08_arma.qmd

# ARMA Models

## ARMA(p,q)

ARMA($p$, $q$) is a mixed autoregressive and moving average process.

$$
y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \dots + \phi_p y_{t-p} +
\epsilon_t + \theta_1\epsilon_{t-1} + \dots + \theta_q\epsilon_{t-q},
$$

or

$$
\phi(L) y_t = \theta(L) \epsilon_t,
$$

where $\{\epsilon_{t}\} \sim \text{WN}(0, \sigma^2)$.

The MA part is always stationary as shown in @prp-maqstat. The
stationarity of an ARMA process solely depends on the AR part. The
condition is the same as @prp-arpstat.

Assume $\phi^{-1}(L)$ exist, then the ARMA($p$,$q$) process can be
reduce to MA($\infty$) process:

$$
y_t = \phi^{-1}(L)\theta(L)\epsilon_t = \psi(L) \epsilon_t,
$$

where $\psi(L) = \phi^{-1}(L)\theta(L)$.

::: callout-tip
## Exercise

Compute the MA equivalence for ARMA(1,1).
:::

## ARIMA(p,d,q)

ARMA($p$,$q$) is used to model stationary time series. If $y_t$ is not
stationary, we can transform it to stationary and model it with an ARMA
model. If the first-order difference $(1-L)y_t = y_t - y_{t-1}$ is
stationary, then we say $y_t$ is **integrated** of order 1. If it
requires $d$-th order difference to be stationary, $(1-L)^dy_t$, we say
it is integrated of order $d$. The ARMA model involves integrated time
series is called ARIMA model:

$$
\phi(L)(1-L)^d y_t = \theta(L)\epsilon_t.
$$