.4: Estimating the Differenced Series with AR(1) Error

The AUTOREG Procedure

Maximum Likelihood Estimates
SSE 0.27107673 DFE 80
MSE 0.00339 Root MSE 0.05821
SBC -226.77848 AIC -231.59192
Regress R-Square 0.0000 Total R-Square 0.0900
Durbin-Watson 1.9268    

Variable DF Estimate Standard Error t Value Approx
Pr > |t|
Intercept 1 0.0293 0.009093 3.22 0.0018
AR1 1 -0.2967 0.1067 -2.78 0.0067

Autoregressive parameters assumed given.
Variable DF Estimate Standard Error t Value Approx
Pr > |t|
Intercept 1 0.0293 0.009093 3.22 0.0018

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