kingwinner
New member
- Joined
- Jul 23, 2011
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- 10
Consider an ARCH(1) model:
Xt = σtZt, where Zt~ i.i.d. N(0,1)
σt2 = w0 + w1 Xt-12
Find (i) E(Xt)
and (ii) the autocovariance function γX(h) for h=0,1,2,3,..., assuming the process is second-order stationary.
Solution:
(i) E(Xt) = E[E(Xt|σt2)] =E[E(σtZt|σt2)]
=E[σtE(Zt|σt2)] = E[σtE(Zt)] = E[σt * 0] = 0
(here I don't understand why E(Zt|σt2)=E(Zt).)
(ii) γX(0)=E(Xt2)
=E(σt2 Zt2)=E(σt2)E(Zt2)
=E(w0 + w1 Xt-12) * 1
= w0 + w1 γX(0)
Solve for γX(0) => γX(0) = w0/(1-w1)
γX(h)=E(XtXt+h)
=E[E(XtXt+h|Zt+h-1,...,Zt+1)] = E[XtE(Xt+h|Zt+h-1,...,Zt+1)]
(here I don't understand why we can pull the Xt out of the expectation.)
=E[XtE(σt+hZt+h|Zt+h-1,...,Zt+1)] = E[Xtσt+hE(Zt+h|Zt+h-1,...,Zt+1)]
(here I don't understand why we can pull the σt+h out of the expectation.)
=E[Xtσt+hE(Zt+h)] = E[Xtσt+h * 0] = 0 for all h>0.
I'm cannot follow the reasoning of the three equalities labelled in red above. Can someone explain why they are true?
Any help would be much appreciated!
(note: also under discussion in Math Help Forum)
Xt = σtZt, where Zt~ i.i.d. N(0,1)
σt2 = w0 + w1 Xt-12
Find (i) E(Xt)
and (ii) the autocovariance function γX(h) for h=0,1,2,3,..., assuming the process is second-order stationary.
Solution:
(i) E(Xt) = E[E(Xt|σt2)] =E[E(σtZt|σt2)]
=E[σtE(Zt|σt2)] = E[σtE(Zt)] = E[σt * 0] = 0
(here I don't understand why E(Zt|σt2)=E(Zt).)
(ii) γX(0)=E(Xt2)
=E(σt2 Zt2)=E(σt2)E(Zt2)
=E(w0 + w1 Xt-12) * 1
= w0 + w1 γX(0)
Solve for γX(0) => γX(0) = w0/(1-w1)
γX(h)=E(XtXt+h)
=E[E(XtXt+h|Zt+h-1,...,Zt+1)] = E[XtE(Xt+h|Zt+h-1,...,Zt+1)]
(here I don't understand why we can pull the Xt out of the expectation.)
=E[XtE(σt+hZt+h|Zt+h-1,...,Zt+1)] = E[Xtσt+hE(Zt+h|Zt+h-1,...,Zt+1)]
(here I don't understand why we can pull the σt+h out of the expectation.)
=E[Xtσt+hE(Zt+h)] = E[Xtσt+h * 0] = 0 for all h>0.
I'm cannot follow the reasoning of the three equalities labelled in red above. Can someone explain why they are true?
Any help would be much appreciated!
(note: also under discussion in Math Help Forum)
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