Arima models

Notations

The backshift, foreshift operators
$B, F$ are defined by $B^k y_t = y_{t-k} \:, \: F^k y_t = y_{t+k}$

We define an ARIMA process as

$$\Delta \left(B \right) \Phi \left(B \right) y_t = \Theta \left(B \right) \epsilon_t$$

where

$\Delta \left(B \right)= 1+ \delta_1 B \cdots + \delta_d B^d$
$\Phi \left(B \right)= 1+ \varphi_1 B \cdots + \phi_p B^p$
$\Theta \left(B \right)= 1+ \theta_1 B \cdots + \theta_q B^q$

are the differencing, auto-regressive and moving average polynomials.

The corresponding stationary ARMA model is defined by

$$\Phi \left(B \right) y_t = \Theta \left(B \right) \epsilon_t$$

Properties of the ARMA model

The Pi-weights are generated by the Rational function $\Pi \left(B\right) = \frac{\Phi \left(B \right)}{ \Theta \left(B \right)}$

and the Psi-weights are generated by the Rational function $\Psi \left(B\right) = \frac{ \Theta \left(B \right)}{ \Phi \left(B \right)}$

We have:

$$\Pi \left(B \right) y_t = \epsilon_t$$

and

$y_t = \Psi \left(B \right) \epsilon_t$ (Wald representation)

The autocovariances of the process are generated by

$$\Psi \left(B \right) \Psi \left(F \right) = \frac{\Theta \left(B \right) \Theta \left(F \right)}{\Phi \left(B \right) \Phi \left(F \right)}$$