Autoregressive Moving Average (ARMA) Models

ARMA models are an extention of AR and MA by combination. ARMA is described as follows

$$ X_t - \phi_1 X_{t-1} - ... - \phi_p X_{t-p} = \beta + a_t - \theta_1 a_{t-1} - ... - \theta_q a_{t-a} $$

where

$\beta = \left( 1 - \phi_1 - \ldots - \phi_p \right) \mu$ and $a_t$ is white noise

and is constained with

  • $\phi_i$ and $\theta_j$ are real valued constants
  • $\phi_i \neq 0$ and $\theta_j \neq 0$
  • $\phi (z)$ and $\theta (z)$ have no common factors

The operator for is

$$ \left( 1 - \phi_1 B - ... - \phi_p B^p \right) X_t = \left( 1 - \theta_1 B - ... - \theta_q B^q \right) $$

And can generally be expressed as

$$ \phi \left( B \right) X_t = \theta \left( B \right) $$

An ARMA model is valid if and only if model is stationary and invertable i.e.

  1. All the roots of $\phi \left( z \right)$ are outside the unit circle
  2. All the roots of $\theta \left( z \right)$ are outside the unit circle

Notation

Since ARMA is a combination of two models (AR and MA), the separate indexes are used to indicate the order of each model. Typically, $p$ refers to the AR order and $q$ refers to the MA order i.e. ARMA(p,q). Additionally, the order of the models are indexed with $i$ and $j$ in formulas, respectively.

  • An MA(q) model is an ARMA(0, q) model
  • An AR(p) model is an ARMA(p, 0) model

References

  • [1] W. Woodward and B. Salder, "Moving Average (MA(q)) and ARMA(p,q) Models", SMU, 2019