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Academic year: 2022

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(5)

Pure seasonal ARMA Models

For P, Q ≥ 0 and s > 0, we say that a time series {Xt} is an ARMA(P,Q)s process if Φ(Bs)Xt = Θ(Bs)Wt, where

Φ(Bs) = 1 −

P

X

j=1

ΦjBjs,

Θ(Bs) = 1 +

Q

X

j=1

ΘjBjs.

It is causal iff the roots of Φ(zs) are outside the unit circle.

It is invertible iff the roots of Θ(zs) are outside the unit circle.

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(6)

Pure seasonal ARMA Models

Example: P = 0, Q = 1, s = 12. Xt = Wt + Θ1Wt12. γ(0) = (1 + Θ21w2 ,

γ(12) = Θ1σw2 ,

γ(h) = 0 for h = 1,2, . . . ,11,13,14, . . ..

Example: P = 1, Q = 0, s = 12. Xt = Φ1Xt12 + Wt. γ(0) = σw2

1 − Φ21 , γ(12i) = σw2 Φi1

1 − Φ21 ,

γ(h) = 0 for other h.

(7)

Pure seasonal ARMA Models

The ACF and PACF for a seasonal ARMA(P,Q)s are zero for h 6= si. For h = si, they are analogous to the patterns for ARMA(p,q):

Model: ACF: PACF:

AR(P)s decays zero for i > P

MA(Q)s zero for i > Q decays

ARMA(P,Q)s decays decays

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(8)

Multiplicative seasonal ARMA Models

For p, q, P, Q ≥ 0 and s > 0, we say that a time series {Xt} is a multiplicative seasonal ARMA model (ARMA(p,q)×(P,Q)s) if Φ(Bs)φ(B)Xt = Θ(Bs)θ(B)Wt.

If, in addition, d, D > 0, we define the multiplicative seasonal ARIMA model (ARIMA(p,d,q)×(P,D,Q)s)

Φ(Bs)φ(B)∇DsdXt = Θ(Bs)θ(B)Wt,

where the seasonal difference operator of order D is defined by

Ds Xt = (1 − Bs)DXt.

(9)

Multiplicative seasonal ARMA Models

Notice that these can all be represented by polynomials

Φ(Bs)φ(B)∇Dsd = Ξ(B), Θ(Bs)θ(B) = Λ(B).

But the difference operators imply that Ξ(B)Xt = Λ(B)Wt does not define a stationary ARMA process (the AR polynomial has roots on the unit

circle). And representing Φ(Bs)φ(B) and Θ(Bs)θ(B) as arbitrary polynomials is not as compact.

How do we choose p, q, P, Q, d, D?

First difference sufficiently to get to stationarity. Then find suitable orders for ARMA or seasonal ARMA models for the differenced time series. The ACF and PACF is again a useful tool here.

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