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I gather from this question and answer, a seasonal series is non-stationary as its mean depends on which month it is.

Suppose I have a series which possibly has a unit root (stochastic trend) but also exhibit strong seasonality (see the chart below).

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Now I think I can remove seasonality first and then use a suitable unit root test. My doubt is, would it be theoretically sound if I directly perform unit root test on the original series.

My feeling is that a strong seasonality, such that a large part of variance in series is coming from seasonal fluctuation, can hide the underlying stochastic trend, causing the unit root test to misidentify the series as having no unit root.

Does this make sense? Further, what if we try to handle the seasonality within the unit root test. For example, in ADF test, we include seasonal dummies (just like we include a trend variable to handle deterministic trend)?. But even if this is correct, we wouldn't have the right critical values for it, right?

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