Does stationarity imply absense of upward or downward trend?

Question

I wonder if a time series being stationary implies that there can be no upward or downward trend. It appears to me that such an implication should hold, since in order to be stationary a time series has to have a constant mean, so in general it should wiggle around the same point. Is my reasoning correct?

EDIT:

So for instance is it possible to get a significant result of the Dickey-Fuller test indicating that a series is stationary, but at the same time get a significant result of the Mann-Kendall trend test, which indicates that there is a trend? In what kind of situation such an outcome may arise?

Answer 1

Unless you specify a qualifier, such as trend-stationary, yes, stationary means there's no trend. Not only the trend, it also means that the variance doesn't change too.

Sometimes, people say trend-stationary meaning that if you remove the trend, then what's left is stationary.

The stock Dickey-Fuller test is rarely used, usually its version called Augmented Dickey Fuller (ADF) is used. Depending on which stat package you have, it may have the option to run it with trend-stationary assumption, e.g. MATLAB adftest function allows this. So, check what options you're using in your stat package for ADF.

As a general observation it's not uncommon for statistical tests to disagree. I'd say that it's unusual to have all tests be in agreement with each other.