In this example, the annual real output series is analyzed over the period 1901 to 1983 (Balke and Gordon, 1986, pp. 581–583). With the following DATA step, the original data are 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; 137.87 139.13 146.10 144.21 155.04 172.97 175.61 161.22 ... more lines ...
proc sgplot data=gnp noautolegend; scatter x=date y=y; xaxis grid values=('01jan1901'd '01jan1911'd '01jan1921'd '01jan1931'd '01jan1941'd '01jan1951'd '01jan1961'd '01jan1971'd '01jan1981'd '01jan1991'd); run;
Output 8.1.1: Real Output Series: 1901 – 1983
The (linear) trendstationary process is estimated using the following form:

where


The preceding trendstationary model assumes that uncertainty over future horizons is bounded since the error term, , has a finite variance. The maximum likelihood AR estimates from the statements that follow 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
Analysis of Real GNP 
Maximum Likelihood Estimates  

SSE  0.23954331  DFE  79 
MSE  0.00303  Root MSE  0.05507 
SBC  230.39355  AIC  240.06891 
MAE  0.04016596  AICC  239.55609 
MAPE  0.69458594  HQC  236.18189 
Log Likelihood  124.034454  Regress RSquare  0.8645 
DurbinWatson  1.9935  Total RSquare  0.9947 
Observations  83 
Parameter Estimates  

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:

where and . In general, the differencestationary process is written as

where is the lag operator. You can observe that the class of a differencestationary process should have at least one unit root in the AR polynomial .
The DickeyFuller 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 DickeyFuller test:

where . The test statistic is the usual t ratio for the parameter estimate , but the does not follow a t distribution.
The following code performs the augmented DickeyFuller test with and we are interesting in the test results in the linear time trend case since the previous plot reveals there is a linear trend.
proc autoreg data = gnp; model y = / stationarity =(adf =3); run;
The augmented DickeyFuller test indicates that the output series may have a differencestationary process. The statistic Tau with linear time trend has a value of and its pvalue is . The statistic Rho has a pvalue of which also indicates the null of unit root is accepted at the 5% level. (See Output 8.1.3.)
Output 8.1.3: Augmented DickeyFuller Test Results
Analysis of Real GNP 
Augmented DickeyFuller Unit Root Tests  

Type  Lags  Rho  Pr < Rho  Tau  Pr < Tau  F  Pr > F 
Zero Mean  3  0.3827  0.7732  3.3342  0.9997  
Single Mean  3  0.1674  0.9465  0.2046  0.9326  5.7521  0.0211 
Trend  3  18.0246  0.0817  2.6190  0.2732  3.4472  0.4957 
The AR(1) model for the differenced series DY is estimated using the maximum likelihood method for the period 1902 to 1983. The differencestationary process is written


The estimated value of is and that of is 0.0293. All estimated values are statistically significant. The PROC step follows:
proc autoreg data=gnp; model dy = / nlag=1 method=ml; run;
The printed output produced by the PROC step is shown in Output 8.1.4.
Output 8.1.4: Estimating the Differenced Series with AR(1) Error
Analysis of Real GNP 
Maximum Likelihood Estimates  

SSE  0.27107673  DFE  80 
MSE  0.00339  Root MSE  0.05821 
SBC  226.77848  AIC  231.59192 
MAE  0.04333026  AICC  231.44002 
MAPE  153.637587  HQC  229.65939 
Log Likelihood  117.795958  Regress RSquare  0.0000 
DurbinWatson  1.9268  Total RSquare  0.0900 
Observations  82 
Parameter Estimates  

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 