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There are many time series which are obviously not stationary (I mean, by definition), some of them are even determinstic, yet the unity root tests like ADF/PP test, etc., happily reject the existence of an unity root in these processes.

So my question is what kind of methods you used to determine whether a process is a truly stationly stochastic process with reasonable confidence?

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    $\begingroup$ Unit-root tests do not test for stationarity - they test the existence of a particular kind of non-stationarity (i.e. of whether a stochastic trend exists). So if the series is constructed as a deterministic upward trend, say, as a function of time, the tests you mention do very well to reject the null of the existence of a "unit root" -that does not mean that they tell us that the series is stationary. $\endgroup$ – Alecos Papadopoulos Jan 20 '14 at 2:10
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Some tests might test if there exist deterministic trend which makes series non-stationary in itself but stationary around trend.

Some techniques which I have used includes:

1) Autocorrelation function. If autocorrelation shape is such that decline is very slow and autocorrelation at lag 1 is close to unity then it means that series depends greatly on previous value.
2) Spectral density function. If most of power is in low frequencies, then it means that series is driven by long-run trends.
3) Eye-ball regression, just visually examine series.

My statistics professor used to say that we economists are often concentrated to study only certain kind of theoretical models and do not allow other forms of non-stationarity. And this might make it difficult to apply these models for real world processes! :)

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