# Stationarity tests for time series

I am currently working on time series modeling, especially on stationarity tests. For this purpose, I am extensively using Pfaff's book "Analysis of integrated and cointegrated time series with R" and I have some questions :

1. On page 63, there is a nice ordinogram (Figure 3.3) explaining how all the ADF tests are related, and what should be the underlying decision tree. First of all, one needs to estimate the ADF equation with a linear trend and test for $\pi=0$ (this statistic is called tau3 in the associated package urca). If we reject the null hypothesis, then there is no unit root. If we cannot reject, we test for $\beta_2=0$ given $\pi=0$ (this statistic is called phi3 in urca). If we reject, then Pfaff writes "test again for a unit root using a standardized normal" with no further explanation.

Does anyone understand what he is talking about? Does this "normal test" appear somewhere in the urca implementation?

2. Suppose the test tau3 for $\pi=0$ is rejected. Then the conclusion should be that there is no unit root but a trend in the series (the series is trend stationary). I have at disposal the underlying linear regression result given by ur.df() from the package urca. Is it correct to conclude that there is actually no trend when the p-value of the t-statistic for the trend coefficient is significant?

• If a unit root cannot be rejected you should follow the test statistics and critical values $\pi=0$ and $\beta=0$. If a unit root is rejected, then you can follow the standard $t$-statistics that would be returned by lm and reported in the top part of summary(ur.df(x)). – javlacalle Dec 23 '14 at 14:13