# What is the real meaning of null hypothesis in unit-root test for a AR(p) process?

There are functions in R (e.g., PP.test and adf.test) which have null hypothesis of unit-root in the process ($H_0$: there is a unit root). Does this null hypothesis mean "the process is difference stationary"? If yes, what is the order of difference required to make it stationary?

Well, for an $AR(1)$ process, yes, a unit root means the process is first order difference stationary. How about for a $AR(p)$ process?

• Thanks again for your answer. Then do you think there is a relationship between PP.test(x) and kpss.test(diff(x))? I am thinking that PP.test(x) is testing difference stationary. That is also what 'kpss.test(diff(x))' does. – yanfei kang Nov 23 '13 at 14:24
Consider the process $y_t=1/2y_{t-1}+1/2y_{t-2}+\epsilon_t$.
Now, if $y_t$ is an $ARIMA(p,1,q)$ we can write $\Delta y_t$ as an $ARMA(p,q)$. Clearly, the process is an $ARMA(2,0)$ process. To check whether it is an $ARIMA(1,1,0)$ we need to look at the $AR$ polynomial $\phi(z)=1-1/2z-1/2z^2$. We can write this as $\phi(z)=(1-z)(1+1/2z)$ so that indeed $d=1$. We have $\Delta y_t=-1/2\Delta y_{t-1}+\epsilon_t$.