Consider the AR( p ) process defined by Y t +1 = φ 1 Y t + φ 2 Y t −1 +· · ·+ φ p

State-space representation of a causal AR(p) process

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Consider the AR(p) process defined by

Yt+1 = φ1Ytφ2Yt−1 +· · ·+φpYtp+1 + Zt+1, t= 0,±1, . . . ,

where {Zt} ∼ WN(02), and φ(z) := 1 − φ1z−· · ·−φpzpis nonzero for |z| ≤ 1. To express {Yt} in state-space form we simply introduce the state vectors

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