chap3 - ARMA Models (Chapter 3) Motivation: For `most' time...

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1 1 Motivation: For ‘most’ time series { X t } , Wold Decomposition { X t } is a linear TS X t = ψ j Z t-j , { Z t } ~ WN(0, σ 2 ), = ψ(Β) Z t , where ψ(Β) = 1 + ψ 1 B + ψ 1 B 2 + . . . We approximate ψ(Β) using a ratio of polynomials which leads us to the class of ARMA models. ARMA Models (Chapter 3) j = 0 2 Write ARMA equations as: φ(Β) X t = θ(Β) Z t where φ( z) and θ( z) are the polynomials φ( z) = 1 −φ 1 z - - φ p z p , θ( z) = 1 1 z + + θ q z q . DEFINTION: { X t } is an ARMA(p,q) process if { X t } is stationary and if for every t, X t −φ 1 X t-1 -- φ p X t-p = Z t 1 Z t-1 + + θ q Z t-q where { Z t } ~ WN(0, σ 2 ). 3.1 ARMA(p,q) Processes L L L L
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2 3 Causality: Solution { X t } is causal if we can write X t = ψ j Z t-j , with | ψ j | < . { X t } causal φ( z ) = 0 for |z | < 1 ψ(Β) = ( see p.85) Existence and Uniqueness: The ARMA eqns φ(Β) X t = θ(Β) Z t have a stationary solution (which is unique) if and only if φ( z ) = 0 for |z| = 1 j = 0 j = 0 8 θ( B) φ (B) 4 Invertibility: Solution { X t } is invertible if we can write Z t = π j X t-j , with | π j | < . { X t } invertible θ( z ) = 0 for |z | < 1 π(Β) = ( see p.86) j = 0 j = 0 φ( B) θ (B)
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3 5 Ex 3.1.1 (An ARMA(1,1) process). X t −.5 X t-1 = Z t + .4 Z t-1 , { Z t } ~ WN(0, σ 2 ), AR polynomial : φ( z)=1 .5z (zero at z=2, so process is causal).
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chap3 - ARMA Models (Chapter 3) Motivation: For `most' time...

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