ARMA Process

this a brief outline for understanding the ARMA process $\{ X_t \}$

Take-Away

$$\phi(B) X_t = \theta(B) W_t \quad | \quad X_t = \psi(B) W_t$$

• $\psi(B)=\frac{\theta(B)}{\phi(B)}$

  1. computed by coefficient matching
  2. has no unit root ($z_0 \: solves \:\psi(z_0) = 0, |z_0| = 1$) $\Rightarrow$ Stationary

• $\phi(B)$

  $\phi(z)$ does not have any roots $z_0$ with $|z_0| \leq 1$ $\Rightarrow$ Casual

• $\theta(B)$

  $\theta(z)$ does not have any roots $z_0$ with $|z_0| \leq 1$ $\Rightarrow$ Invertible

Definition

$\{ X_t\}$ is an ARMA process if:

• $\{ X_t\}$ is stationary

• $\{ X_t\}$ satistifes the difference equation $\phi(B) X_t = \theta(B) W_t$

  $\phi(B)$ is polynomial of degree p; $\theta(B)$ is polynomial of degree q.

  $W_t$ is white noise

   

Requirements

typical ARMA process should satisfy 3 properties : Stationary, Casuality & Invertibility.

• Stationary

  autocovariance $\gamma(h)$ function is independent of time t

• Casuality

  ${X_t}$ is determined by history process $X_{t-1}, X_{t-2}...$

  Mathematical:

  $$\exist \: \psi(z) = \psi_0 + \psi_1z + \psi_2 z^2 + ... \:, \sum_{i=1}^\infty |\psi_i | \leq +\infty \\ X_t = \psi(B) W_t$$

   

• Invertibility

  a white noise $W_t$ could be reversely represented by process ${X_k: k \leq t}$

this property further assures uniqueness of the process $X_t$, i.e., $X_t$ could only be determined by one white noise $W_t$

Mathematical:

I. Stationary

determined by $\psi(z)$

ARMA $X_t$ has a unique stationary solution if and only if the polynomial $\psi(z)$ has no roots having magnitude exactly equal to one.

II. Casuality

determined by $\phi(z)$

Casual only if $\phi(z)$ does not have any roots $z_0$ with $|z_0| \leq 1$

III. Invertibility

determined by $\theta(z)$

Invertible only if $\theta(z)$ does not have any roots $z_0$ with $|z_0| \leq 1$

Coefficient Matching

use the difference equation to iteratively solve $\psi(z)$' s coefficient $\psi_i$

match the coefficient of term z from RHS and LHS:

• constant

  $$\psi_0 = 1$$

   

• $z$

  $$(-\phi_1 \psi_0 + \psi_1) z = \theta_1 z$$

• $z^2$

  ...