Evaluating the importance of a unit-root

Question

I have a monthly time series and I'm trying to determine if such set of data is stationary or not; the dataset is about composed by 160 record.

Specifically, I'm running 2 test found in literature:

1. KPSS: if $H_0$ has been rejected then one cannot assume the time series is stationary;
2. Phillips-Perron test: if $H_0$ has been rejected then one cannot assume that the time series has a unit-root (then it is stationary);

I preferred to implement the Phillips-Perron test in place of the most common Augmented Dickey-Fuller test since the Phillips-Perron test adjusts for the heteroschedasticity and serial correlation.

Here below, one can find the output of such analysis.

The KPSS test returns not significant p-values both for single-mean, implying that you cannot infer that the time series is not stationary; likewise, the Phillips-Perron test returns significant p-values for the single-mean and trend component, but not for the zero-mean case.

How should I consider or interpret such result?

I wonder if one can evaluate the importance and the strength of such unit-root; for instance, in the question the user @ferdi deals with the variance ratio test to argue the framework to evaluate the importance of a unit root in a time series.

Could you suggest some reference about?

I'm currently running the analysis in SAS, but any programming language would be nice.

Thank all in advance!!

Answer 1

Phillips-Perron test: if $H_0$ has been rejected then one cannot assume that the time series has a unit-root (then it is stationary);

This is not entirely correct. Lack of a unit root does not imply stationarity. It only implies lack of a very special kind of nonstationarity, namely, the unit-root kind. Other kinds of nonstationarity are, for example, level shifts, shifts of variance, and other. The PP test might not be sensitive to them.

How should I consider or interpret such result?

On the face value, and provided the test results are correct, you have a time series that does not have a unit root but is not stationary either. Perhaps it can be characterized by one of the kinds of nonstationarity mentioned above, or yet another kind.

I wonder if one can evaluate the importance and the strength of such unit-root; for instance, in the question the user @ferdi deals with the variance ratio test to argue the framework to evaluate the importance of a unit root in a time series.

The answer by Ferdi you quote seems to use some nonstandard/nonstatistical terminology, or at least one that is not widespread. I am not exactly sure what is meant there, but it reminds me of something related I heard in a lecture by prof. Ruey S. Tsay. He noted that a time series $x_t$ could be a sum of a stationary process $y_t$ and a unit-root process $z_t$: $$ x_t:=y_t+z_t. $$ If the variance of the stationary process $y_t$ is large (say, $100$), while the variance of the error term in the unit-root process $z_t$ is small (say, $0.01$), then $x_t$ might behave essentially as a stationary process in a finite sample. It would be very hard to detect the presence of a unit root unless the sample size is very large. Surely, asymptotically $y_t$ would dominate $z_t$, and $x_t$ would show its unit root. But for a fixed period of time that we care about, $y_t$ dominates. Hence, in practice we might treat $x_t$ as stationary. (And in the long run we are all dead.) Perhaps this is similar to what Ferdi meant in his answer.