Stationarity

Stationarity in time series analysis refers to the statistical properties of the data, such as the mean and variance, remaining constant over time. A stationary time series has no trends, seasonal patterns, or other systematic changes in the mean or variance. Analyzing stationary time series is important because it allows for the use of standard statistical techniques for modeling and forecasting.

Weak v.s. Strict (Strong) Stationarity

Weak Stationarity (Second-Order Stationarity)

If a time series is weak stationary, the mean is constant and independent of time location; the autocorrelation of any two data points in the time series will only depend on the time difference but not on the time locations of the data points.

For A time series $\{X_t\}$ is weakly stationary if the following conditions hold:

1. Constant Mean: $\mathbb{E}[X_t]=\mu ,\mathrm{\forall }t$
2. Constant Variance: $Var(X_t)={\sigma }^2$
3. Autocovariance depends only on the lag: For any $s,t$, the autocovariance depends only on the lag $\tau =|t-s|$ as: $Cov(X_t,X_s)=\gamma (\tau )=\gamma (t-s)$, where the $\gamma (\tau )$ is the autocovariance function, independent of $t$ and $s$, depending only on $\tau$.

For strict stationarity, the joint probability of the time series data points is the same when time shifts.

Strict Stationarity

A time series $\{X_t\}$ is strictly stationary if the joint distribution of $\{X_{t1},X_{t2},...,X_{tk}\}$ is invariant under time shifts for any $t_1,t_2,...,t_k$ and any integer $h$:

for all $h,k$, and $t_1,t_2,...,t_k$.