Suppose that { Z t } is a causal stationary GARCH (p, q) process Zt _ √ h t e t , where { e t } ∼…

Suppose that {Zt} is a causal stationary GARCH(p, q) process Zt _ √htet, where {et} ∼ IID(0,1),

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Suppose that { Z t } is a causal stationary GARCH (p, q) process Zt _ √ h t e t , where { e t } ∼…
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htα0 + α1Z2t−1 +· · ·+αpZ2tpβ1ht−1 +· · ·+βqhtq .

a. Show that E(Z2t |Z2t−1,Z2t−2, . . .) ht .

b. Show that the squared process {Z2t } is an ARMA(m, q) process satisfying

the equations

Z2tα0 + 1 + β1)Z2t−1 +· · ·+m βm)Z2tmUt − β1Ut−1 −· · ·−βqUtq ,

where= max{p, q}, αj = 0 forj > pβj = 0 forj > q, andUt Z2t ht is white noise if EZ4t < ∞.

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