Weak vs. Strong Stationarity¶
1. Weak (Covariance) Stationarity¶
Weak stationarity, also known as stationarity in a wide sense, is the most practical assumption in time series analysis. It focuses on the first two moments (mean and variance) rather than the entire distribution.
Core Requirements:
A process is weakly stationary if its first and second moments are unaffected by a change of time origin. Specifically:
* Constant Mean: \(E(Y_{t}) = \mu\) for all \(t\).
* Constant Variance: \(Var(Y_{t}) = \sigma^{2} < \infty\) for all \(t\).
* Lag-Dependent Covariance: \(Cov(Y_{t}, Y_{t+k}) = \gamma_{k}\) for all \(t\). The covariance must depend only on the lag \(k\) and be completely free from the specific time point \(t\).
* Lag-Dependent Correlation: \(Corr(Y_{t}, Y_{t+k}) = \rho_{k}\) for all \(t\).
Note: When a researcher says, "I am analyzing a stationary time series," they are almost always referring to a weakly stationary series.
2. Examples of Stationarity Testing¶
Example 2: White Noise Process
Let \(Y_{t} = e_{t}\), where \(e_{t} \sim \text{IID}(0, \sigma^{2})\).
* \(E(Y_{t}) = E(e_{t}) = 0\) (Constant)
* \(Var(Y_{t}) = Var(e_{t}) = \sigma^{2}\) (Constant)
* \(Cov(e_{t}, e_{t+k}) = 0\) for \(k \neq 0\) due to the IID property.
* Result: \(Y_{t} = e_{t}\) is stationary.
Example 3: Moving Average Process
Let \(Y_{t} = e_{t} + 0.5 e_{t-1}\), where \(e_{t} \sim \text{IID}(0, \sigma^{2})\).
* \(E(Y_{t}) = E(e_{t}) + 0.5 E(e_{t-1}) = 0\)
* \(Var(Y_{t}) = Var(e_{t}) + 0.25 Var(e_{t-1}) + 2 Cov(e_{t}, 0.5e_{t-1}) = \sigma^{2} + 0.25\sigma^{2} + 0 = 1.25\sigma^{2}\).
* The variance is finite and free from \(t\).
* Result: This process is stationary (as the covariance \(Cov(Y_t, Y_{t+k})\) will also be independent of \(t\)).
Example 4: Random Walk
Let \(Y_{t} = e_{1} + e_{2} + \dots + e_{t}\).
* \(E(Y_{t}) = 0\).
* \(Var(Y_{t}) = t\sigma^{2}\).
* Result: Since the variance is time-dependent, \(Y_{t}\) is not stationary.
Example 5: Deterministic Trend
Let \(Y_{t} = a + bt + e_{t}\).
* \(E(Y_{t}) = E(a + bt + e_{t}) = a + bt\).
* Result: Since the mean is time-dependent, \(Y_{t}\) is not stationary.
3. Comparing Strong and Weak Stationarity¶
| Feature | Strict (Strong) Stationarity | Weak (Covariance) Stationarity |
|---|---|---|
| Requirement | Joint Distribution is time-invariant. | First two moments are time-invariant. |
| Focus | Distributional sense. | Moment sense. |
Key Relationships:¶
- Strong \(\not\implies\) Weak: A process can be strictly stationary but not weakly stationary. For example, an IID Cauchy process is strictly stationary, but because its mean and variance are undefined (do not exist), it cannot be weakly stationary.
- Functions of Processes: A nonlinear function of a strictly stationary variable is still strictly stationary. However, it may not be weakly stationary; if \(Y_{t}\) is strictly stationary, \(Y_{t}^{2}\) is also strictly stationary, but its variance may not exist.
- Weak \(\implies\) Strong (The Gaussian Exception): For a Gaussian Time Series, weak stationarity does imply strong stationarity. This is because a multivariate normal distribution is completely characterized by its first two moments.
4. Non-stationarity¶
A process is non-stationary if it lacks statistical equilibrium. This usually stems from three main issues:
1. Trend: Long-term upward or downward movement.
2. Seasonality: Cyclical or repetitive patterns over fixed periods.
3. Heteroscedasticity: Changing variance over time.