Stationary Time Series

A stationary time series is one whose statistical properties do not change over time.

Formally, a time series $y_t$ is stationary if:

• The mean is constant: $E[y_t]=\mu$ for all $t$.
• The variance is constant: $\text{Var}(y_t)={\sigma }^2$ for all $t$.
• The autocovariance depends only on the lag $k$, not on the specific time $t$: $\text{Cov}(y_t,y_{t+k})={\gamma }_k$

This means the process has the same behavior regardless of when you observe it.

Types of stationarity

1. Strict stationarity: The entire distribution of the process is invariant to shifts in time (all moments remain constant).
2. Weak (or covariance) stationarity: Only the first two moments (mean, variance, covariance) are invariant. This weaker definition is often sufficient for models like ARIMA.

Examples

Stationary: White noise (mean = 0, variance constant).

Non-stationary:

• Series with a trend (e.g., increasing sales over time).
• Series with changing variance (e.g., volatility clustering in finance).
• Series with seasonality (patterns repeating over time).

Why it matters

Many classical time series models (e.g., ARIMA, SARIMA) assume stationarity. Non-stationary data can lead to misleading results. Common Data Transformation to achieve stationarity include:

• Differencing: $y_t-y_{t-1}$
• Log transformation (to stabilize variance)
• Detrending or deseasonalising