Proc ARIMA to test Stationary.

Stationarity is a mathematical and statistical property of time series data. Much of the probability theory of time series analysis is based on the assumption that time series are stationary. If either the mean or the variance is not constant, then the time series is not stationary.

If the characteristic equation has a unit root, e.g., 1-pi1*B-pi2*B^2=1 for AR(2), it means that B=1 is a solution so then the mean drops out and the series is non-stationary. For ARIMA models, non-stationary is equivalent to one or more unit roots in the characteristic polynomial.

First, a systematic change in mean. If a series does not seem to have a constant mean (part of the definition of stationary) when graphed, that is a visible symptom of non-stationarity.

Second, a systematic change in variance. Log transformation could be able to make the variance constant. And also, log transformations are for removing exponential trends and other types of nonlinear behavior in time series data.

Third, a seasonal or periodic variation. The symptom had to do with the plot of the lag j correlation against the lag number j. Symbolizing the series value at time t by Yt, the lag j correlation is the correlation between Yt-j and Yt. If the plot of the estimated autocorrelations dies down very slowly with increasing j, this provides a second visual symptom of nonstationarity. Meanwhile, if your original time series has a sine-wave pattern, you may find it useful to use sinusoidal components as predictor variables. You can use spectural analysis to detect the presence of sinusoidal components.

There are three methods of dealing with non-stationary time series listed above: differencing, using log transformations, using sinusoidal components. Differencing is the most common method of handing non-stationary means, however, other methods are preferable for some types of series.

Differencing includes first difference, second difference, span-12 difference(for monthly data), span-4 difference(for quarterly data), etc. In many of these cases, it appeared that the series of first differences Yt - Yt-1 did seem to be stationary. Sometimes you may mistakenly take a difference when the data so not require it, which is over-differencing. If the inverse auto-correlation function (IACF) plot from the identification phase output of PROC ARIMA tails off very slowly, you may have over-differencing.

Note that if is a pure AR series, then the IACF of Yt is the same as the ACF of a pure MA process and will cut off after a few lags, i.e. the IACF behaves like the PACF of Yt .On the other hand, if is a pure MA series, then the IACF of Yt is the same as the ACF of a pure AR process and IACF will decline exponentially, i.e. the IACF again behaves like the PACF of Yt.

The Dickey-Fuller Unit Root test involves a regression of delta_Y on Y_{t-1}-Y^bar, delta_{Y-1}, ..., delta_{Y-p}, where delta indicates a first difference, i.e., Yt-Yt-1 on Y{t-1}-Y^bar, Yt-1-Yt-2, Yt-2-Yt-3,..., Yt-p-Yt-p-1. The t statistic for the parameter estimate of Y_{t-1}-Y^bar provides the statistical test of the stationary hypothesis and the distribution is call Tau. If the parameter estimate is negative and significant, the series is stationary. Otherwise, the series is not stationary, that is, it has a unit root and need to take difference.

proc arima data=predicted_series ;
i var=severity nlag=12 stationarity=(ADF=(0,1));
run;

/*
Augmented Dickey-Fuller Unit Root Tests

Type Lags Rho Pr < Rho Tau Pr < Tau F Pr > F

Zero Mean 0 0.1562 0.7175 0.11 0.7167
1 0.7651 0.8663 0.82 0.8868
Single Mean 0 -7.2238 0.2519 -2.10 0.2460 2.72 0.3774
1 -3.7934 0.5563 -1.56 0.4996 2.42 0.4546
Trend 0 -66.2436 0.0004 -6.92 <.0001 24.02 0.0010
1 -54.6745 0.0003 -5.24 0.0002 13.79 0.0010

*/

Zero Mean indicates Yt-Yt-1 on t, Yt-1, ..., with noint of Proc reg. It is assumed that the series mean is zero whether or not there are unit roots. The null hypothesis is that the time series is non-stationary, while the alternative hypothesis is that the series is stationary. Non-significant value indicates non-stationary.
Single Mean indicates Yt-Yt-1 on t, Yt-1, ..., of Proc reg with intercept. Because the model, under the alternative, has the single mean. If you simply subtracts the estimated mean for all observations, then check zero mean results.
Trend indicates Yt-Yt-1 on t, Yt-1, ..., of Proc reg with intercept.

In each case it is the coefficient of Yt-1 that indicates whether differencing needs to be done with a 0 coefficient indicating differencing. In each case, the usual t formula applies but the distribution is nonstandard. Differencing would not be appropriate as it would induce a unit root of moving average term. On the other hand, if the error is a unit root process then differencing is appropriate.

The remaining question is whether the nonstandard distribution is the same in each case. Unfortunately the answer is no, but fortunately the distributional results needed are available. For the SAS user, it is enough to know that the test computations in PROC ARIMA use the proper distributions so their p-values can be trusted.