ARMA Process and Wide-Sense Stationarity (WSS)
Core Concept:
An ARMA (AutoRegressive Moving Average) process is Wide-Sense Stationary (WSS) if and only if the system's autoregressive part is stable and the input is a stationary white noise process.
An ARMA (AutoRegressive Moving Average) process is Wide-Sense Stationary (WSS) if and only if the system's autoregressive part is stable and the input is a stationary white noise process.
1. The ARMA(p, q) Model Equation
A stochastic process {Xt} follows an ARMA(p, q) model if it satisfies the following linear difference equation:
Xt =
φ1Xt-1 + φ2Xt-2 + ... + φpXt-p
+ at
+ θ1at-1 + θ2at-2 + ... + θqat-q
Where:
- φ1, ..., φp: Autoregressive (AR) parameters.
- θ1, ..., θq: Moving Average (MA) parameters.
- at: A white noise process (the innovation).
2. Essential Conditions for WSS
For an ARMA process to be classified as Wide-Sense Stationary, it must satisfy two primary technical conditions:
Condition 1: White Noise Input Stability
The driving noise sequence at must be a stationary process with:
- Constant Mean: E[at] = 0 (typically zero-mean).
- Constant Variance: Var(at) = σa2 < ∞.
- Lack of Correlation: Cov(at, as) = 0 for t ≠ s.
Condition 2: The Stability (Stationarity) Condition
The stability of an ARMA process depends exclusively on the AR polynomial. Consider the characteristic equation:
Φ(z) = 1 − φ1z − φ2z2 − ... − φpzp = 0
For the process to be WSS, all roots (z) of this characteristic equation must lie outside the unit circle in the complex plane:
|z| > 1
(Equivalently, the eigenvalues of the system must lie inside the unit circle.)
3. Statistical Properties of a WSS ARMA Process
When the stability conditions are met, the ARMA process exhibits the following time-invariant statistical properties:
- Time-Invariant Mean: The expected value E[Xt] does not change over time.
- Constant Variance: The power or variance Var(Xt) remains finite and constant.
- Lag-Dependent Autocorrelation: The correlation between Xt and Xt-k depends only on the lag k, not on the absolute time t.
4. Stationarity Summary Table
| Process Type | Always WSS? | Stationarity Condition |
|---|---|---|
| MA (Moving Average) | Yes | Always WSS as long as coefficients are finite. |
| AR (AutoRegressive) | Conditional | Roots of AR polynomial must lie outside the unit circle. |
| ARMA | Conditional | Determined strictly by the AR part stability. |
Summary:
While the MA part of an ARMA process affects its invertibility, it has no impact on its stationarity. An ARMA process is Wide-Sense Stationary if and only if the AR component is stable (roots outside the unit circle).
While the MA part of an ARMA process affects its invertibility, it has no impact on its stationarity. An ARMA process is Wide-Sense Stationary if and only if the AR component is stable (roots outside the unit circle).