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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?

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Yes, the null hypothesis for PP and ADF tests is that the process is difference stationary. The order of the difference is one. If you suspect that the order is higher, you should test the differenced series.

For AR(p) process (and generaly for ARMA(p,q)) unit root means first order difference stationary. Hence the notation ARIMA(p,d,q), which means that the d-difference of the process is the ARMA(p,q) process.

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  • $\begingroup$ 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. $\endgroup$ – yanfei kang Nov 23 '13 at 14:24
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Yes.

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 $\Delta y_t=-1/2\Delta y_{t-1}+\epsilon_t$.

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