Author: Eiko

Time: 2025-01-27 20:35:09 - 2025-01-27 20:41:23 (UTC)

Stationary ARMA Processes

• ARMA means autoregressive (AR) moving average (MA).

• This is an important class of processes described by finite difference equations.

• It has an elegant theory of linear projections.

Core Concepts

White Noise

It is not the classical white noise we are talking about, in this context, a white noise is a stationary process $\{Z_t{\}}_{\mathbb{Z}}$ that is componentwise Gaussian, and have covariance (i.e. they don’t have to be independent for this definition).

$$\mathrm{Cov}(Z_s,Z_t)={\sigma }^2{\delta }_{st}.$$

This is expected to generate all the randomness of our system and provide an easy framework for the concept of causuality, similar to what ${\mathcal{F}}_t=\sigma (\{Z_i:i\le t\})$ means.

ARMA

$\{X_t\}$ is ARMA$(p,q)$ if

• $X$ is stationary

• For each $t$, we have

  $$X_t-{\varphi }_1{X}_{t-1}-\cdots -{\varphi }_p{X}_{t-p}=Z_t+{\theta }_1{Z}_{t-1}+\cdots +{\theta }_q{Z}_{t-q}.$$

One can write this difference equation in operator form, writing $B$ for the backward shifting operator, we have polynomials $\varphi (z)=\sum {\varphi }_i{z}^i$ and $\theta (z)=\sum {\theta }_i{z}^i$ with ${\varphi }_0={\theta }_0=1$. We can write the above equation as

$$\varphi (B)X_t=\theta (B)Z_t.$$

Example

• The MA$(q)$ process is defined by $\varphi (z)=1$ and so

  $$X_t=\theta (B)Z_t.$$

  We can compute

  $$\mathrm{Cov}(X_{t+h},X_t)={\sigma }^2\sum _{j=0}^{q-|h|}{\theta }_{j+|h|}{\theta }_j$$

  thus it is stationary.

Existence Of Solution

• For the ARMA$(p,q)$ process, if $0\notin \varphi (\partial D(0,1))$, then it has a unique stationary solution given by $X_t=\sum _{j=0}^{\mathrm{\infty }}{\psi }_j{Z}_{t-j}$ where $\psi (z)=\frac{\theta (z)}{\varphi (z)}$, converging on some ring domain $r^{-1}<|z|<r$.

• The solution is causal if $\psi (z)$ does not involve negative powers of $z$, only depends on the past.

• If $\varphi (D(0,1))$ does not contain $0$, then there is a causal solution given by the same formula.