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Would there be a case that a time series does have seasonality but, ADF test fails to point it out. I want to be sure of it being stationary so that I can use it in a regression and be sure that the results are not spurious.

Richard Hardy
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whisperer
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    You seem to equate a statistical test with a property of an underlying process. Although the distinction may seem subtle, it's important. So, what exactly do you mean by "be stationary"? – whuber Nov 27 '19 at 22:52
  • Can I use it to regress against y (definitely stationary) and be sure that the regression results are not spurious. Thanks for pointing it out. Will update the questions accordingly. – whisperer Nov 27 '19 at 22:55
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    A time series with a seasonal component is non-stationary. If your target variable is stationary, it doesn't have a seasonal component, and therefore it seems unlikely to me that a seasonal time series is really an appropriate regressor. The fact that you think it is causes me to wonder, along with @whuber, what exactly you mean by "be stationary". See https://stats.stackexchange.com/questions/131092/does-a-seasonal-time-series-imply-a-stationary-or-a-non-stationary-time-series?rq=1, which your question may be a duplicate of. – jbowman Nov 27 '19 at 23:10
  • I checked that question and the two comments gave opposite views. The time series is stationary judging by the ADF test. From a design point of view, I agree that it is a less likely predictor. However during the non seasonal spikes it should be based on the relation between them. – whisperer Nov 27 '19 at 23:24
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    Note that the first answer, despite being accepted, got a pretty negative score and a lot of disagreement. However, mea culpa; this https://stats.stackexchange.com/questions/174741/can-a-time-series-be-stationary-if-the-formula-for-the-mean-level-depends-on-t one is a better quick reference. – jbowman Nov 28 '19 at 01:03
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    Why not refer to Rob Hyndman's explanation on https://otexts.com/fpp2/stationarity.html#fn14? *"A stationary time series is one whose properties do not depend on the time at which the series is observed. Thus, time series with trends, or with seasonality, are not stationary — the trend and seasonality will affect the value of the time series at different times."* There's a footnote to this definition though. – Isabella Ghement Nov 28 '19 at 05:25
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    @whisperer, if I read your last comment correctly, you seem to read the ADF test as an omnibus test against all sorts of possible violations of stationarity (e.g. the ones mentioned by Isabella). It definitely cannot do that. It is, broadly speaking, a test of the null of a stochastic trend against the alternative of mean reversion. – Christoph Hanck Nov 28 '19 at 08:50
  • Thanks all, for your comments and references! As far as I understand, the answer seems to be yes it is possible to pass ADF test (declaring ADF stationary) and still have seasonality. – whisperer Nov 29 '19 at 14:24

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