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I have a time series $X_t$ (shown below) with a structure break. The stationary test kpss.test() says it has a unit root. How to explain this? Why does $X_t$ have a unit root? Sure it is not constant in mean, so it is non-stationary. But I can not relate its non-stationarity to the concept of unit-root.

x=c(rnorm(1000,0,1),rnorm(1000,10,1))
kpss.test(x)

The $p$-value of the test is 0.01, so we reject the null hypothesis of a stationary process.

For example, a random walk has a unit root but it is constant in mean. So any relationship between unit root and constant-in-mean? Any comments about this?

enter image description here

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  • $\begingroup$ Note that a random walk does not have a constant mean; its first difference does. $\endgroup$ – Scortchi Nov 21 '13 at 9:41
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The null hypothesis for KPSS is stationarity. If the null hypothesis is rejected this means that the series is not stationary, which in your case it is clear that it is not. This does not mean that the series is unit-root, although the test was designed in a way to suggest that.

All the tests have their assumptions (usually technical mathematical ones) and it is not uncommon than not the null hypothesis is rejected because of the failure of the assumptions, not because the alternative hypothesis is true. In this particular case KPSS was not designed to safeguard against the structural breaks. If you suspect that there are structural breaks in your data use Zivot-Andrews unit-root test from the package urca, function ur.za.

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  • $\begingroup$ Thanks for you answer. When I use PP.test to this time series. It gives me $p=0.01$, which means a wrong assumption of unit root. If I use 'adf.test', it gives $p>0.1$, so we don't have enough evidence to reject the null hypothesis of unit root. A little bit confusing here. Do you have any comments please? $\endgroup$ – yanfei kang Nov 22 '13 at 4:09
  • $\begingroup$ Again, ADF and PP tests are not designed to guard against structural breaks. Hence they should not be used. $\endgroup$ – mpiktas Nov 22 '13 at 8:41

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