I have a question about the stationarity and I wonder if you can offer some words of wisdom. I have a time series X, which passed the Augmented Dickey-Fuller test and the Kwiatkowski, Phillips, Schmidt and Shin (KPSS) test with p-values as large as 40%. I consider X as stationary. However, the logged X cannot pass the Kwiatkowski, Phillips, Schmidt and Shin (KPSS) test with p-values close to 5 or 6%.
1) If X is stationary, should all transformation of X, such as log(X) or sqrt(X) or X^(1/3), be stationary as well?
2) Should I use some other different stationarity tests for the non-linear transformation log(X) or sqrt(X)?
Definitely it is possible that a non-linear transformation of a stationary time series is non-stationary and even it is possible with linear transformation. For example, we have the white noise (stationary process) as input signal of the integrator (linear element). So in this case the output signal is Wiener process which is non-stationary.
Thanks, Oleg, for your answer.
As you wrote, if a trend term is added to a white noise, the process becomes non-stationery. (The process is still trend-stationary, though.) Something I am not sure is whether a Box-Cox transformation of a white noise is stationery.
The following is a link for Box-Cox transformation: http://www.isixsigma.com/tools-templates/normality/making-data-normal-using-box-cox-power-transformation/
Be careful that if you test your time series first, then apply a model based on the results of these tests, the inference you could make from the model will be biased (significantly or not, I am not sure).
Look into data-instigated model, or data-snooping.
@Ralf Time independent transformations of a strictly stationary variable are strictly stationary. If a variable is trend stationary then it will not be stationary if transformed. Note that if a variable is I(1) it may not be I(1) when transformed. In my earlier comment i was thinking about the effect of transformations of the dependent variable on residual disturbances.
If f() is a real function of a single real variable and X(t) is a stationary time series, then the transformed time series Y(t) = f(X(t)) is always stationary, too. So for your three examples of transformation functions, f(X)=log(X), f(X)=sqrt(X), and f(X)=X^(1/3), this is the case, i.e. the stationarity won't get lost. If a test finds a different result, the test is definitely wrong. This could be due to a wrong parametrization or due to the fact that all the statistical moments (variance and higher moments) will change as a result of the transformation. Also, some tests can get difficulties from an inhomogeneous scaling of the transform. For example, the log(X) transformation will "compress" large values of X, but leave low X values uncompressed. Maybe you can try other methods (e.g. recurrence plots, available in the free version of the software package Dataplore, see below) for identifying instationarity.
Good luck and best regards, Ralf
http://dataplore.com
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