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Arima model

Notations

The backshift, foreshift operators
B,F are defined by Bkyt=ytk,Fkyt=yt+k

We define an ARIMA process as

Δ(B)Φ(B)yt=Θ(B)ϵt

where

Δ(B)=1+δ1B+δdBd Φ(B)=1+φ1B+ϕpBp Θ(B)=1+θ1B+θqBq

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

The corresponding stationary ARMA model is defined by

Φ(B)yt=Θ(B)ϵt

Properties of the ARMA model

The Pi-weights are generated by the Rational function Π(B)=Φ(B)Θ(B)

and the Psi-weights are generated by the Rational function Ψ(B)=Θ(B)Φ(B)

We have:

Π(B)yt=ϵt

and

yt=Ψ(B)ϵt (Wold representation)

The autocovariances of the process are generated by

Ψ(B)Ψ(F)=Θ(B)Θ(F)Φ(B)Φ(F)

Pseudo-spectrum (or spectral density)