How to diagnose and achieve stationarity
The statistical properties, such as the mean, variance, or autocorrelation, of a stationary time series are independent of the period—that is, they don't change over time. Thus, stationarity implies that a time series does not have a trend or seasonal effects. Furthermore, it requires that descriptive statistics, such as the mean or the standard deviation, when computed for different rolling windows, are constant or do not change significantly over time. A stationary time series reverts to its mean, and the deviations have a constant amplitude, while short-term movements are always alike in a statistical sense.
More formally, strict stationarity requires the joint distribution of any subset of time-series observations to be independent of time with respect to all moments. So, in addition to the mean and variance, higher moments such as skew and kurtosis also need to be constant, irrespective of the lag between different observations...
