Stationarity

In time series analysis, stationarity refers to a condition where the statistical properties of a time series—such as its mean, variance, and autocorrelation—remain constant over time. A stationary time series does not have trends, changing variability, or evolving seasonal patterns. This stability makes it easier to model and forecast, which is why many time series methods, including ARIMA, require the data to be stationary.

There are two main types:

• Strict stationarity: The entire distribution of the process remains unchanged over time.
• Weak (or second-order) stationarity: Only the mean, variance, and autocorrelation structure are constant. This form is more commonly used in practice.

The graphs below show two time series:

• The left panel displays a stationary time series (constant mean and variance).
• The right panel shows a non-stationary time series (a drifting mean).

Most real-world time series data are not stationary. Trends, seasonal cycles, or changing volatility can make a series non-stationary. To address this, analysts often apply a process called differencing—subtracting each value from the previous one—to remove trends or seasonality and transform the data into a stationary form. In some cases, multiple rounds of differencing may be needed.

For example, monthly sales data that steadily increase over time are not stationary due to the upward trend. But by applying differencing, the resulting series may fluctuate around a constant mean, making it suitable for models like ARIMA that assume stationarity. Identifying and correcting non-stationarity is a critical first step in reliable time series analysis.