Consider an ARCH(1) model:(adsbygoogle = window.adsbygoogle || []).push({});

X_{t}= σ_{t}Z_{t}, where Z_{t}~ i.i.d. N(0,1)

σ_{t}^{2}= w_{0}+ w_{1}X_{t-1}^{2}

Find (i) E(X_{t})

and (ii) the autocovariance function γ_{X}(h) for h=0,1,2,3,..., assuming the process is second-order stationary.

Solution:

(i) E(X_{t}) = E[E(X_{t}|σ_{t}^{2})] =E[E(σ_{t}Z_{t}|σ_{t}^{2})]

=E[σ_{t}E(Z_{t}|σ_{t}^{2})] = E[σ_{t}E(Z_{t})] = E[σ_{t}* 0] = 0

(here I don't understand why E(Z_{t}|σ_{t}^{2})=E(Z_{t}).)

(ii) γ_{X}(0)=E(X_{t}^{2})

=E(σ_{t}^{2}Z_{t}^{2})=E(σ_{t}^{2})E(Z_{t}^{2})

=E(w_{0}+ w_{1}X_{t-1}^{2}) * 1

= w_{0}+ w_{1}γ_{X}(0)

Solve for γ_{X}(0) => γ_{X}(0) = w_{0}/(1-w_{1})

γ_{X}(h)=E(X_{t}X_{t+h})

=E[E(X_{t}X_{t+h}|Z_{t+h-1},...,Z_{t+1})] = E[X_{t}E(X_{t+h}|Z_{t+h-1},...,Z_{t+1})]

(here I don't understand why we can pull the X_{t}out of the expectation.)

=E[X_{t}E(σ_{t+h}Z_{t+h}|Z_{t+h-1},...,Z_{t+1})] = E[X_{t}σ_{t+h}E(Z_{t+h}|Z_{t+h-1},...,Z_{t+1})]

(here I don't understand why we can pull the σ_{t+h}out of the expectation.)

=E[X_{t}σ_{t+h}E(Z_{t+h})] = E[X_{t}σ_{t+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! :)

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# Time Series: ARCH model properties

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