Arima model
Notations
The backshift, foreshift operators
B,F
are defined by
Bkyt=yt−k,Fkyt=yt+k
We define an ARIMA process as
Δ(B)Φ(B)yt=Θ(B)ϵtwhere
Δ(B)=1+δ1B⋯+δdBd Φ(B)=1+φ1B⋯+ϕpBp Θ(B)=1+θ1B⋯+θqBqare the differencing, auto-regressive and moving average polynomials.
The corresponding stationary ARMA model is defined by
Φ(B)yt=Θ(B)ϵtProperties 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=ϵtand
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)