Arima model
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
\(y_t = \Psi \left(B \right) \epsilon_t\) (Wold representation)
The autocovariances of the process are generated by
\[\Psi \left(B \right) \Psi \left(F \right) = \frac{\Theta \left(B \right) \Theta \left(F \right)}{\Phi \left(B \right) \Phi \left(F \right)}\]