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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

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

where

The preceding trend-stationary model assumes that uncertainty over future horizons is bounded since the error term, , 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 ApproxPr > |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 ApproxPr > |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 y0=0. In general, the difference-stationary process is written as

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 .

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:

where . The test statistic is the usual t ratio for the parameter estimate , but the does not follow a t distribution.

The %DFTEST macro computes the test statistic 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

The estimated value of is -0.297 and that of 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 ApproxPr > |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 ApproxPr > |t| Intercept 1 0.0293 0.009093 3.22 0.0018

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